Fri, Jan 26

Geometry and Topology Seminar
Katherine Goldman (Ohio State University)
CAT(0) and cubulated Shepherd groups
4:00PM, 122 Mathematics Building

Shephard groups are common generalizations of Coxeter groups, Artin groups, graph products of cyclic groups, and (certain) complex reflection groups. We extend a well-known result that Coxeter groups are CAT(0) to a class of Shephard groups that have “enough” finite parabolic subgroups. We also show that in this setting, if the underlying Coxeter diagram is type FC, then the Shephard group is cubulated (i.e., acts properly and cocompactly on a CAT(0) cube complex). This provides many new non-trivial examples of CAT(0) and cubulated groups. Our method of proof combines the works of Charney-Davis on the Deligne complex for an Artin group and of Coxeter on the classification and properties of the regular complex polytopes. Along the way we introduce a new criteria (based on work of Charney) for a simplicial complex made of simplices of shape \(A_3\) to be CAT(1).

Mon, Jan 29

Algebra Seminar
Artan Sheshmani, Harvard/BIMSA

4:00PM, Zoom (please email achirvas@buffalo.edu)

We elaborate on construction of a derived Lagrangian intersection theory on moduli spaces of divisors on compact Calabi Yau threefolds. Our goal is to compute deformation invariants associated to a fixed linear system of divisors in CY3. We degenerate the CY3 into a normal crossing singular variety composed of Fano threefolds meeting along aK3. The deformation invariance arguments, together with derived Lagrangian intersection counts over the special fiber of the induced moduli space degeneration family, provides one with invariants of the generic CY fiber. This is report on several joint projects in progress with Ludmil Katzarkov, 
Tony Pantev, Vladimir Baranovsky and 
Maxim Kontsevich

Fri, Feb 2

Geometry and Topology Seminar
Tam Cheetam-West (Yale)
Profinite rigidity of some fibered, hyperbolic 3-manifold groups
 

4:00PM, 122 Mathematics Building

For an infinite, residually finite group, it is interesting to ask what properties of the group are captured by its finite quotients. We will discuss how to use ideas of Bridson-McReynolds-Reid-Spitler to show, for example, that the fundamental group of zero surgery on the knot \(6_2\) is completely determined (among all residually finite groups) by the collection of its finite quotients.

Fri, Feb 9

Applied Math Seminar
Zechuan Zhang, UB
Inverse spectral theory for the periodic self-adjoint Zakharov-Shabat and IST for the defocusing NLS equation with periodic BCs.

3:00PM, Math 122

The inverse spectral theory for a self-adjoint one-dimensional Dirac operator associated periodic potentials is formulated via a Riemann-Hilbert problem approach. The resulting formalism is also used to solve the initial value problem for the nonlinear Schrödinger (NLS) equation. A uniqueness theorem for the solutions of the Riemann-Hilbert problem is established, which provides a new method for obtaining the potential from the spectral data. Two additional, scalar Riemann-Hilbert problems are also formulated that provide conditions for the periodicity in space and time of the solution generated by arbitrary sets of spectral data. The formalism applies for both finite-genus and infinite-genus potentials. The formalism also shows that only a single set of Dirichlet eigenvalues is needed in order to uniquely reconstruct the potential of the Dirac operator and the corresponding solution of the defocusing NLS equation, in contrast with the representation of the solution of the NLS equation via the finite-genus formalism, in which two different sets of Dirichlet eigenvalues are used.

Mon, Feb 12

Algebra Seminar
David Hernandez, Université Paris Cité
A new Weyl group action and a cluster structure for representations of shifted quantum groups 
4:00PM, Zoom (please email achirvas@buffalo.edu)

Shifted quantum affine algebras and their truncations emerged from the study of quantized Coulomb branches. I will report on a joint work with Geiss and Leclerc : we show that the Grothendieck ring of the category O for the shifted quantum affine algebras has the structure of a cluster algebra. The cluster variables of a class of distinguished initial seeds are certain formal power series defined from a new Weyl group action     introduced  in a joint work with Frenkel. These cluster variables satisfy a system of functional relations called QQ-system. We will also discuss a connection with a recent work of Koroteev-Zeitlin involving q-opers.

Wed, Feb 14

Analysis Seminar
Jintao Deng, SUNY at Buffalo
The maximal rational Novikov conjecture and the coarse embeddability
4:00PM, 250 Math Building

Let C be the smallest collection of countable, discrete groups that contains all coarsely embeddable groups and is closed under inductive limits and extensions. This collection contains many groups, including the non-coarsely embeddable groups constructed by Arzhantseva-Tessera and the Gromov's monster group. In this talk, I will talk about the result that the maximal rational Novikov conjecture holds for each group in C. I will also talk about the applications of the maximal rational Novikov conjecture in geometry. This is based on a recent result with G. Tian, Z. Xie and G. Yu.

Fri, Feb 16

Geometry and Topology Seminar
Luya Wang (Stanford)
Deformation inequivalent symplectic structures and Donaldson's four-six question

4:00PM, via Zoom

Studying symplectic structures up to deformation equivalences is a fundamental question in symplectic geometry. Donaldson asked: given two homeomorphic closed symplectic four-manifolds, are they diffeomorphic if and only if their stabilized symplectic six-manifolds, obtained by taking products with \(CP^1\) with the standard symplectic form, are deformation equivalent? I will discuss joint work with Amanda Hirschi on showing how deformation inequivalent symplectic forms remain deformation inequivalent when stabilized, under certain algebraic conditions. This gives the first counterexamples to one direction of Donaldson’s “four-six” question and the related Stabilizing Conjecture by Ruan. Time permitting, I will also discuss our upcoming work giving more supporting evidence in the other direction of the “four-six” question.

Mon, Feb 19

Algebra Seminar
Peter Koroteev, UB
The Diamond of Integrability 
4:00PM, : Zoom (please email achirvas@buffalo.edu)

I shall review recent progress in understandingthe connections between representation theory, enumerative geometry, andintegrable systems inspired by a certain class of physical theories (so-calledN=2* theories). Conveniently one can arrange most of known results into theDiamond\url{https://nam12.safelinks.protection.outlook.com/?url=https%3A%2F%2Fmath.berkeley.edu%2F~pkoroteev%2Fdiamond33.pdf&data=05%7C02%7Cmahacker%40buffalo.edu%7C80da21a7a9714eec0dea08dc2c68ccaa%7C96464a8af8ed40b199e25f6b50a20250%7C0%7C0%7C638434075826068027%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C0%7C%7C%7C&sdata=AKIKLeW33io0iwQSAuEJEFp13Pp%2FoNALAJpxs1V2K%2Bo%3D&reserved=0}.I will describe some of its corners as well as highlight open questions.

Wed, Feb 21

Analysis Seminar
Jingbo Xia, SUNY at Buffalo
Fock space: a bridge between Fredholm index and the quantum Hall effect
4:00PM, 250 Math Building

We identify the quantized Hall conductance of Landau levels with a Fredholm index, by using the theories of Helton-Howe-Carey-Pincus, and Toeplitz operators on the classic Fock space and higher Fock spaces. The index computations reduce to the single elementary one for the lowest Landau level. This brings new insights to the extraordinarily accurate quantization of the Hall conductance as measured in quantum Hall experiments.

Wed, Feb 28

Analysis Seminar
Mohan Ramachandran, SUNY at Buffalo
Maximal Spectrum, Stone Cech, and Stone Weierstrass theorems
4:00PM, 250 Math Building

In this talk I will use the maximal spectrum with Zariski Topology to give simple proofs of the theorems in the title. We use ideas from a paper of M H Stone from 1937 to give these simple proofs.

Mon, Mar 4

Algebra Seminar
Minh-Tam Quang Trinh, MIT
Hecke Traces, Braid Varieties, and Springer Actions 
4:00PM, 250 Mathematics Building

The HOMFLYPT polynomial is a link invariant that Jones–Ocneanu constructed using a family of traces on the Hecke algebras of the symmetric groups. It is categorified by triply-graded Khovanov–Rozansky(KhR) homology, a richer invariant that can be constructed by applying trace functors to complexes of Soergel bimodules, or equivalently, certain complexes of perverse sheaves on flag varieties. Seeking to make KhR more explicitly geometric, I introduced a generalized Steinberg variety for any positive braid, and showed that the KhR homology of its link closure is an isotypic summand of the cohomology of the variety under a Springer-type action. These Steinberg braid varieties are closely related to the twisted wild character varieties studied by Boalch–Yamakawa, and refine the Richardson braid varieties of recent interest in algebraic combinatorics. At least for periodic braids, they should be related via nonabelian Hodge theory to the homogeneous affine Springer fibers studied by Lusztig–Smelt, Sage, Sommers, and many others. I will explain where these varieties come from, and what predictions they offer for affine Springer theory.

Mon, Mar 4

Geometry and Topology Seminar
Marie Trin (Institut Mathématiques de Rennes)
Counting arcs of the same type

4:00PM, 122 Mathematics Building

Two closed curves on a hyperbolic surface are said to be of the same type if they differ from a mapping class. The question of counting curves of the same type with bounded length has been studied by M.Mirzakhani who showed that the counting is polynomial into the lenght. Mirzakhani's results were recovered and extended by Erlandsson-Souto proving convergence theorems for certain sequences of measures. In 2022, N.Bell obtained results similar to those of Mirzakhani for arcs of the same type in surfaces with boundary. We will introduce the method based on the convergence of measures for curves counting and then look at a way to adapt it to the case of arcs.

Wed, Mar 6

Analysis Seminar
Jintao Deng, SUNY at Buffalo
The maximal rational Novikov conjecture and the coarse embeddability, Part II
4:00PM, 250 Math Building

Title: The maximal rational Novikov conjecture and the coarse embeddability, Part II

Wed, Mar 6

Special Event
Jintao Deng, SUNY at Buffalo
The maximal rational Novikov conjecture and the coarse embeddability, Part II
4:00PM, 250 Math Building

Title: The maximal rational Novikov conjecture and the coarse embeddability, Part II

Fri, Mar 8

Applied Math Seminar
Esteban Vargas Bernal, Arizona State University
InfoMap for absorbing random walks.

3:00PM, Math 122

Community structure is a network feature in which densely connected sets of nodes (that are called communities) are sparsely connected to other densely connected set of nodes. This feature can significantly impact dynamics on the network. For example, community structure can affect attributes of disease dynamics such as outbreak size, outbreak duration, and outbreak peak. In turn, attributes of dynamics can affect community structure. For example, the recovery rates of a disease can be a barrier for the transmission of the disease within a community. In this context, we can interpret the recovery rates as "absorption" of a random walker representing the transmission of the disease. We give an algorithm that outputs communities that are informed by both the edges of the network and node absorption. To this end, we adapt the random-walk based algorithm InfoMap. Intuitively, InfoMap produces communities such that one step of a random walker of a non-absorbing random walk is more likely to occur within a community than between two distinct communities. We use absorption-scaled graphs and Markov time sweeping to adapt InfoMap to absorbing random walks. We apply our algorithm to study the effect on susceptible-infected-recovered dynamics of communities informed by absorption.

Mon, Mar 11

Algebra Seminar
Chengze Duan, University of Maryland
College Park
Good-position braids, transversal slices andaffine Springer fibers 
4:00PM, Zoom (please email achirvas@buffalo.edu)

Let \(G\) be a reductive group over analgebraically closed field and \(W\) its Weyl group. Using Coxeter elements,Steinberg constructed cross-sections of the adjoint quotient of \(G\) which alsoyield transversal slices of regular unipotent classes. In 2012, He and Lusztigconstructed transversal slices using minimal-length elements in ellipticconjugacy classes in \(W\), yielding transversal slices of basic unipotentclasses. In this talk, we generalize minimal-length elements to good-positionbraids in the associated braid monoid of \(W\) and use these elements to constructtransversal slices of all unipotent classes in \(G\). Moreover, these newelements are also connected to affine Springer fibers.

Fri, Mar 15

Applied Math Seminar
Anastassiya Semenova, University of Washington
Water Waves from Stokes to Now.

3:00PM, Math 250

The study of ocean surface waves is essential for predicting and preparing for natural disasters such as tsunamis, storm surges, and the occurrence of rare events like freak waves. Although ocean waves naturally occur in three dimensions, they can often be effectively analyzed in a two-dimensional framework. For example, waves moving away from the epicenter of a storm can be considered unidirectional after traveling a significant distance. I primarily focus on the examination of periodic traveling waves at the free surface of an ideal two-dimensional fluid.
Specifically, my main interest is the stability properties of surface waves of permanent shape commonly referred to as Stokes waves. The stability of such waves is examined by linearization of the nonlinear equations of motion about these Stokes waves and by studying the resulting spectral problem numerically. In this talk, I demonstrate the spectrum of Stokes waves of various amplitudes. Additionally, I present the growth rate of the dominant instability and discuss the Benjamin-Feir, high-frequency and superharmonic instabilities associated with these waves. Such instabilities swiftly disintegrate wave crests of large-amplitude Stokes waves, which explains why long-lived large-amplitude oceanic swell is not observed in the ocean.

Fri, Mar 15

Geometry and Topology Seminar
Johanna Mangahas (UB)
(Non-)Recognizing Spaces for Stable Subgroups

4:00PM, 122 Mathematics Building

Stability for subgroups of finitely generated groups generalizes the property of quasiconvexity for subgroups of hyperbolic groups: they are quasi-isometrically embedded, and ambient-group quasi-geodesics between points in the subgroup fellow travel each other (from which it follows that the subgroup is hyperbolic, whereas the ambient group generally is not).  We are interested in when this property is recognized by an action of the larger group on some hyperbolic space, by which we mean the stable subgroup quasi-isometrically embeds into that space.  In well-studied settings such as mapping class groups, right-angled Artin groups, or more generally hierarchically hyperbolic groups, the hyperbolic space admitting the group's largest acylindrical action provides such a recognizing space for all stable subgroups.  Sometimes the corresponding result is true for relatively hyperbolic groups admitting largest acylindrical actions, but we provide a counterexample to show it is not true in general.  This is joint work with Sahana Balasubramanya, Marissa Chesser, Alice Kerr, and Marie Trin.

Mon, Mar 25

Algebra Seminar
Samuel DeHority, Columbia
Flat connections and toroidal algebras 
4:00PM, Zoom (please email achirvas@buffalo.edu)

Toroidal algebras are higher-(functional-)dimension analogues of affine algebras. Following Billig, we discuss representations of toroidal algebras using techniques in vertex operator algebras as well as certain extensions of them to superalgebras, and also to Hopf algebras which play the role of a Yangian of a toroidal algebra extended by divergence-free vector fields for some groups. We will also discuss a flat connection analogous to the trigonometric Casimir connection of a semisimple Lie algebra and explain how properties of the connection, including a tensor product structure and the quasimodularity connection, are related to properties of the Hopf algebra

Fri, Mar 29

Geometry and Topology Seminar
Carolyn Abbott (Brandeis University)
Morse boundaries of CAT(0) cubical groups
4:00PM, 122 Mathematics Building

The visual boundary of a hyperbolic space is a quasi-isometry invariant that has proven to be a very useful tool in geometric group theory. When one considers CAT(0) spaces, however, the situation is more complicated, because the visual boundary is not a quasi-isometry invariant. Instead, one can consider a natural subspace of the visual boundary, called the (sublinearly) Morse boundary. In this talk, I will describe a new topology on this boundary and use it to show that the Morse boundary with the restriction of the visual topology is a quasi-isometry invariant in the case of (nice) CAT(0) cube complexes. This result is in contrast to Cashen’s result that the Morse boundary with the visual topology is not a quasi-isometry invariant of CAT(0) spaces in general. This is joint work with Merlin Incerti-Medici.

Wed, Apr 3

Analysis Seminar
Liang Guo, East China Normal University
Hilbert-Hadamard spaces and the equivariant coarse Novikov conjecture
4:00PM, 250 Math Building

The equivariant coarse Novikov conjecture synthesizes all the Novikov-type conjectures, including the strong Novikov conjecture for groups and the coarse Novikov conjecture for metric spaces. It has fruitful applications in topology and geometry. In a recent work of Sherry Gong, Jianchao Wu, and Guoliang Yu, a notion of Hilbert-Hadamard space is introduced to study the Novikov conjecture for specific groups, which can be seen as an infinite-dimensional Hadamard manifold. To generalize their idea to the equivariant coarse Novikov conjecture, in this talk, we study a dynamic system that admits an equivariant coarse embedding into an admissible Hilbert-Hadamard space. I will start with several applications of the equivariant Novikov conjecture and show that the equivariant coarse Novikov conjecture holds for such a dynamic system. This is based on a joint work with Qin Wang, Jianchao Wu, and Guoliang Yu.

Wed, Apr 10

Algebra Seminar
Joe Kramer-Miller, Lehigh University
Transcendental properties of the Artin-Hasse exponential modulo p
3:00PM, 250 Mathematics Building

The Artin-Hasse exponential is a p-adic analogue to the classical exponential function. It is ubiquitous in p-adic analysis, where it plays a pivotal role in the construction of Witt vectors and in Dwork theory. The miracle of the Artin-Hasse exponential is that its Taylor expansion has p-integral coefficients, and thus may be reduced modulo p. In this talk I will discuss recent work on the transcendental properties of the Artin-Hasse exponential mod p. We give two proofs that theArtin-Hasse exponential mod p is transcendental, answering a long-outstanding question posed by Thakur. We also explain several algebraic independence results for the Artin-Hasse exponential evaluated at different polynomials.

Wed, Apr 10

Analysis Seminar
Wencai Liu, Texas A&M University
Algebraic geometry, complex analysis and combinatorics in spectral theory of periodic graph operators
4:00PM, 250 Math Building

In this talk, we will discuss the significant role that the algebraic and analytic properties of complex Bloch and Fermi varieties play in the study of periodic operators. I will begin by highlighting recent discoveries about these properties, especially their irreducibility. Then, I will show how we can use these findings, together with techniques from complex analysis and combinatorics, to study spectral and inverse spectral problems arising from periodic operators.

Thu, Apr 11

Colloquium
Indira Chatterji (University of Côte d'Azur / Fields Institute)
Groups and geometry
4:00PM, 250 Mathematics Building

Hyperbolicity is a geometric phenomenon observed in several contexts like the sound under water. In the setting of graphs or metric spaces, hyperbolicity can be defined in terms of thinness of triangles. A product of hyperbolic spaces is no longer hyperbolic but retains some features of hyperbolicity, that I shall discuss through the lens of group actions, and random walks.

Fri, Apr 12

Applied Math Seminar
Alexandr Chernyavskiy, UB
Dark-bright soliton perturbation theory for the Manakov system.

3:00PM, Math 122

A direct perturbation  method for studying dynamics of
dark-bright solitons of the Manakov system in the presence of
perturbations is presented. We combine multiscale expansion method,
perturbed conservation laws, and a boundary layer approach, which breaks
the problem into an inner region, where the bulk of the soliton resides,
and an outer region, which evolves independently of the soliton. We show
that a shelf develops around the dark soliton component, with speed of the
shelf proportional to the background intensity. Conservation laws of
the Manakov system are used to determine the properties of the shelf and
perturbed solutions. Our analytical predictions are corroborated by
numerical simulations..

Fri, Apr 12

Geometry and Topology Seminar
Indira Chatterji (University of Côte d'Azur / Fields Institute)
Property T versus aTmenability
Abstract : A group has property T if any action on a Hilbert space has a fixed point, and a group is called aTmenable if it admits a proper action on a Hilbert space by affine isometries. I will review what classical groups have property T, or aTmenability, or neither, how those notions appeared and what they are good for.
4:00PM, 122 Mathematics Building

Title: Property T versus aTmenability
Abstract : A group has property T if any action on a Hilbert space has a fixed point, and a group is called aTmenable if it admits a proper action on a Hilbert space by affine isometries. I will review what classical groups have property T, or aTmenability, or neither, how those notions appeared and what they are good for.

Mon, Apr 15

Algebra Seminar
Dani Szpruch, Open University of Israel
An analog of the Hasse-Davenport product relation for \(\epsilon\)-factors and an application 
3:00PM, 250 Mathematics Building

The classical Hasse-Davenport product relation is an identity involving products of Gauss sums defined over a finite field. In this talk we shall introduce some generalizations of this classical result for Tate \(\epsilon\)-factors and closely related arithmetic factors defined over a p-adic field. We will then show that these generalizations are equivalent to a certain representation-theoretic identity involving an analog of Shahidi local coefficients for covering groups.

Wed, Apr 17

Analysis Seminar
Andy Zucker, University of Waterloo
Ultracoproducts and weak containment for flows of topological groups
4:00PM, 250 Math Building

We develop the theory of ultracoproducts and weak containment for flows of arbitrary topological groups. This provides a nice complement to corresponding theories for p.m.p. actions and unitary representations of locally compact groups. We isolate a new class of topological groups, which we call Fubini groups, for which iterated ultracopowers of certain G-flows behave nicely. Among the Fubini groups are the class of locally Roelcke precompact groups, for which the theory is especially rich. For these groups, we can define for certain families of G-flows a suitable compact space of weak types. When G is locally compact, all G-flows belong to one such family, yielding a single compact space describing all weak types of G-flows.

Fri, Apr 19

Applied Math Seminar
Anna Vainchtein, University of Pittsburgh
Supersonic fronts and pulses in a lattice with hardening-softening interactions.

3:00PM, Math 122

This talk is based on recent joint work with Lev Truskinovsky (ESPCI ParisTech). We consider a version of the classical Hamiltonian Fermi-Pasta-Ulam problem with nonlinear force-strain relation in which a hardening response is taken over by a softening regime above a critical strain value. We show that in addition to pulses (solitary waves) this discrete system also supports non-topological and dissipation-free fronts (kinks). Moreover, we demonstrate that these two types of supersonic traveling wave solutions belong to the same family. Within this family, solitary waves exist for continuous ranges of velocity that extend up to a limiting speed corresponding to kinks. As the kink velocity limit is approached from above or below, the solitary waves become progressively more broad and acquire the structure of a kink-antikink bundle. We investigate stability of the obtained solutions via Floquet analysis and direct numerical simulations. To motivate and support our study of the discrete problem we also analyze a quasicontinuum approximation with temporal dispersion. We show that this model captures the main effects observed in the discrete problem.

Mon, Apr 22

Algebra Seminar
Thomas Creutzig, Edmonton/Erlangen
Representation theory of affine VOAs 
4:00PM, Zoom (please email achirvas@buffalo.edu)

An affine Lie algebra is a central extension of the loop algebra of a finite dimensional Lie algebra. The vacuum modules at any complex level \(k\) of the affine Lie algebra themselves carry an interesting algebraic structure, namely that of a vertex operator algebra (VOA). There presentation theory of affine VOAs is rather rich, e.g. suitable categories of modules form ribbon categories and there are exciting connections to geometry, quantum groups, physics and much more.

I will give an overview on the state of the art in this area.

Tue, Apr 23

Colloquium
Tomasz Mrowka, MIT
2023-24 Myhill Lecture #1
4:00PM

Title: 2023-24 Myhill Lecture #1

Wed, Apr 24

Colloquium
Tomasz Mrowka, MIT
2023-24 Myhill Lecture #2
4:00PM

Title: 2023-24 Myhill Lecture #2

Thu, Apr 25

Colloquium
Tomasz Mrowka, MIT
2023-24 Myhill Lecture #3
4:00PM

Title: 2023-24 Myhill Lecture #3

Mon, Apr 29

Algebra Seminar
Vasily Krylov, MIT
Around the Hikita-Nakajima conjecture 
4:00PM, Zoom (please email achirvas@buffalo.edu

Symplectic duality predicts that symplectic singularities should come in pairs with matching properties. For instance, Nakajima quiver varieties are conjecturally dual to the BFN Coulomb branches of the corresponding quiver gauge theories, while Slodowy varieties should be dual to (covers of) nilpotent orbits. In this talk, I will discuss the Hikita-Nakajima conjecture that relates dual varieties. I will explain a possible approach to proving this conjecture when both symplectic singularities admit symplectic resolutions and illustrate the approach in examples. The talk is based on the joint work with Pavel Shlykov (arXiv:2202.09934) and the work in progress with Do Kien Hoang and Dmytro Matvieievskyi.

Wed, May 1

Analysis Seminar
Xin Ma, Fields Institute
Soficity, Amenability, and LEF-ness for topological full groups
4:00PM, 250 Math Building

Topological full groups, as an algebraic invariant, were introduced to study continuous orbit equivalence relations by Giordano, Putnam, and Skau. Then, these groups have been found applications to geometric group theory by providing interesting examples with certain properties such as simplicity, soficity, amenability, and LEF_ness. In this talk, I will show methods of establishing the soficity and LEF-ness for topological full groups. Moreover, I will explain how one can obtain amenability from the sofic approximations when the acting group is amenable and the action is distal.

Thu, May 2

Colloquium
Dr Willy Hereman, Colorado School of Mines
Symbolic computation of solitary wavesolutions and solitons through homogenization of degree
4:00PM, Mathematics Building room 250

A simplified version of Hirota's method for thecomputation of solitary waves and solitons of nonlinear PDEs will be presented.The approach requires a change of dependent variable so that the transformedPDE is homogenous of degree in the new variable.

The resulting homogenous PDE does not have tobe quadratic and the method still applies if its bilinear form is not known.Solitons are then computed using a perturbation scheme involving linear andnonlinear operators. For soliton equations the scheme terminates after a finitenumber of steps. To illustrate the approach, solitons are computed for a classof fifth-order KdV equations due to Lax, Sawada-Kotera, and Kaup-Kupershmidt.

Homogenization of degree also allows one tofind solitary wave solutions of nonlinear PDEs that are not completelyintegrable. Examples include the Fisher and FitzHugh-Nagumo equations, and acombined KdV-Burgers equation. When applied to a wave equation with a cubicsource term, the method leads to a `bi-soliton' solution which describes thecoalescence of two wavefronts.

The method is largely algorithmic andimplemented in Mathematica. A demonstration of the software packagePDESolitonsSolutions will be given.

Fri, May 3

Applied Math Seminar
Willy Hereman, Colorado School of Mines
Symbolic computation of conservation laws of nonlinear partial differential equations.

3:00PM, Math 250

A method will be presented for the symbolic computation of conservation laws of nonlinear partial differential equations (PDEs) involving multiple space variables and time.

Using the scaling symmetries of the PDE, the conserved densities are constructed as linear combinations of scaling homogeneous terms with undetermined coefficients. The variational derivative is used to compute the undetermined coefficients. The homotopy operator is used to invert the divergence operator, leading to the analytic expression of the flux vector.

The method is algorithmic and has been implemented in Mathematica. The software is being used to compute conservation laws of nonlinear PDEs occurring in the applied sciences and engineering.

The software package will be demonstrated for PDEs that model shallow water waves, ion-acoustic waves in plasmas, sound waves in nonlinear media, and transonic gas flow. Equations featured in this talk include the Korteweg-de Vries and Zakharov-Kuznetsov equations.

Mon, May 6

Algebra Seminar
Ivan Loseu, Yale
Harish-Chandra centers for affine Kac-Moody algebras in positive characteristic Abstract : This talk is based on a joint work in progress with Gurbir Dhillon. A remarkable theorem of Feigin and E. Frenkel from the early 90s describes the center of the universal enveloping algebra of an(untwisted) affine Kac-Moody Lie algebra at the so-called critical level proving a conjecture of Drinfeld: the center in question is the algebra of polynomial functions on an infinite-dimensional affine space known as the space of opers. In our work we study a part of the center in positive characteristic \(p\) at an arbitrary non-critical level. Namely, we prove that the algebra of loop-group invariants in the completed universal enveloping algebra is still the algebra of polynomials on an infinite-dimensional affine space that is ``\(p\) times smaller than the Feigin-Frenkel center''. In my talk I will introduce all necessary notions, state the result, explain motivations and examples. 
4:00PM, Zoom (please email achirvas@buffalo.edu)

Title: Harish-Chandra centers for affine Kac-Moody algebras in positive characteristic

Abstract : This talk is based on a joint work in progress with Gurbir Dhillon. A remarkable theorem of Feigin and E. Frenkel from the early 90s describes the center of the universal enveloping algebra of an(untwisted) affine Kac-Moody Lie algebra at the so-called critical level proving a conjecture of Drinfeld: the center in question is the algebra of polynomial functions on an infinite-dimensional affine space known as the space of opers. In our work we study a part of the center in positive characteristic \(p\) at an arbitrary non-critical level. Namely, we prove that the algebra of loop-group invariants in the completed universal enveloping algebra is still the algebra of polynomials on an infinite-dimensional affine space that is ``\(p\) times smaller than the Feigin-Frenkel center''. In my talk I will introduce all necessary notions, state the result, explain motivations and examples.

Fri, May 10

Applied Math Seminar
Stephen Anco, Brock University
Integrable multi-component nonlinear evolution equations and their bi-Hamiltonian
structure in the AKNS formalism.
3:00PM, Math 250 - note the room change

The AKNS formalism provides a well-known framework for obtaining a Lax pair and setting up an inverse scattering transform for several prominent integrable systems: 
KdV and mKdV equations, NLS equation, Hirota (complex mKdV) equation. All of these systems  are closely connected to the Lie algebras \(sl(2,R)\) and \(sl(2,C)\)and are known to possess a bi-Hamiltonian structure which generates an infinite hierarchy of symmetries and
conservation laws. 
An open question has been how to derive their bi-Hamiltonian structure in a systematic
fashion based solely on the structure of the underlying Lie algebras. In this talk, I will
show that there is a natural bi-Hamiltonian structure encoded in symmetric Lie algebras and
that this structure gives rise to the standard bi-Hamiltonian operators of the integral
systems coming from the AKNS formalism. In particular, when the Lax pair is formulated in a
suitable way, it automatically yields associated bi-Hamiltonian operators. 
These main results will be illustrated for \(sl(2,R)\) and \(sl(2,C)\) and extended to all
symmetric Lie algebras. A summary of the various types of vector NLS and real/complex vector
mKdV systems will be given from this point of view. As further examples, a new integrable
nonlocal NLS equation will be derived from \(so(5)\) and \(so(4,1)\).

Mon, May 20

Algebra Seminar
Professor Jie Du, University of New South Wales
Canonical bases for quantum gl_{m|n}
4:00PM, Mathematics Building room 250

Using quantum differential operators, we construct a quantum gl_{m|n}supermodule structure on a certain polynomial superalgebra. We then extend this representation to its formal power series algebra which contains a submodule isomorphic to the regular representation of this quantum super group .In this way, we obtain a new presentation for quantum gl_{m|n} by a basis together with explicit multiplication formulas by generators. This allows us to introduce the canonical basis for the +-part.
On the other hand, we extend the supermodule structure on the polynomial superalgebra to a (partial) Laurent polynomial superalgebra which induces the regular representation of the modified quantum gl_{m|n}. In this way, a new presentation and its canonical basis theory are developed for this modified case.

Fri, Aug 30

Geometry and Topology Seminar
G&T Seminar
4:00PM, 122 Mathematics Building

Fri, Sep 6

Geometry and Topology Seminar
Tomasz Maszczyk (University of Warsaw)
Foliations with rigid Godbillon-Vey class
4:00PM, 122 Mathematics Building

Under the technical condition of separability of some topological space of cohomology, we show that the Godbillon-Vey number of a foliation of codimension one on a compact orientable 3-fold is topologically rigid if and only if the foliation admits a projective transversal structure. Here by the rigidity of the Godbillon-Vey number we mean that it is constant under the infinitesimal singular deformations of the foliation, and a foliation admits a transversal projective structure whenever it can be glued from the level sets of locally defined functions related by fractional linear transition maps.

Mon, Sep 9

Algebra Seminar
Tomasz Maszczyk, University of Warsaw
Hochschild cohomology for abstract convexity and Shannon entropy
4:00PM, 250 Math building

Shannon entropy was introduced as a statistical measure of information loss but appeared in other fields of mathematics as well. We plan to sketch its relations with polylogarithms and motives after Cathelineau, Dupont, Bloch, Goncharov, Elbaz-Vincent—Gangl, a cohomological interpretation by Kontsevich, and the information cohomology after Baudot--Bennequin. In the latter approach, Shannon’s entropy is a one-cocycle. Next, we survey the Faddeev algebraic characterization theorem and the fundamental information functional equation after Tverberg, Kendall, and Lee. Then, we will sketch Gromov’s program and comment on the categorical interpretation by Baez-Fritz-Leinster. Finally, we plan to present another cohomological derivation of Shannon’s entropy based on a new kind of Hochschild cohomology we construct for abstract convexity. The latter admits a cohomological interpretation of extensions of convex bodies by vector spaces parallel to Hochschild extensions of associative algebras by square-zero ideals. Now, the Shannon entropy arises as a two-cocycle which can be understood as an analog of a first Chern class of the one-cocycle from information cohomology.

Wed, Sep 11

Analysis Seminar
Janusz Wysoczanski, University of Wroclaw
Finitely Generated Weakly Monotone C*-algebras
4:00PM, 250 Math Building

In this talk I will present construction and properties of C*-algebras generated by finite number of creation/annihilation operators, acting on (weakly monotone) subspace of the full Fock space, related to Pusz-Woronowicz Twisted Canonical Communication Relations. They happen to be quotients of Cuntz-Krieger algebras and are related to Hong-Szymanski's C*-algebras of quantum odd dimensional spheres. Graph algebra's point of view will be presented. Maximal abelian sub-algebra and its spectrum will be considered and identified.

Mon, Sep 16

Algebra Seminar
Anna Wysoczańska-Kula, Uniwersytet Wrocławski
Free resolution of universal unitary quantum groups
4:00PM, 250 Mathematics Building

Hochschild cohomology is a classical invariant of algebras, which can be used in particular to distinguish objects. A possible way of computing the Hochschild cohomology is via (projective) resolutions. I will present a method to compute such a resolution and then the Hochschild cohomology for universal unitary quantum groups \(U_F^+\), \(F\in GL_n(\mathbb{C})\). For that purpose we exhibit a free-glued product structure of \(U_F^+\), and use the projective resolution of \(O_E^+\) given by J.Bichon (2013). This is a joint result with I.Baraquin, U.Franz, M.Gerhold and M.Tobolski.

Wed, Sep 18

Analysis Seminar
Wenbo Sun, Virginia Tech
Geometry Ramsey Conjecture over finite fields
4:00PM, 250 Math Building

The Geometry Ramsey Conjecture is a question raised by Graham in 1994, which says that given any finite configuration X which lies on a sphere,for any finite coloring of the Euclidean space, there always exists a monochromatic congruent copy of tX for any large enough scalar t. One can also formulate a similar question for the finite field setting. While the study of the Geometry Ramsey Conjecture in literature focuses on the harmonic analysis approach, in this talk, we will explain how the higher order Fourier analysis method can be used to answer the Geometry Ramsey Conjecture in the finite field setting.

Fri, Sep 20

Geometry and Topology Seminar
Matthew Stoffregen (Michigan State University)
Pin(2) Floer homology and the Rokhlin invariant
4:00PM, 122 Mathematics Building

In this talk, we describe a family of homology cobordism invariants that can be extracted from Pin(2)-equivariant monopole Floer homology (using either Manolescu or Lin's definitions), that have some properties in common with both the epsilon and upsilon invariants in knot Floer homology.  We'll show a relationship of this family to questions about torsion in the homology cobordism group, and to triangulation of higher-dimensional manifolds.  This is joint work in progress with Irving Dai, Jen Hom, and Linh Truong.

Mon, Sep 23

Algebra Seminar
Janusz Wysoczański, University of Wroclaw
Hecke algebras on homogeneous trees and relationswith Hankel and Toeplitz matrices
4:00PM, 250 Mathematics Building

A homogeneous tree of degree \(q+1\) (a positive integer) is a connected graph, with no loops and with each vertex having exactly \(q+1\) neighbours. The distance \(d(x,y)\) between vertices \(x\) and \(y\) is the length of the uniquely defined geodesic connecting them.  In particular there are \((q+1)q^{n-1}\)vertices at distance \(n>0\) from a given one. In this talk we consider distance-dependent two-variable functions (kernels) \(f(x,y)\), defined on pairs of vertices.

The Hecke algebra on homogeneous tree is a commutative algebra, spanned by particular kernels, defined on pairs of vertices \((x,y)\)and indexed by non-negative integers \(f_n(x,y)\). Each of these kernels depends on the distance between the vertices and vanishes if the distance is not equal to their index, otherwise it equals 1. We will show that the Hecke algebra is generated by \(f_1\), which satisfies quadratic (Hecke) equation. Our main interest is in showing that the Hecke algebra is MASA (i.e. Maximal Abelian SubAlgebra) in some bigger algebra.

If \(q>1\) then a geometric trick of a Y-turn on the tree will do the job. If \(q=1\), which corresponds to the tree of integers, the      Y-turn is not possible, and we introduce some additional (Banach space)structure and show that the Hecke algebra is not MASA, but its commutant decomposes as a direct sum of Hankel and Toeplitz (double-infinite) matrices.

Wed, Sep 25

Analysis Seminar
Jakob Streipel, SUNY at Buffalo
Stechkin's trick
4:00PM, 250 Math Building

In this talk we'll discuss a somewhat forgotten inequality from the 1970's due to S. B. Stechkin, and how it can be used to improve zero-free regions of L-functions by combining it with the standard approach due to de la Vallée Poussin in 1896. As we will demonstrate, this simple inequality lets one improve any zero-free region argument that uses a so-called explicit formula, and as an example we will talk about recent and ongoing joint work with Steven Creech, Alia Hamieh, Simran Khunger, Kaneenika Sinha, and Kin Ming Tsang where we use this trick (and other tools) to find an explicit zero-free region for L-functions of modular forms.

Fri, Sep 27

Applied Math Seminar
Mohammad-Ali Miri, Queens College CUNY
Analog Optical Computing: Combinatorial Optimization and Learning Unitary Operations.
3:00PM, Math 250

The growing demand for faster, more energy-efficient, and scalable information processing has spurred interest in novel unconventional computing methods, with photonics emerging as a particularly promising platform due to its potential for ultra-high bandwidth, low power consumption, and inherent parallelism. In this talk, I will present two main directions of my research activities focusing on optical computing. In the first part of this talk, I will discuss the mapping of combinatorial optimization problems onto coherent networks of photonic oscillators. Next, I will present numerical results on integrating such nonlinear dynamical systems for finding fixed points that efficiently approximate the global minima of hard combinatorial optimization problems. In the second part of this talk, I will discuss a novel factorization and parametrization of discrete linear unitary operators that enables the realization of photonic chips performing unitary matrix operations on light. This factorization is based on alternating a fixed NxN unitary operator with an N-parameter diagonal phase operator for N+1 layers. In both parts, I will close the discussion with open problems.

Mon, Oct 7

Algebra Seminar
Vasudevan Srinivas, UB
Two results on \'etale fundamental groups in characteristic p 
4:00PM, Mathematics Building room 250

This talk will discuss two results on \'etale fundamental groups of varieties over an algebraically closed field of characteristic \(p>0\), based on joint work with Hèl\'ene Esnault and other coauthors. The first, along with Mark Schusterman, is that the tame fundamental group is finitely presented for such a variety which is the complement of an SNC divisor in a smooth projective variety. The second, along with Jakob Stix, is to give an obstruction for a smooth projective variety to admit a lifting to characteristic 0, in terms of the structure of its \'etale fundamental group as a profinite group.

Wed, Oct 16

Analysis Seminar
Jingbo Xia, SUNY at Buffalo
The Helton-Howe trace formula for the Drury-Arveson space
4:00PM, 250 Math Building

The famous Helton-Howe trace formula was originally established for antisymmetric sums of Toeplitz operators on the Bergman space of the unit ball. We prove its analogue on the Drury-Arveson space.

Fri, Oct 18

Applied Math Seminar
Jia Zhao, Binghamton University
General Numerical Framework for Structure-Preserving Reduced Order Models of Thermodynamically Consistent Reversible-Irreversible PDEs
3:00PM, Math 250

In this talk, I will present a newly developed numerical framework to derive structure-preserving reduced order models for thermodynamically consistent PDEs. Our approach focuses on two key aspects: (a) a systematic method for generating reduced order models that respect the underlying thermodynamic principles of the original PDE systems, and (b) a strategy for constructing accurate, efficient, and structure-preserving numerical algorithms to solve these reduced order models. The framework’s generality allows it to be applied to a wide range of PDE systems governed by thermodynamic laws. We will demonstrate the effectiveness of this approach through several numerical examples. This is a joint work with Zengyan Zhang from Binghamton University.

Fri, Oct 18

Geometry and Topology Seminar
Christopher Karpinski (McGill University)
A weak Tits alternative for groups acting on buildings.


4:00PM, 122 Mathematics Building

Buildings are highly symmetrical non-positively curved simplicial complexes introduced by Jacques Tits in the 1950s to study semisimple algebraic groups. Over the years, buildings have garnered interest among geometric group theorists due to their non-positively curved structure and close connection to Coxeter groups. We prove that groups acting properly and cocompactly on buildings satisfy an algebraic dichotomy, commonly encountered among groups with non-positive curvature features, known as the weak Tits alternative: either the group is virtually abelian or it contains a nonabelian free subgroup. This is joint work with Damian Osajda and Piotr Przytycki.

Wed, Oct 23

Analysis Seminar
Raphael Ponge, Sichuan University
Noncommutative Geometry, Semiclassical Analysis, and Weak Schatten Classes
4:00PM, 250 Math Building

In this talk, I will present new results regarding semiclassical Weyl's laws in the setup of Connes' noncommutative geometry. They provide precise asymptotics for the counting functions of Schroedinger operators under the semiclassical limit. This improves and simplifies previous results of McDonald-Sukochev-Zanin. This provides a bridge between semiclassical analysis and noncommutative geometry. Thanks to the Birman-Schwinger principle and old results of Birman-Solomyak this reduces to establishing various weak Schatten class properties for the operators at stake. This has a number of applications. We shall present two of them. First, we recover previously known semiclassical Weyl's laws on Euclidean domains and closed manifolds. These results were proved in 60s and 70s. However, thanks to our setup, they can be deduced results of Minakshisundaram and Pleijel on short time heat kernel asymptotics for Laplacians that were established in the late 40s. Second, we obtain semiclassical Weyl laws for noncommutative tori for any dimension. These laws were conjectured by Ed McDonald and the speaker.

Fri, Oct 25

Applied Math Seminar
Deniz Bilman, University of Cincinnati
General rogue waves of infinite order: exact properties, asymptotic behaviour, and effective numerical computation.
3:00PM, Math 250

In this talk we will present results from a comprehensive analysis of a family of solutions of the focusing nonlinear Schrödinger equation called general rogue waves of infinite order. These solutions have recently been shown to describe various limit processes involving large-amplitude waves, and they have also appeared in some physical models not directly connected with nonlinear Schrödinger equations. We establish the following key property of these solutions: they are all in \(L^2(\mathbb{R})\) with respect to the spatial variable but they exhibit anomalously slow temporal decay. In this talk, we will define these solutions, establish their basic exact and asymptotic properties, and describe computational tools for calculating them accurately. This is joint work with Peter D. Miller.

Fri, Nov 1

Applied Math Seminar
Tharusha Bandara, UB
Mathematical Model on HIV and Nutrition
3:00PM, Math 250

HIV continues to be a significant global health issue, having claimed millions of lives in the last few decades. While several empirical studies support the idea that proper nutrition is beneficial in the fight against HIV, very few studies have focused on developing and utilizing mathematical modeling approaches to assess the association between HIV, the human immune response to the disease, and nutrition. In this presentation, we introduce novel within-host models for HIV that capture the dynamic interactions among HIV, the immune system, and nutrition. We explore the relationship between serum protein levels and key parameters such as viral loads, viral production rates, and the enhancement rate of protein by the virus in HIV-infected individuals. Additionally, we will discuss the correlation between dietary protein intake and serum protein levels in individuals with HIV. We will conclude the presentation with the introduction of a novel epidemiological model on HIV, which can be integrated with the aforementioned within-host model, considering economic and nutritional aspects.

Fri, Nov 8

Applied Math Seminar
Chun Liu, Illinois Institute of Technology
Active Fluids and Applications.
3:00PM, Math 250

Almost all biological activities involve chemical reactions and active materials. In this talk I will develop a general theory for active fluids which convert  chemical energy into various type of mechanical energy. This is the extension of the classical energetic variational approaches for mechanical systems. The methods will cover a wide range of both chemical reaction kinetics  and mechanical processes. This is a joint project with many collaborators, in particular, Bob Eisenberg, Yiwei Wang and Tengfei Zhang.

Mon, Nov 11

Algebra Seminar
Adam Sikora, University at Buffalo
Stated skein algebras and a geometric approach to quantum groups 
4:00PM, Mathematics Building room 250

We introduce a theory of stated \(SL(n)\)-skein algebras of surfaces, which provides a geometric/combinatorial interpretation for the quantum function algebras \(O_q(SL(n))\) and other related notions from quantum algebra. They also quantize the \(SL(n)\)-character varieties of surfaces, are examples of quantum cluster algebras, and are closely related to Reshetikhin-Turaev quantum invariants of links, factorization homology, and the lattice gauge theory.

Wed, Nov 13

Analysis Seminar
Joseph Leung, Rutgers University
Second moment for GL(3) L-functions in the critical line
4:00PM, 250 Math Building

We discuss the second moment for the GL(3) standard L-functions on the critical line. When the GL(3) form is specialized as the Eisenstein series, this is the infamous sixth moment of the zeta function. In a joint work with Matthew Young and Agniva Dasgupta, we obtain a nontrivial upper bound for the moment. This work is inspired by the short second moment result achieved by Aggarwal, Leung, and Munshi. We will discuss the key ideas of the proof and its applications, which include an improvement on the Rankin-Selberg problem.

Fri, Nov 15

Applied Math Seminar
Nicholas Ossi, UB
Time-periodic breather solutions of the discrete defocusing Ablowitz-Ladik equation with large background amplitude.
3:00PM, Math 250

This talk will cover the results of a recent project on the so-called Kuznetsov-Ma (KM) breather solutions of the defocusing Ablowitz-Ladik equation (an integrable spatial discretization of the nonlinear Schrödinger equation) with large background amplitude. These KM solutions are periodic in time and localized in space, and can be obtained via the inverse scattering transform method. While they can generically experience singularity in finite time, conditions under which KM breathers remain regular will be discussed. Additionally, Darboux transformations are employed to construct multi-KM solutions. This is joint work with Barbara Prinari and Evans Boadi from UB, Stathis Charalampidis from San Diego State University, and Panos Kevrekidis from UMass Amherst.

Mon, Nov 18

Algebra Seminar
Jakob Streipel, UB

4:00PM, Mathematics Building, room 250

Getting bounds for L-functions is along-standing problem in number theory, both as an ``easier'' (but still fiendishly hard) substitute for the Riemann Hypothesis and as a type of result which in its own right has interesting applications to arithmetic, algebra, and geometry.  One compelling reason to work on this problem is that it comes with a built-in way of measuring progress: there is always a ``trivial'' bound coming essentially from complex analysis, with no use of the arithmetic structure of the problem, known as the convexity bound (which is \(\text{conductor}^{1/4}\)). Beyond the convexity bound there are natural barriers that for some mysterious reason seem to crop up in all problems of this kind, called Burgess-type bounds (\(\text{conductor}^{3/16}\))and the Weyl-type bounds (\(\text{conductor}^{1/3}\)). On \(GL(2)\), a Weyl-type bound has been known for over 40 years due to Anton Good.  In this talk we present joint work in progress with Roman Holowinsky, Ritabrata Munshi, and Prahlad Sharma where we for the first time cross the Weyl barrier on \(GL(2)\).

Fri, Feb 3

Geometry and Topology Seminar
Morgan Weiler (Cornell University)
ECH cobordism maps and infinite staircases of 4D symplectic embeddings
4:00PM, 122 Mathematics Building

The ellipsoid embedding function of a symplectic manifold measures the amount by which the symplectic form must be scaled in order to fit an ellipsoid of a given eccentricity. It generalizes the Gromov width and ball packing numbers. In 2012 McDuff and Schlenk computed the ellipsoid embedding function of the ball, showing that it exhibits a delicate piecewise linear pattern known as an infinite staircase. Since then, the embedding function of many other symplectic four-manifolds have been studied, and not all have infinite staircases. We classify those symplectic Hirzebruch surfaces whose embedding functions have an infinite staircase. We will emphasize the relationship between the geometric motivation for cobordism maps coming from embedded contact homology (ECH) and McDuff's methods for obstructing ellipsoid embeddings using Taubes' Gromov-Seiberg-Witten invariants of symplectic 4-manifolds. Based on work with Magill and McDuff and work in progress with Magill and Pires.

Mon, Feb 13

Applied Math Seminar
Jiyoung Kang, Pukyong National University
Brain Dynamics and its Control: Computational Approaches

2:00PM, Math 250 and on Zoom - contact mbichuch@buffalo.edu for link

The brain is a complex nonlinear system, and it is challenging to understand its dynamics. In addition, the brain possesses plasticity, which frequently results in a temporary improvement followed by a decline after brain disease treatment. We have been developing computational modeling techniques for brain dynamics analysis to address these issues. In this presentation, several computational modeling studies utilizing electrophysiology data, such as calcium imaging and voltage-sensitive dye imaging data, along with energy landscape analysis studies using fMRI data, will be discussed. I will conclude by discussing our recent computational framework for brain control, which can account for both brain dynamics and plasticity.

Mon, Feb 13

Algebra Seminar
Yiqiang Li, University at Buffalo
Quantum groups and edge contraction

4:00PM, 250 Math Bldg

Edge contraction is a simple operation on graphs that produces a new graph by merging two vertices on a given graph along an edge. In this talk, I will report recent studies on the behaviors of representation-theoretic objects attached to graphs under an edge contraction operation.

Thu, Feb 16

Special Event
Solitons and the inverse scattering transform: an overview
 Solitons and the inverse scattering  transform: an overview.Abstract: An exciting and extremely active area of research investigation is the study of solitons and the nonlinear partial differential equations that describe them. In this talk, we will discuss what solitons are, and what makes them so special. We will see when the first solitons were observed, and when the first math that describe them appeared. We will introduce ourselves to integrable systems, and we will describe how the technique of the inverse scattering transform is applied in soliton theory.  If time permits, we will give some examples of integrable systems and we will discuss their applications.  
4:00PM

Title: Solitons and the inverse scattering  transform: an overview.

Abstract: An exciting and extremely active area of research investigation is the study of solitons and the nonlinear partial differential equations that describe them. In this talk, we will discuss what solitons are, and what makes them so special. We will see when the first solitons were observed, and when the first math that describe them appeared. We will introduce ourselves to integrable systems, and we will describe how the technique of the inverse scattering transform is applied in soliton theory.  If time permits, we will give some examples of integrable systems and we will discuss their applications. 

Fri, Feb 17

Geometry and Topology Seminar
Yvon Verberne (University of Toronto)
Automorphisms of the fine curve graph

4:00PM, 122 Mathematics Building

The fine curve graph of a surface was introduced by Bowden, Hensel and Webb. It is defined as the simplicial complex where vertices are essential simple closed curves in the surface and the edges are pairs of disjoint curves. We show that the group of automorphisms of the fine curve graph is isomorphic to the group of homeomorphisms of the surface, which shows that the fine curve graph is a combinatorial tool for studying the group of homeomorphisms of a surface. This work is joint with Adele Long, Dan Margalit, Anna Pham, and Claudia Yao.

Mon, Mar 6

Algebra Seminar
Bangming Deng, Tsinghua U
Fourier transforms on Ringel-Hall algebras
4:00PM, Zoom - contact achirvas@buffalo.edu for link

We study Fourier transforms on the double Ringel-Hall algebra of a quiver and make a comparison between Lusztig's symmetries and the isomorphisms defined by Sevenhant and Van den Bergh via combining BGP-reflection isomorphisms and Fourier transforms on the double Ringel-Hall algebra.

Mon, Mar 13

Applied Math Seminar
Weinan Wang, University of Arizona
Recent progress on the well-posedness theory for some kinetic models

2:00PM, Math 250 and on Zoom - contact mbichuch@buffalo.edu for link

The Boltzmann and Landau equations are two fundamental models in kinetic theory. They are nonlocal and nonlinear equations for which (large data) global well-posedness is an extremely difficult problem that is nearly completely open. In this talk, I will discuss two more tractable and related questions: (1) Local well-posedness for the Boltzmann equation and (2) Schauder estimates and their application to uniqueness of solutions to the Landau equation. At the end of the talk, I will discuss some open problems and future work. This is based on joint work with Christopher Henderson.

Fri, Mar 17

Geometry and Topology Seminar
Assaf Bar-Natan (Brandeis University)
How the Thurston metric on Teichmuller space is (not) like L^(infty)

4:00PM, 122 Mathematics Building

The Thurston Metric, introduced by Thurston in 1986, is an asymmetric metric on Teichmuller space, which measures distance between surfaces using the Lipschitz constant of maps between them. In this talk, I will tell you what I know about geodesics in this metric. Specifically, I will tell you about the geodesic envelope, its shape (and how the Thurston metric is similar to L^(infty)), and its width (and how the Thurston metric is not similar to L^(infty)). We'll finish up with a theorem which gives sufficient conditions for geodesics between two points to be "essentially unique" (ie, uniformly bounded diameter from each other) for low complexity surfaces.

Mon, Mar 27

Algebra Seminar
Guanglian Zhang, Shanghai Jiao Tong University
Every type-A quiver locus is a Kazhdan-Lusztig variety

9:00AM, Note unusual time. On Zoom (please email achirvas@buffalo.edu)

The Zariski orbit closures of the representations of type-A Dynkin quivers under the action of general linear groups are related in deep ways to Schubert varieties. In this paper, we construct a scheme-theoretic isomorphism from a type-A quiver locus to the intersection of some opposite Schubert cell and Schubert variety, also known as a Kazhdan-Lusztig variety in geometric representation theory. This isomorphism is a generalization, and also an unification, of the Zelevinsky maps on equioriented type-A quiver loci and bipartite type-A quiver loci which are respectively presented by A. V. Zelevinsky in 1985 and by R. Kinser and J. Rajchgot in 2015. This result provides a more direct and natural connection between type-A quiver loci and Schubert varieties than prior similar work.

Mon, Mar 27

Applied Math Seminar
Qingguo Hong, Penn State
A priori error analysis and greedy training algorithms for neural networks solving PDEs.

2:00PM, Math 250 and on Zoom - contact mbichuch@buffalo.edu for link

We provide an a priori error analysis for methods solving PDEs using neural networks. We show that the resulting constrained optimization problem can be efficiently solved using greedy algorithms, which replaces stochastic gradient descent. Following this, we show that the error arising from discretizing the energy integrals is bounded both in the deterministic case, i.e. when using numerical quadrature, and also in the stochastic case, i.e. when sampling points to approximate the integrals. This innovative greedy  algorithm is tested on several benchmark examples to confirm its efficiency and robustness.

Fri, Mar 31

Geometry and Topology Seminar
Yuan Yao (UC Berkeley)
Computing embedded contact homology in Morse-Bott settings

4:00PM, 122 Mathematics Building

Embedded contact homology (ECH) is a Floer theory defined for contact 3-manifolds with generators periodic Reeb orbits and differential defined by counts of J-holomorphic curves. It has been shown to be isomorphic to a versions of monopole Floer homology and Heegard Floer homology. It has many applications to symplectic and contact geometry (e.g. symplectic embedding problems, dynamics of Reeb vector fields). In this talk we will first review the definition of ECH; then we will discuss how to define ECH in the Morse-Bott setting. Our main tools will be  1) the intersection theory of J-holomorphic curves, 2) understanding how J-holomorphic curves degenerate into geometric objects called cascades when the background contact form degenerates to a Morse-Bott contact form, as well as 3) a gluing theorem that tells us how to glue cascades back into J-holomorphic curves.

Mon, Apr 3

Algebra Seminar
Mariusz Tobolski, University of Wroclaw
Cohomology of free unitary quantum groups 
4:00PM, Mathematics Building room 250

In this talk, I will present the Hochschild and bialgebra cohomology with 1-dimensional coefficients of the \(*\)-algebras associated with free universal unitary quantum groups. The result is based on the free resolution of the counit of the free orthogonal quantum groups found by Collins, Härtel, and Thom which was then generalized by Bichon to the case of quantum groups associated with a nondegenerate bilinear form. In fact, we compute cohomology groups of the universal cosovereign Hopf algebras, which generalize free unitary quantum groups and are connected to quantum groups of non-degenerate bilinear forms. This is a joint work with U. Franz, M. Gerhold,A. Wysocza\'nska-Kula, and I. Baraquin.

Mon, Apr 10

Algebra Seminar
Robert Corless, Western University
Bohemian Matrix Geometry 
4:00PM, Mathematics Building room 250

A Bohemian matrix family is a set of matrices all of whose entries are drawn from a fixed, usually discrete and hence bounded, subset of a field of characteristic zero. Originally these were  integers---hence the name, from the acronym BOunded HEight Matrix of Integers(BOHEMI)---but other kinds of entries are also interesting. Some kinds of questions about Bohemian matrices can be answered by numerical computation, but sometimes exact computation is better. In this paper we explore some Bohemianfamilies                  (symmetric, upper Hessenberg, or Toeplitz) computationally, and answer some (formerly) open questions posed about the distributions of eigenvalue densities.

This work connects with several disparate areas of mathematics, including dynamical systems, combinatorics, probability and statistics, and number theory.  Because the thinking about the topic is so recent, most of the material is still quite exploratory, and this talk will be accessible to students as well as to faculty.  Several open problems remain open, and I would welcome your thoughts on them.

This is joint work with several people, including EuniceY.S. Chan, Leili Rafiee Sevyeri, Neil J. Calkin, Piers W. Lawrence, Laureano Gonzalez-Vega, Dan Piponi, Juana Sendra, and Rafael Sendra.

Fri, Apr 14

Geometry and Topology Seminar
Nima Hoda (Cornell University)
Normed polyhedral complexes and nonpositive curvature
4:00PM, 122 Mathematics Building

Much recent work in geometric group theory has involved the study of groups acting on metric spaces of nonpositive curvature modeled on \(\ell^1\) (median graphs and metric median spaces), \(\ell^2\) (CAT(0) spaces) and \(\ell^{\infty}\) (Helly graphs and injective spaces).  All three of these cases appear in CAT(0) cube complexes depending on the choice of metric (\(\ell^1\), \(\ell^2\) or \(\ell^{\infty}\)) placed on the cubes.  It is natural to ask if there are metric nonpositive curvature conditions that can be modeled on more general normed spaces.  In this talk, I will discuss recent work with Thomas Haettel and Harry Petyt in which we prove Busemann-convexity of CAT(0) cube complexes whose cubes are given an \(\ell^p\) metric, with \(1 < p < \infty\).  I will also discuss conditions we proved for normed polyhedral complexes to be Busemann-convex and strongly bolic.

Mon, Apr 17

Algebra Seminar
Jacopo Zanchettin, SISSA
Hopf algebroids and twists for quantum projectivespaces 
4:00PM, Mathematics Building room 250

The Ehresmann-Schauenburg (E-S) bialgebroid associatedwith a Hopf-Galois extension is the noncommutative analog of the gauge groupoidassociated with a principal bundle. As for a Hopf algebra, a Hopf algebroid isa bialgebroid with an invertible antipode. In this talk, after recalling somebasic notions about rings, coring, and bialgebroids, we first show how twists(a sub-group of characters) of a bialgebroid are related to antipodes in thegeneral case. Eventually, after a short introduction to Hopf-Galois extensions,we characterize them for the E-S bialgebroid. Finally, we work out the exampleof a family of \(O(U(1))\)-extensions over quantum projective spaces. This talkis based on joint work with L. Dabrowski and G. Landi arXiv:2302.12073

Fri, Apr 21

Geometry and Topology Seminar
Vasudevan Srinivas (Tata Institute)
What is the Hodge Conjecture?

4:00PM, 122 Mathematics Building

The Hodge Conjecture is one of the famous unsolved problems in algebraic geometry over the complex number field. This talk will give an accessible introduction to this problem, meant for the non-expert. At the end, I will briefly discuss some related work of mine with A. Rosenschon.

Mon, Apr 24

Algebra Seminar
Ana Agore, Max Planck Institut and Simion Stoilow Institute of Mathematics
Universal constructions for Poisson algebras. Applications. 
9:00AM, Zoom (please email achirvas@buffalo.edu)

We introduce the universal algebra of two Poisson algebras \(P\) and \(Q\) as a commutative algebra \(A := \mathcal{P}(P, Q)\) satisfying a certain universal property. The universal algebra is shown to exist for any finite-dimensional Poisson algebra \(P\) and several of its applications are highlighted. For any Poisson \(P\)-module \(U\), we construct a functor \(U\otimes-: {}_A\mathcal{M} \to {}_Q\mathcal{PM}\) from the category of \(A\)-modules to the category of Poisson \(Q\)-modules which has a left adjoint whenever \(U\) is finite-dimensional. Similarly, if \(V\) is an \(A\)-module, then there exists another functor \(-\otimes V:{}_P\mathcal{PM}\to {}_Q\mathcal{QM}\) connecting the categories of Poisson representations of \(P\) and \(Q\) and the latter functor also admits a left adjoint if \(V\) is finite-dimensional. If \(P\) is\(n\)-dimensional, then \(\mathcal{P}(P) := \mathcal{P}(P, P)\) is the initial object in the category of all commutative bialgebras coacting on \(P\). As an algebra,\(\mathcal{P}(P)\) can be described as the quotient of the polynomial algebra\(k[X_{ij} | i, j = 1, · · · , n]\) through an ideal generated by \(2n^3\)non-homogeneous polynomials of degree \(\le 2\). Two applications are provided. The first one describes the automorphisms group\(\mathrm{Aut}_{\mathrm{Poiss}}(P)\) as the group of all invertible group-like elements of the finite dual \(\mathcal{P}(P)^{\circ}\).  Secondly, we show that for an abelian group\(G\), all \(G\)-gradings on \(P\) can be explicitly described and classified in terms of the universal coacting bialgebra \(\mathcal{P}(P)\). Joint work with G.Militaru.

Mon, Apr 24

Applied Math Seminar
Boaz Ilan, UC Merced
NLS equations: solitons, dispersive shocks and singularity formation.

2:00PM, Zoom - contact mbichuch@buffalo.edu for link

The Nonlinear Schrödinger (NLS) equation is a universal model for nonlinear dispersive waves. It describes the mean field superfluidic dynamics of a dilute gas of bosons near the absolute zero temperature, called a Bose-Einsten condensate, intense laser propagation through matter, and many other nonlinear systems. NLS equations and their generalizations possess a rich variety of special solutions, such as solitons, dispersive shocks, and singularity formation. I will discuss recent analytical and computational results and open problems with application to BECs.

Wed, Apr 26

Analysis Seminar
Min Woong Ahn, SUNY at Buffalo
The error-sum function of Pierce expansions
4:00PM, 250 Math Building

The notion of the error-sum function was first studied by Ridley and Petruska in the context of the regular continued fraction expansion. The Pierce expansion is another classical representation of a real number. In this talk, I will introduce the error-sum function of Pierce expansions and discuss the basic properties of the function and the fractal property of the graph of the function.

Thu, Apr 27

Colloquium
Bena Tshishiku (Brown University)
Mapping class groups and Nielsen realization problems

4:00PM, 250 Mathematics Building

The mapping class group \(\mathit{Mod}(M)\) of a smooth manifold \(M\) is the group of diffeomorphisms of \(M\), modulo isotopy. The study of mapping class groups interacts with many areas, including geometric topology, group theory, dynamics, and algebraic geometry. We explain some of these connections from the point-of-view of the Nielsen realization question. This problem, versions of which were posed by Nielsen (1932) and Thurston (1977), asks when a subgroup \(G<\mathit{Mod}(M)\) can be lifted to \(\mathrm{Diff}(M)\) under the natural projection \(\mathit{Diff}(M)\to \mathit{Mod}(M)\). 

Fri, Apr 28

Geometry and Topology Seminar
Bena Tshishiku (Brown University)
Pseudo-Anosov theory in the Goeritz group

4:00PM, 122 Mathematics Building

For a surface \(S_g\) of genus \(g\), the Goeritz group is the subgroup of the mapping class group \(\mathit{Mod}(S_g)\) consisting of isotopy classes that extend to the handlebodies in the genus-g Heegaard splitting of the 3-sphere. There are many open questions about the algebra of this group, including whether or not it's finitely generated when \(g\geq 4\). This talk will focus on geometric aspects of the genus-2 Goeritz group. I will explain a refinement of the Nielsen-Thurston classification for this group and will show that its purely pseudo-Anosov subgroups are convex cocompact, which answers a question of Farb-Mosher in a special case. 

Mon, May 1

Algebra Seminar
Michael Brannan, University of Waterloo
Ulam stability for quantum groups 
4:00PM, 250 Mathematics Building

In recent years, there has been a growing interest in the study of approximate representations of various algebraic structures. This is due to some very deep connections with (1) approximation properties for groups and (2) questions about robustness in quantum information theory. The basic question that we are interested in is the following: If we are given a linear map from an algebra (or group) into the bounded operators on a Hilbert space that is “almost” multiplicative, under what conditions can we guarantee that this map is a small perturbation of an actual representation of the algebra? I will describe some of the history around this problem as well as some on going work with Junichiro Matsuda (Kyoto) and Jennifer Zhu (Waterloo), where we investigate the Ulam (=operator norm) stability of approximate representations for compact and discrete quantum groups.

Wed, May 3

Analysis Seminar
Daxun Wang. SUNY at Buffalo
Boundary actions of groups and their C*-algebras
4:00PM, 250 Math Building

Pure infiniteness of C*-algebras plays an important role in the classification of C*-algebras. In this talk, we will talk about boundary actions of some popular groups that arise in geometric group theory such as right angled Artin groups, right angled Coxeter groups and graphs of groups, and show that the reduced crossed product C*-algebras of these boundary actions are purely infinite. This is a joint work with Xin Ma.

Fri, May 12

Geometry and Topology Seminar
Adam Sikora (University at Buffalo)
On skein modules of rational homology spheres

4:00PM, 122 Mathematics Building

The Kauffman bracket skein module S(M) of a 3-manifold M classifies polynomial invariants of links in M satisfying Kauffman bracket skein relations. Witten conjectured that the skein module is finite dimensional for any closed M. This conjecture was proved by Gunningham, Jordan, and Safronov, however their work does not lead to an explicit computation of S(M). In fact, S(M) has been computed for a few specific families of closed 3-manifolds so far only. We introduce a novel method of computing these skein modules for certain rational homology spheres. (This is joint work with R. Detcherry and E. Kalfagianni.)

Wed, Sep 6

Analysis Seminar
Jintao Deng, SUNY at Buffalo
The equivariant coarse Baum-Connes conjecture
4:00PM, 250 Mth Building

The equivariant coarse Baum-Connes conjecture claims that a certain assembly map from the equivariant K-homology of a metric space with a group action to the K-theory of the Roe algebras is an isomorphism. It has important applications in the study of the existence of Riemannian metric with positive scalar curvature. In this talk, I will talk about the concept of Roe algebras which encode the large-scale geometry of a metric space and group actions. The higher index of an elliptic operator is an element of the K-theory of this algebra. The equivariant coarse Baum-Connes conjecture provides an algorithm to compute its K-theory. I will talk about our recent result that the equivariant coarse Baum-Connes conjecture holds for a metric space with a group action under the conditions that the group is amenable and the associated quotient space is coarsely embeddable into Hilbert space. This is a joint work with Qin Wang and Benyin Fu.

Thu, Sep 7

Colloquium
Demonstration of new GRADER app
4:00PM, Room 250

Mon, Sep 11

Algebra Seminar
Peter Koroteev, University at Buffalo
Opers and integrability
4:00PM, University at Buffalo, Buffalo, NY 14260, USA

I will introduce (\(q\)-)opers on a projective line in the presence of twists and singularities and will discuss the space of such opers. We will see how Bethe Ansatz equations for quantum spin chains and energy level equations of classical soluble models of Calogero-Ruijsenaars type naturally appear from the oper construction. Both can also be described in terms of so-called \(QQ\)-systems, which have their origins in algebra and representation theory. Our construction is universal and works for any simple, simply-connected complex Lie group \(G\).

Wed, Sep 13

Analysis Seminar
Jintao Deng, SUNY at Buffalo
The equivariant coarse Baum-Connes conjecture Part II
4:00PM, 250 Math Building

Title: The equivariant coarse Baum-Connes conjecture Part II

Mon, Sep 18

Algebra Seminar
Tomasz Maszczyk, University of Warsaw
Quantum symmetries of Frobenius algebras
4:00PM, University at Buffalo, Buffalo, NY 14260, USA

We introduce the notions of "quantum support" and "quantum fundamental cycle" for a Frobenius algebra. We realize the Pareigis Hopf algebra, which encodes the monoidal structure of the category of complexes (via the Pareigis transform which is the identity on objects), as a universal quantum symmetry of the dual numbers algebra. We show that under the Pareigis transform the category of corresponding equivariant quasicoherent sheaves on the double point is equivalent to the category of complexes with square zero homotopies. In particular, the Pareigis transform of the algebra of dual numbers is the terminal object of the extended Hinich category of local pseudo-compact algebras. We prove that the Pareigis transform of the Frobenius support of the algebra of dual numbers is a closed graded trace of dimension\(-1\) on the terminal Hinich algebra, being a boundary of the Pareigis transform of the augmentation of dual numbers. This can be understood as a DGA model of the empty set with a homologically trivial \((-1)\)-dimensional fundamental cycle. We also study symmetries of other truncated polynomial algebras and relate them to the representation theory of \(SL(2)\), Hamiltonian moment maps, Fourier transforms, and the Springer resolution of the singularity of the nilpotent cone of \(SL(2)\).

Wed, Sep 20

Analysis Seminar
Min Woong Ahn, SUNY at Buffalo
Hausdorff dimensions in Pierce expansions
4:00PM, 250 Math Building

The Pierce expansion is one of many real number representation systems. Shallit (1986) established the law of large numbers, the central limit theorem, and the law of the iterated logarithm of the digits of the Pierce expansions. Additionally, it was shown that the series of iterates under a mapping that yields the Pierce expansion converges Lebesgue-almost everywhere. In this talk, I will discuss the Hausdorff dimensions of such sets with Lebesgue measure zero.

Fri, Sep 22

Applied Math Seminar
Jiuhua Hu, University of Wisconsin at Madison
Wavelet-based Edge Multiscale Parareal Algorithm for Parabolic Equations with Heterogeneous Coefficients.

3:00PM, Math 122 and on Zoom - contact mbichuch@buffalo.edu for link

In this talk, I will start with an introduction of some multiscale methods to solve multiscale problems. Then I will talk about the Wavelet-based Edge Multiscale Parareal Algorithm to solve parabolic equations with heterogeneous coefficients. This algorithm combines the merits of multiscale methods that can deal with heterogeneity in the spatial domain effectively, with the robust capabilities of parareal algorithms for speeding up time evolution problems. The convergence rate of this algorithm can also be derived.  To illustrate the algorithm's performance, I will present some numerical tests. This is a joint work with Guanglian Li (The University of Hong Kong).

Mon, Sep 25

Algebra Seminar
Gigel Militaru, University of Bucharest
The set-theoretic quantum Yang-Baxter equation: new perspectives and strategies
4:00PM, Zoom only (please email achirvas@buffalo.edu)

We provide an answer, in a special case, to the(extremely difficult) problem of Drinfel'd by proving that the category of solutions of the set-theoretic Yang-Baxter equation of Frobenius-Separability(FS) type is equivalent to the category of pointed Kimura semigroups. As applications, all involutive, idempotent, nondegenerate, surjective, finiteorder, unitary or indecomposable solutions of FS type are classified. Forinstance, if \(|X| = n\), then the number of isomorphism classes of all such solutions on \(X\) that are (a) left non-degenerate, (b) bijective, (c) unitary or (d) indecomposable and left-nondegenerate is: (a) the Davis number \(d(n)\),(b) \(\sum_{m|n} \, p(m)\), where \(p(m)\) is the Euler partition number, (c)\(\tau(n) + \sum_{d|n}\left\lfloor \frac d2\right\rfloor\), where \(\tau(n)\) is the number of divisors of \(n\), or (d) the Harary number \(\mathfrak{c} (n)\).

Joint work with Ana Agore and Alexandru Chirvasitu.

Thu, Sep 28

Colloquium
Xuhua He, University of Hong Kong
Machine learningassisted exploration for affine Deligne-Lusztig varieties Speaker host: Yiqiang Li
10:00AM, Via zoom link: https://buffalo.zoom.us/j/98322986435?pwd=OE5OZTVEcE1BZmNuZmYrdmhUNVk4UT09

Speaker: XuhuaHe,  University of Hong Kong

Title: Machine learningassisted exploration for affine Deligne-Lusztig varieties

Speaker host: Yiqiang Li

Fri, Sep 29

Applied Math Seminar
Ming Zhong, Illinois Institute of Technology
Learning Collective Behaviors from Observation.

3:00PM, Math 122 and on Zoom - contact mbichuch@buffalo.edu for link

Collective behavior (clustering, flocking, milling, swarming and synchronization, etc), also known as self organization, is the emergence of global patterns from local interactions of agents in complex systems. It can be observed in physics (super conductivity), chemistry (liquid crystals), biology (protein folding), sociology (herd behavior), etc. It is challenging to understand such behaviors using rigorous mathematical formulas.  We present a series of machine learning methods to construct dynamical systems to explain these behaviors from observation data.  We demonstrate the efficiency and effectiveness of our methods not only through extensive numerical testings, but also via thorough theoretical analysis.  Future directions of enhancements and extensions are also discussed.

Fri, Oct 13

Applied Math Seminar
Nick Moore, Colgate University
TBA.

3:00PM, Math 122 and on Zoom - contact mbichuch@buffalo.edu for link

TBA.

Fri, Oct 13

Geometry and Topology Seminar
Ivan Dynnikov (Steklov Mathematical Institute of RAS)
An algorithm for comparing Legendrian links
 

4:00PM, 122 Mathematics Building

The talk is based on my joint works with Maxim Prasolov and Vladimir Shastin, where we studied the relation between rectangular diagrams of links and Legendrian links. This relation allows for a complete classification of exchange classes of rectangular diagrams in terms of equivalence classes of Legendrian links and their symmetry groups. Since all rectangular diagrams of given complexity can be searched, this yields a method to algorithmically compare Legendrian links. Of course, the general algorithm has too high complexity for a practical implementation, but in some situations, the most time consuming parts can be bypassed, which allows us to confirm the non-equivalence of Legendrian knots in several previously unresolved cases.

Mon, Oct 16

Algebra Seminar
Gurbir Dhillon, Yale
Sheaves on semi-infinite flag manifolds and Langlands duality Abstract : Let \(G\), \(G^L\) be Langlands-dual reductive groups. The geometric Satake equivalence is the wonderful fact that one can realize the monoidal category of \(G^L\)-representations as a category of constructible sheaves on a space built from \(G\), namely its affine Grassmannian. After recalling this story, we will present similar realizations of the representations of every parabolic and Levi subgroup of \(G^L\) as sheaves on the semi-infinite partial flag manifolds of \(G\). Time permitting, we will describe some natural extensions, including the realizations of the representations of various quantum groups. The contents of this talk build on work and conjectures of many people, notably Arkhipov, Bezrukavnikov, Braverman, Feigin, Finkelberg, Frenkel, Gaitsgory, Lusztig, Mirkovic, Vilonen, and Raskin, and is the subject of work in progress with Campbell, Chen, Lysenko, and Achar-Riche. 
4:00PM, Mathematics Building, room 250

Title: Sheaves on semi-infinite flag manifolds and Langlands duality

Abstract : Let \(G\), \(G^L\) be Langlands-dual reductive groups. The geometric Satake equivalence is the wonderful fact that one can realize the monoidal category of \(G^L\)-representations as a category of constructible sheaves on a space built from \(G\), namely its affine Grassmannian. After recalling this story, we will present similar realizations of the representations of every parabolic and Levi subgroup of \(G^L\) as sheaves on the semi-infinite partial flag manifolds of \(G\). Time permitting, we will describe some natural extensions, including the realizations of the representations of various quantum groups.

The contents of this talk build on work and conjectures of many people, notably Arkhipov, Bezrukavnikov, Braverman, Feigin, Finkelberg, Frenkel, Gaitsgory, Lusztig, Mirkovic, Vilonen, and Raskin, and is the subject of work in progress with Campbell, Chen, Lysenko, and Achar-Riche.

Tue, Oct 17

Applied Math Seminar
Ruth López Fajardo, Florida State
From Data to Insights: Permeability Estimation with a Direct Filter
 
Abstract : In this study, we implement a computational method for the estimation of permeability values within a defined spatial domain based on observed fluid pressure distributions. Our methodology frames the problem as an inverse problem where the Forward Model, responsible for generating the observed data, is solved using a Locally Conservative Enriched Galerkin Solver for Elliptic and Parabolic partial differential equations (PDEs). The parameter estimation process is then addressed with a Bayesian framework, treating it as an on-line problem, which is subsequently solved by the Direct filter method.
2:00PM, Math 122

Title: From Data to Insights: Permeability Estimation with a Direct Filter
 
Abstract : In this study, we implement a computational method for the estimation of permeability values within a defined spatial domain based on observed fluid pressure distributions. Our methodology frames the problem as an inverse problem where the Forward Model, responsible for generating the observed data, is solved using a Locally Conservative Enriched Galerkin Solver for Elliptic and Parabolic partial differential equations (PDEs). The parameter estimation process is then addressed with a Bayesian framework, treating it as an on-line problem, which is subsequently solved by the Direct filter method.

Wed, Oct 18

Analysis Seminar
Francesc Perera, Universitat Autonoma de Barcelona
Traces on ultrapowers of C*-algebras
4:00PM, 250 Math Building

Every sequence of traces on a C*-algebra A induces a limit trace on a free ultrapower of A. Using Cuntz semigroup techniques, we characterize when these limit traces are dense. Quite unexpectedly, we obtain as an application that every simple C*-algebra that is (m, n)-pure in the sense of Winter is already pure. This is joint work with Ramon Antoine, Leonel Robert, and Hannes Thiel.

Thu, Oct 26

Colloquium
Lei Yang, Institute for Advanced Study
Effective versions of Ratner’s equidistribution theorem
4:00PM, Mathematics Building, room 250

I will talk about recent progress in the study of quantitative equidistribution of unipotent orbits in homogeneous spaces, namely, effective versions of Ratner's equidistribution theorem. In particular, I will explain the proof of unipotent orbits in SL(3, R)/SL(3, Z). The proof combines new ideas from harmonic analysis and incidence geometry. In particular, the quantitative behavior of unipotent orbits is closely related to a Kakeya model.  

Mon, Oct 30

Algebra Seminar
Bach Nguyen, Xavier University
: Poisson geometry and the representation theory of cluster algebras 
4:00PM, Mathematics Building room 250

The relationship between Poisson geometry and cluster algebras was first studied by M. Gekhtman, M. Shapiro, and A.Vainshtein. Following their work, we study the global geometry picture of the affine Poisson varieties associated with a cluster algebra and its quantizations,  root-of-unity quantum cluster algebras. In particular, we prove that the spectrum of the upper cluster algebra, endowed with the GSV Poisson structure, has a Zariski-open orbit of symplectic leaves and give an explicit description of it. Our result provides a generalization of the Richardson divisor of Schubert cells in flag varieties. Further, we describe the fully-Azumaya loci of the root-of-unity upper quantum cluster algebras, using the theory of Poisson orders. This classifies their irreducible representations of maximal dimension. This is joint work with Greg Muller, Kurt Trampel and Milen Yakimov.

Wed, Nov 1

Analysis Seminar
Jinmin Wang, Texas A&M University
Stoker's problem and index theory on manifolds with polytope singularities
4:00PM, 250 Math Building

The Stoker problem states that the dihedral angles of a convex Euclidean polyhedron determine the angles of each faces. In this talk, I will present joint works with Zhizhang Xie and Guoliang Yu that answer positively to Stoker's problem, and prove a more general dihedral rigidity for manifolds with polytope singularities. I will briefly introduce our approach, the index theory of Dirac-type operators on manifolds with polytope singularities under certain boundary conditions. One of the key observations is the essential self-adjointness of the Dirac-type operators near conical singularities.

Fri, Nov 3

Applied Math Seminar
Yijun Sun, UB
Machine learning approaches to cancer progression modeling and cancer driver gene detection.

3:00PM, Math 122 and on Zoom - contact mbichuch@buffalo.edu for link

Cancer evolution has been studied for over 50 years. However, due to the difficulty in obtaining time-series data, there is currently no established cancer progression model. The lack of time-series data has been recognized by the field as the central problem in studying cancer evolution. We recently developed a novel computational strategy that overcomes the existing sampling limitations and enables the construction of cancer progression models by using massive static sample data. The application of the developed computational pipeline to the TCGA and METABRIC datasets resulted in the first working model of breast cancer progression that covers the entire disease process. To demonstrate the validity of the constructed model, we performed a series of internal and external validations. The replication of the bifurcating structure in additional 25 independent datasets and the post-construction association analysis of genetic and clinical variables and matched primary and metastatic tumor samples provided substantial support for the model. Our study shed light on some longstanding issues regarding the origins of molecular subtypes and their possible progression relationships. Built upon our progression modeling analysis, we also developed novel strategies to detect cancer driver genes and pathways directly based on their definitions derived from cancer evolution theory and visualize identified changes in a cancer development roadmap. A computational approach that can overcome the existing sampling limitations and thereby enables the leveraging of accumulating data represents a major advance with respect to the application of bioinformatics methodology to the study of progressive human diseases.

Mon, Nov 6

Algebra Seminar
Leonid Rybnikov, MIT/Montreal
The Gaudin model in the Deligne category Rep\(GL_t\) 
4:00PM, Mathematics Building Room 250

Deligne's category \(D_t\) is a formal way to define the category of finite-dimensional representations of the group \(GL_n\)with \(n=t\) being a formal parameter (which can be specialized to any complex number). I will show how to interpolate the construction of the higher Hamiltonians of the Gaudin magnet chain associated with the Lie algebra\(\mathfrak{gl}_n\) to any complex \(n\), using \(D_t\). Next, according to Feigin and Frenkel, Bethe ansatz equations in the Gaudin model are equivalent to no-monodromy conditions on a certain space of differential operators of order \(n\) on the projective line. We also obtain interpolations of these no-monodromy conditions to any complex \(n\) and prove that the relations in the algebra of higher Gaudin Hamiltonians for complex \(n\) are generated by our interpolations of the no-monodromy conditions. I will explain the relation of this to the Bethe ansatz for the Gaudin model associated with the Lie superalgebra\(\mathfrak{gl}_{m|n}\). This is joint work with Boris Feigin and Filipp Uvarov, https://nam12.safelinks.protection.outlook.com/?url=https%3A%2F%2Farxiv.org%2Fabs%2F2304.04501&data=05%7C01%7Cmahacker%40buffalo.edu%7C5ae4961e3a474d4ecf8c08dbd577137c%7C96464a8af8ed40b199e25f6b50a20250%7C0%7C0%7C638338479620977552%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=KME1sbxoHoM4C8QUJZS3xVq4Ka3BrqZPM088%2Be3UXfU%3D&reserved=0.

Fri, Nov 10

Applied Math Seminar
Dmitry Pelinovsky, Mcmaster University
Instability of peaked waves in hydrodynamical models.

3:00PM, Math 122 and on Zoom - contact mbichuch@buffalo.edu for link

Stokes' wave is a traveling (periodic or solitary) wave with a stagnation point at its crest, where the surface of fluids has a peaked singularity. Stokes waves have been considered within Euler's equations and have been modeled by using reduced equations of motion such as the Camassa-Holm equation, the rotation-modified Ostrovsky equation, and other systems with wave breaking. I will overview the recent analysis of nonlinear, linear, and spectral instability of the traveling peaked waves in some reduced models such as the b-family of the Camassa-Holm equations. Well-posedness of the initial-value problem for this family holds in the energy space as long as the first spatial derivative (the wave slope) is bounded. We show that the travelling peaked waves are unstable due to wave breaking when the wave slope becomes unbounded in a finite time. This instability can also be studied by using the spectral stability analysis within the linearized equations of motion obtained consistently with the well-posedness results.

Mon, Nov 13

Algebra Seminar
Changlong Zhong, Albany
Elliptic cohomology and the Fourier-Mukai transform 
4:00PM, Mathematics Building Room 250

Recently, there has been rapid development in using equivariant elliptic cohomology in the context of enumerative geometry, representation theory, and mathematical physics. In particular, the elliptic stable envelope for symplectic varieties is defined, and a 3d mirror symmetry statement is given by Okounkov. For cotangent bundles of flag varieties the stable envelope is constructed by Rimanyi-Weber using algebraic geometry, and they are transformed by the newly defined elliptic Demazure-Lusztig (DL)operators. These operators can be thought of as rational sections of a certain vector bundle over \(A\times A^\vee\), where \(A\) is (the spectrum of) thee quivariant elliptic cohomology of a point and \(A^\vee\) is its dual abelian variety. Classically, there is an equivalence between the derived categories of\(A\) and \(A^\vee\), defined by the Poincare line bundle, called the Fourier Mukaitrans form. The module category over the elliptic affine Hecke algebra, defined by Ginzburg-Kapranov-Vasserot, is a subcategory. In this talk I will define an equivalence between this module category and that for the Langlands dual system. This functor is constructed by using an algebra over \(A\times A^\vee\) determined by the elliptic DL operators. This is joint work with G. Zhao.

Fri, Nov 17

Applied Math Seminar
Giselle Sosa Jones, Oakland University
Discontinuous Galerkin discretizations of multiphase flow problems in porous media.

3:00PM, Math 122 and on Zoom - contact mbichuch@buffalo.edu for link

Modeling the flow of liquid, aqueous, and vapor phases through porous media is a complex and challenging task that requires solving nonlinear coupled partial differential equations. In this talk, we propose a second-order accurate and energy-stable time discretization method for the three-phase flow problem in porous media. We prove the convergence of the subiterations to resolve the nonlinearity, and show that the time-stepping method mimics the energy balance relation that the continuous problem satisfies. Our spatial discretization uses an interior penalty discontinuous Galerkin method, for which we establish the well-posedness of the discrete problem and provide error estimates under certain conditions on the data. We validate our method through numerical simulations, which show that our approach achieves the expected theoretical convergence rates. Furthermore, the numerical examples highlight the advantages of our time discretization over other time discretizations.

Mon, Nov 27

Algebra Seminar
Pablo Boixeda Alvarez, Yale
The center of the small quantum group and affineSpringer fibers Abstract : The quantum group \(U_q\) is a Hopf algebraintroduced by Lusztig deforming the enveloping algebra. The representationtheory of this algebra is particularly interesting at \(l^{th}\) roots of unity,where it includes a finite-dimensional subalgebra known as the small quantumgroup.  In joint work with Bezrukavnikov,Shan and Vasserot we construct an injective map to the center of this algebrafrom the cohomology of a certain affine Springer fiber \(\cF{l}_{ts}\) for aregular semisimple element \(s\). In recent progress we check that this map issurjective in type A and get a bound on dimension in general types related tothe diagonal coinvariant algebra. We also give an algebro-geometric descriptionof the spectrum of the cohomology of the Springer fiber. The work relies onunderstanding the representation category through a filtration coming fromintersection with \(G[[t]]\)-orbits in \(\cF{l}_{ts}\) . In this talk I willpresent the result and related properties of this filtration of the category.
4:00PM, Mathematics Building room 250

Speaker:  PabloBoixeda Alvarez, Yale

Title: The center of the small quantum group and affineSpringer fibers

Abstract : The quantum group \(U_q\) is a Hopf algebraintroduced by Lusztig deforming the enveloping algebra. The representationtheory of this algebra is particularly interesting at \(l^{th}\) roots of unity,where it includes a finite-dimensional subalgebra known as the small quantumgroup.  In joint work with Bezrukavnikov,Shan and Vasserot we construct an injective map to the center of this algebrafrom the cohomology of a certain affine Springer fiber \(\cF{l}_{ts}\) for aregular semisimple element \(s\). In recent progress we check that this map issurjective in type A and get a bound on dimension in general types related tothe diagonal coinvariant algebra. We also give an algebro-geometric descriptionof the spectrum of the cohomology of the Springer fiber. The work relies onunderstanding the representation category through a filtration coming fromintersection with \(G[[t]]\)-orbits in \(\cF{l}_{ts}\) . In this talk I willpresent the result and related properties of this filtration of the category.

Mon, Dec 4

Algebra Seminar
Hunter Dinkins, Northeastern
q-Hypergeometric Functions and the Geometry of Quiver Varieties 
4:00PM, Mathematics Building room 250

Vertex functions are special functions associated to Nakajima quiver varieties. They generalize basic hypergeometric functions, which are q-deformations of classical hypergeometric functions, whose study began with Gauss. In recent years, conjectures originating in physics known as "3d mirror symmetry" have uncovered new properties of these functions. Our main result relates the vertex function of a type Aquiver variety with that of the cotangent bundle of a complete flag variety. Asa consequence, we are able to prove 3d mirror symmetry of vertex functions for a certain class of type A quiver varieties. Time permitting, we will explain ongoing work to extend these results to bow varieties. No prior knowledge of these topics will be assumed.

Mon, Dec 11

Algebra Seminar
Yan Soibelman, Kansas State
Perturbative expansions in Chern-Simons theory from the point of view of Holomorphic Floer Theory 
4:00PM, Zoom (please email achirvas@buffalo.edu)

About 10 years ago together with Maxim Kontsevich we started a project which we called "Holomorphic FloerTheory" (HFT for short).  In HFT one deals with complex symplectic manifolds instead of real ones.  Then, generically, the Floer differential vanishes. As a result, one has to develop a new theory and ask new questions. I am going to explain some of them in the simplest example of the Morse theory of a holomorphic Morse function.

HFT has many applications e.g. the unifying point of view on the Riemann-Hilbert correspondence.  I am going to explain an application to exponential integrals in finite and infinite dimensions.  In the case of Chern-Simons theory this leads to several interesting conjectures, including the one about the existence of a mixed Hodge structure of infinite rank associated to a complexified Chern-Simons functional.

Tue, Feb 22

Applied Math Seminar
Applied Math Seminar: Alexander Korotkevich (UNM)
Numerical Verification of the 6-Wave 1D Kinetic Equation.Speaker: Alexander Korotkevich (University of New Mexico, Department of Math&Stat)
4:00PM, Zoom

We consider wave kinetic equation (WKE) for quintic nonlinear Schroedinger equation (qNLSE),corresponding to 6-waves 3-to-3 interaction. WKE was derived for periodic boundary conditions. Wepropose conditions of applicability of WKE for description of qNLSE. These conditions were confirmedduring comparison of simulation of dynamical equation (qNLSE) and WKE. We observed convergence ofsolution of qNLSE to solution of WKE with increase of the period of the system. If we stay inproposed range of applicability, good correspondence between WKE and qNLSE is observed.Simulations for different boundary conditions (Dirichlet and Neumann) were performed. Wedemonstrate, that conditions of applicability derived for periodic boundary conditions are notdirectly applicable and have to be corrected. It should be noted, that in laboratory wave tankexperiments boundary conditions are usually different from periodic ones. At the same time,researchers expect to observe good correspondence with WKE derived for infinite domain.6-waves qNLSE appears in some of the applications, e.g. if one considers one dimensional opticalfiber communication line and would like to consider turbulence of the waves propagating in it. Nextorder correction on intensity, with respect to the classical NLSE, would result in qNLSE, aftercanonical transformation eliminating 4-waves nonlinear term. Also 6-waves WKE appears in descriptionof waves on Kelvin vortices in superfluid Helium.

Fri, Feb 25

Special Event
Makoto Ozawa (Komazawa University) via Zoom only Friday

4:00PM

Geometry Topology Seminar


VIA ZOOM

Handlebody decompositions of 3-manifolds and polycontinuous patterns

Tue, Mar 1

Applied Math Seminar
Applied Math Seminar
Denis Silantyev (UCCS)
Generalized Constantin-Lax-Majda Equation: Collapse vs. Blow Up and Global ExistenceSpeaker: Denis Silantyev (UC Colorado Springs, Department of Mathematics)
4:00PM, Zoom

We investigate the behavior of the generalized Constantin-Lax-Majda (CLM) equation which is a 1D model for the advection and stretching of vorticity in a 3D incompressible Euler fluid. Similar to Euler equations the vortex stretching term is quadratic in vorticity, and therefore is destabilizing and has the potential to generate singular behavior, while the advection term does not cause any growth of vorticity and provides a stabilizing effect. We study the influence of a parameter a which controls the strength of advection, distinguishing a finite time singularity formation (collapse or blow-up) vs. global existence of solutions. We find a new critical value ac=0.6890665337007457... below which there is finite time singularity formation that has a form of self-similar collapse, with the spatial extent of blow-up shrinking to zero, and above which up to a=1 we have an expanding blow up solutions. We identify the leading order complex singularity for general values of a which controls the leading order behavior of the collapsing solution. We also rederive a known exact collapsing solution for a=0 and we find a new exact analytical collapsing solution at a=1/2. For ac<a≤1, we find a blow-up solution on the real line in which the spatial extent of the blow-up region expands infinitely fast at the singularity time. For a>1, we find that the solution exists globally with exponential-like growth of the solution amplitude in time.

Tue, Mar 8

Applied Math Seminar
Dr. Kai Yang, Florida International University
Numerical methods for the KdV-type equations
4:00PM, Zoom: for link see email announcement or contact sergeyd at buffalo dot edu

We review several numerical approaches for KdV-type equations, including the generalized KdV and Benjamin-Ono equations as well as the KdV equation with fractional Laplacian. The spatial discretization is achieved by using the Fourier spectral method for fast decay solutions (e.g., in KdV equation), or the spectral method from the Wiener rational basis functions for both fast and slow decay solution cases. Both of these two spatial discretizations preserve the Hamiltonian in the spatial discrete sense. We also discuss the arbitrarily high order Hamiltonian conservative schemes that are constructed by applying the Scalar Auxiliary Variable (SAV) reformulation with the symplectic Runge-Kutta method in the time evolution.

Thu, Mar 10

Colloquium
Cary Malkiewich, Binghamton University
Brave new fixed-point theory
4:00PM, Zoom: for link see email announcement or contact badzioch at buffalo dot edu

The development of algebraic topology in the 20th century could be characterized as a gradual passage from combinatorial, numerical invariants such as the Betti numbers to group-valued invariants such as homology, and then from homological algebra to spectral or "brave new" algebra in the latter half of the century. In brave new algebra, integers are replaced by maps of spheres. What is amazing is that just about every concept in algebra can be transformed to accommodate this new paradigm.
The effects of this development continue to ripple through topology and nearby subjects, including the very classical topic of Nielsen fixed-point theory. I'll explain how work of Dold in the 1970s and of Ponto in the 2000s and 2010s infused the "brave new" perspective into fixed point theory. The result is that, amazingly, we can now work with the Lefschetz number and its generalizations without ever triangulating or mentioning homology groups, and the definitions now generalize easily to parametrized families of fixed-point problems.

Mon, Mar 14

Algebra Seminar
Benjamin Passe, United States Naval Academy
Boundary representations and isolated points
4:00PM, Zoom. Contact achirvas AT buffalo DOT edu for link.

Operator systems provide a way to examine convexity (and a generalization called matrix convexity) in operator algebraic terms. Extreme points in particular match up with a special class of irreducible representations, called boundary representations, defined by Arveson. While a single operator system \(S\) may have many different concrete forms, the boundary representations from each form nonetheless generate the same \(C^*\)-envelope of \(S\). Similar analogues of exposed points and isolated extreme points exist within the boundary representations, though these subsets are not as commonly used. In joint work with Ken Davidson, we determined exactly when a smallest concrete presentation of a separable operator system exists, as well as how it can be identified up to unitary equivalence using a restricted class of boundary representations.

Tue, Mar 15

Applied Math Seminar
Pavel Lushnikov, University of New Mexico
Conformal mappings and integrability of surface dynamics
4:00PM, Zoom: for link contact sergeyd@buffalo.edu

A Hamiltonian formulation of the time dependent potential flow of ideal incompressible fluid with a free surface is considered in two-dimensional geometry. It is well known that the dynamics of small to moderate amplitudes of surface perturbations can be reformulated in terms of the canonical Hamiltonian structure for the surface elevation and Dirichlet boundary condition of the velocity potential. Arbitrary large perturbations can be efficiently characterized through a time-dependent conformal mapping of a fluid domain into the lower complex half-plane. We reformulate the exact Eulerian dynamics through a non-canonical nonlocal Hamiltonian system for the pair of new conformal variables. The corresponding non-canonical Poisson bracket is non-degenerate, i.e., it does not have any Casimir invariant. Any two functionals of the conformal mapping commute with respect to Poisson bracket. We also consider a generalized hydrodynamics for two components of superfluid Helium which has the same non-canonical Hamiltonian structure. In both cases the fluid dynamics is fully characterized by the complex singularities in the upper complex half-plane of the conformal map and the complex velocity. Analytical continuation through the branch cuts generically results in the Riemann surface with infinite number of sheets. An infinite family of solutions with moving poles are found on the Riemann surface.Residues of poles are the constants of motion. There constants commute with each other in the sense of underlying non-canonical Hamiltonian dynamics which provides an argument in support of the conjecture of complete Hamiltonian integrability of surface dynamics.

Fri, Mar 18

Geometry and Topology Seminar
Subhankar Dey, University of Alabama
Detection results in link Floer homology

4:00PM, 122 Mathematics Building

 In this talk I will briefly describe link Floer homology toolbox and its usefulness. Then I will show how link Floer homology can detect links with small ranks, using a rank bound for fibered links by generalizing an existing result for knots, and some very useful spectral sequences. I will also show that stronger detection results can be obtained in a sense that the knot Floer homology can be shown to detect \(T(2,8)\) and \(T(2,10)\), and that link Floer homology detects \((2,2n)\)-cables of trefoil and the figure eight knot. This talk is based on joint work with Fraser Binns.

Thu, Mar 31

Colloquium
Colloquium: Michael Brannan (University of Waterloo)Via Zoom
4:00PM

Fri, Apr 1

Geometry and Topology Seminar
Hong Chang, University at Buffalo
 Efficient geodesics in the curve complex and their dot graphs

4:00PM, 122 Mathematics Building

 The notion of {\em efficient geodesics} in \(\mathcal{C}(S_{g>1})\), the complex of curves of a closed orientable surface of genus \(g\),  was first introduced in "Efficient geodesics and an effective algorithm for distance in the complex of curves".  There it was established that there exists (finitely many) efficient geodesics between any two vertices, \( v_{\alpha} , v_{\beta} \in \mathcal{C}(S_g)\), representing homotopy classes of simple closed curves, \(\alpha , \beta \subset S_g\). The main tool for used in establishing the existence of efficient geodesic was a {\em dot graph}, a booking scheme for recording the intersection pattern of a {\em reference arc}, \(\gamma \subset S_g\), with the simple closed curves associated with the vertices of geodesic path in the zero skeleton, \(\mathcal{C}^0(S_g)\).  In particular, it was shown that any curve corresponding to the vertex that is distance one from \(v_\alpha\) in an efficient geodesic intersects any \(\gamma\) at most \(d -2\) times, when the distance between \(v_\alpha\) and \(v_\beta\) is \(d \geq 3\).  In this note we make a more expansive study of the characterizing ``shape'' of the dot graphs over the entire set of vertices in an efficient geodesic edge-path.  The key take away of this study is that the shape of a dot graph for any efficient geodesic is contained within a {\em spindle shape} region.

Thu, Apr 7

Special Event
Colloquium: Gino Biondini, University at Buffalo
Two adventures in integrable systems: thenonlinear Schrodinger equation with non-trivial boundary conditions 
4:00PM, Room 250 Math Building, North Campus

A significant advance in mathematical physics in thesecond half of the twentieth century was the development of the theory ofmodern integrable systems. These systems are nonlinear evolution equations ofphysical significance that provide the nonlinear counterpart to the classicalPDEs of mathematical physics.

One such equation, and in some respects the mostimportant one, is the nonlinear Schrödinger (NLS) equation. The NLS equation isa universal model for weakly nonlinear dispersive wave packets, and arises in avariety of physical settings, including deep water, optics, acoustics, plasmas,condensed matter, etc. In addition, the NLS equation is a completelyintegrable, infinite-dimensional Hamiltonian system, and as a result itpossesses a remarkably deep and beautiful mathematical structure. At the rootof many of these properties is the existence of Lax pair, namely the fact thatthe NLS equation can be written as the compatibility condition of anoverdetermined pair of linear ODEs. The first half of the Lax pair for the NLSequation is the Zakharov-Shabat scattering problem, which is equivalent to aneigenvalue problem for a one-dimensional Dirac operator.

Even though the NLS equation has been extensively studiedthroughout the last sixty years, it continues to reveal new phenomena and offermany surprises. In particular, the focusing NLS equation with nontrivialboundary conditions has received renewed attention in recent years. This talkis devoted to presenting two recent results in this regard. Specifically, Iwill discuss: (i) A characterization of the universal nonlinear stage ofmodulational instability, achieved by studying the long-time asymptotics ofsolutions of the NLS equation with non-zero background; (ii) A characterizationof a two-parameter family of elliptic finite-band potentials of thenon-self-adjoint ZS operator, which are associated with purely real spectrum ofHill’s equation (i.e., the time-independent Schrodinger equation with periodiccoefficients) with a suitable complex potential.

Fri, Apr 8

Geometry and Topology Seminar
Sahana Hassan Balasubramanya, University of Münster
Actions of solvable groups on hyperbolic spaces
4:00PM, Zoom

(joint work with A.Rasmussen and C.Abbott) Recent papers of Balasubramanya and Abbott-Rasmussen have classified the hyperbolic actions of several families of classically studied solvable groups. A key tool for these investigations is the machinery of confining subsets of Caprace-Cornulier-Monod-Tessera. This machinery applies in particular to solvable groups with virtually cyclic abelianizations.

Mon, Apr 11

Algebra Seminar
Xiuping Su, University of Bath
Kac's Theorem for a class of string algebras of affine type \(\mathbf {C}\).

4:00PM, Contact achirvas@buffalo.edu for zoom link

Quiver representation theory and Lie theory are closely related via Gabriel's Theorem, Kac's Theorem and Dlab-Ringel's Theorem.  When a quiver (i.e. an oriented simply laced graph) \(Q\) is of finite representation type, Gabriel proved that a quiver is of finite representation type if and only if \(Q\) is Dykin and the map sending a representation \(X\) to its dimension vector provides a one-to-one correspondence between the isomorphism classes of indecomposable representations of \(Q\) and the positive roots of \(Q\) . The latter was generalized by Kac to any quiver. For non-simply laced graphs of finite or affine types, Dlab-Ringel generalized Gabriel's Theorem, using modulated (or valued) quivers.
Recently, Geiss-Leclerc-Schroer introduced a class of Iwanaga-Gorenstein algebras \(H\) via quivers \(Q\) with relations associated with symmetrizable Cartan matrices and studied \(\tau\)-locally free\(H\)-modules. Among other things, they proved that when the Cartan matrix is of finite type, there is a one-to-one correspondence between the dimension vectors of indecomposable \(\tau\)-locally free \(H\)-modules and the positive roots of the associated Lie algebra and conjectured that Kac's Theorem holds for any \(H\).
In this talk, I will describe the Auslander-Reiten quivers of some string algebras of affine type \(\mathbf{C}\), which are Iwanaga-Gorenstein algebras \(H\) associated to Cartan matrices of affine type \(\mathbf{C}\) and confirm GLS-conjecture for this case.

Tue, Apr 12

Applied Math Seminar
Dmitry Zakharov, Central Michigan U
Lump chains in the KP-I equation


4:00PM, Zoom - contact sergeydy@buffalo.edu for link

The Kadomstev--Petviashvili equation is one of the fundamental equations in the theory of integrable systems. The KP equation comes in two physically distinct forms: KP-I and KP-II. The KP-I equation has a large family of rational solutions known as lumps. A single lump is a spatially localized soliton, and lumps can scatter on one another or form bound states. The KP-II equation does not have any spatially localized solutions, but has a rich family of line soliton solutions that form evolving polygonal patterns.
I will discuss two new families of solutions of the KP-I equation, obtained using the Grammian form of the tau-function. The first is the family of lump chain solutions. A single lump chain consists of a linear arrangement of lumps, similar to a line soliton of KP-II. More generally, lump chains can form evolving polygonal arrangements whose structure closely resembles that of the line soliton solutions of KP-II. I will also show how lump chains and line solitons may absorb, emit, and reabsorb individual lumps.
Joint work with Andrey Gelash, Charles Lester, Yury Stepanyants, and Vladimir Zakharov.

Thu, Apr 14

Colloquium
Peter Thomas (Case Western U)
Phase and phase-amplitude reduction for stochastic oscillators
4:00PM, 250 Math Bldg, also accessible via Zoom - contact badzioch@buffalo for link

Phase reduction is a powerful and widely used tool for studying synchronization,
entrainment, and parametric sensitivity of limit cycle oscillators. The
classical phase reduction framework goes back at least 50 years for
deterministic ODE systems. In both natural and engineered systems, however,
stochastic dynamics are ubiquitous. For stochastic systems, the appropriate
analog of phase reduction remains a matter of debate. In 2013 Schwabedal and
Pikovsky introduced a notion of phase reduction for stochastic oscillators based
on a first-passage-time analysis. In 2014 Thomas and Lindner introduced an alternative
asymptotic phase for Markovian stochastic oscillators based on a spectral
decomposition of the Koopman operator (a.k.a. the generator of the Markov process,
or the adjoint Kolmogovor operator). I will report on recent advances in
understanding and expanding these ideas, including (i) reformulation of the
first-passage-time (FPT) phase in terms of the solution of a partial
differential equation with nonstandard boundary conditions, (ii) quantitative
comparison of the FPT and spectral phase for planar stochastic systems, and
(iii) extension of the spectral phase to a novel "phase-amplitude" reduction for
stochastic oscillators.
This is joint work with Benjamin Lindner (Humboldt University, Dept. of Physics
and Bernstein Center for Computational Neuroscience) and Alberto Perez-Cervera
(Complutense University of Madrid, Dept. of Applied Mathematics).

Fri, Apr 15

Geometry and Topology Seminar
Daxun Wang, University at Buffalo
Boundary action of CAT(0) groups and their \(C^\ast\)-algebras.

4:00PM, 122 Mathematics Building

(joint with Xin Ma) Boundaries of certain CAT(0) spaces and group actions on them play important roles in geometric group theory. In this talk, we will talk about boundary actions of CAT(0) spaces from a point of view of topological dynamics and \(C^\ast\)-algebras. In particular, we will describe the actions of right angled Coxeter groups and right angled Artin groups on certain boundaries. This provides some pure infiniteness results for reduced crossed product \(C^\ast\)-algebra of these actions. Next, we will talk about the action of fundamental groups of graph of groups on the visual boundaries of their Bass-Serre trees. This provides a new method in identifying \(C^\ast\)-simple generalized Baumslag-Solitar groups.

Tue, Apr 19

Applied Math Seminar
Bernard Deconinck, U of Washington
The water wave pressure problem

4:00PM, Zoom - contact sergeyd@buffalo.edu for link

I will discuss a new method to recover the water-wave surface elevation from pressure data obtained at the bottom of a fluid. The new method requires he numerical solution of a nonlocal nonlinear equation relating the pressure and the surface elevation which is obtained from the Euler formulation of the water-wave problem without approximation. From this new equation, a variety of different asymptotic formulas are derived. The nonlocal equation and the asymptotic formulas are compared with both numerical data and physical experiments, demonstrating excellent agreement, significantly beyond what is obtained using Archimedes' p=rho g h. 

Fri, Apr 22

Geometry and Topology Seminar
Matt Durham, UC Riverside/Cornell University
Local quasicubicality and sublinear Morse geodesics in mapping class groups and Teichmuller space

4:00PM, 122 Mathematics Building

Random walks on spaces with hyperbolic properties tend to sublinearly track geodesic rays which point in certain hyperbolic-like directions. Qing-Rafi-Tiozzo recently introduced the sublinear-Morse boundary to more broadly capture these generic directions.In joint work with Abdul Zalloum, we develop the geometric foundations of sublinear-Morseness in the mapping class group and Teichmuller space. We prove that their sublinearly-Morse boundaries are visibility spaces and admit continuous equivariant injections into the boundary of the curve graph. Moreover, we completely characterize sublinear-Morseness in terms of the hierarchical structure on these spaces.Our techniques include developing tools for modeling sublinearly-Morse rays via CAT(0) cube complexes. Part of this analysis involves establishing a direct connection between the geometry of the curve graph and the combinatorics of hyperplanes in these cubical models.

Mon, Apr 25

Algebra Seminar
Daniel Sage, LSU
The Deligne–Simpson problem for connections on \(\mathbb{G}_m\) with a maximally ramified singularity 
4:00PM, Mathematics Building Room 250

A natural question in the theory of systems of meromorphic differential equations (or equivalently, meromorphic connections) on \(\mathbb{P}^1\) is whether there exists a global connection with specified local  behavior at a collection of singular points.  The Deligne-Simpson problem is concerned with a variant of this question.  The classical additive Deligne–Simpson problem is the existence problem for Fuchsian connections whose residues at the singular points are contained in specified adjoint orbits.  Crawley-Boevey solved this problem in 2003 by reinterpreting it in terms of quiver varieties. A more general version of the problem, solved by Hiroe in 2017, allows additional unramified irregular singularities. We apply the theory of fundamental and regular strata due to Bremer and Sage to formulate a version of the Deligne–Simpson problem in which certain ramified singularities are allowed. These ramified singular points are called toral singularities; they are singularities whose leading term with respect to a Moy–Prasad filtration is regular semisimple. We solve this problem in a special case that plays an important role in recent work on the geometric Langlands program: connections on \(\mathbb{G}_m\) with a maximally ramified singularity at 0 and possibly an additional regular singular point at infinity. We also give a complete characterization of all such connections that are rigid, under the additional hypothesis of unipotent monodromy at infinity.

Tue, Apr 26

Applied Math Seminar
Svetlana Roudenko, Florida International University
The gKdV world thru the NLS lens
4:00PM, Zoom, contact sergeyd@buffalo.edu for link

In this talk we discuss the family of generalized KdV equations borrowing tools and approaches from the NLS equation. We address the wellposedness (for any power of nonlinearity), show formation and behavior of solitons (and thus, soliton resolution) as well as solutions behavior in the \(L^2\)-critical and supercritical settings including the finite time blow-up and the description of its dynamics.

Thu, Apr 28

Colloquium
Juanita Pinzón Caicedo, University of Notre Dame
Four-manifolds and knot concordance

4:00PM, 250 Math Bldg. Also via Zoom - contact badzioch@buffalo.edu for link.

 The study of 4-dimensional objects is special: a manifold can admit infinitely many non-equivalent smooth structures,  and manifolds can be homeomorphic but not diffeomorphic. This difference between topological and smooth structures,  can be addressed in terms of the study of knots as boundaries of surfaces embedded in 4D space. In this talk I will focus  on some knot operators known as satellites and will show that satellites can bound very different surfaces in the smooth  and topological category.
This talk is organized jointly by the Math Department and the UB chapter of AWM. 

Fri, Apr 29

Geometry and Topology Seminar
Juanita Pinzon Caicedo, University of Notre Dame
Satellite Operations that are not homomorphisms.

4:00PM, 122 Mathematics Building

Two knots \(K_0\) and \(K_1\) are said to be smoothly concordant if the connected sum \(K_0\#m({K_1}^r)\) bounds a disk smoothly embedded in the 4-ball. Smooth concordance is an equivalence relation, and the set \(\mathcal{C}\) of smooth concordance classes of knots is an abelian group with connected sum as the binary operation. Satellite operations, or the process of tying a given knot P along another knot K to produce a third knot P(K), are powerful tools for studying the algebraic structure of the concordance group. In this talk I will describe conditions on the pattern P that suffice to conclude that the function \(P:\mathcal{C}\to \mathcal{C}\) is not a homomorphism. This is joint work with Tye Lidman and Allison Miller.

Mon, May 2

Special Event
Nicolle González, UCLA
A skein theoretic \(A_{q,t}\) algebra 
4:00PM, Mathematics Building room 250

The \(A_{q,t}\) algebra first arose in connection to the celebrated proof of the shuffle theorem given by Carlsson and Mellit. This algebra is given as an extension of two copies of the affine Hecke algebra by certain raising and lowering operators. Its polynomial representation, which played a critical role in the proof given by Carlsson and Mellit, was later realized geometrically by Carlsson-Mellit and Gorsky in the context of parabolic flag Hilbert schemes. In this talk I will present a skein theoretic formulation of this representation given by certain skein-Heisenberg diagrams on a punctured annulus. This formulation recovers the original algebraic description of Carlsson and Mellit, but given the simplicity of the diagrams allows many computations to be more straightforward and intuitive. More interestingly, this diagrammatic presentation is primed for a direct categorification via the category of Soergel bimodules. This is joint work with Matt Hogancamp.

Fri, May 6

Geometry and Topology Seminar
Ciprian Manolescu, Stanford University
A knot Floer stable homotopy type
Knot Floer homology (introduced by Ozsváth–Szabó and Rasmussen) is an invariant whose definition is based on symplectic geometry, and whose applications have transformed knot theory over the last two decades. Starting from a grid diagram of a knot, I will explain how to construct a spectrum whose homology is knot Floer homology. Conjecturally, the homotopy type of the spectrum is an invariant of the knot. The construction does not use symplectic geometry, but rather builds on a combinatorial definition of knot Floer homology. We inductively define models for the moduli spaces of pseudo-holomorphic strips and disk bubbles, and patch them together into a framed flow category. The inductive step relies on the vanishing of an obstruction class that takes values in a complex of positive domains with partitions. (This is joint work with Sucharit Sarkar.)
4:00PM, Zoom

Title: A knot Floer stable homotopy type
Knot Floer homology (introduced by Ozsváth–Szabó and Rasmussen) is an invariant whose definition is based on symplectic geometry, and whose applications have transformed knot theory over the last two decades. Starting from a grid diagram of a knot, I will explain how to construct a spectrum whose homology is knot Floer homology. Conjecturally, the homotopy type of the spectrum is an invariant of the knot. The construction does not use symplectic geometry, but rather builds on a combinatorial definition of knot Floer homology. We inductively define models for the moduli spaces of pseudo-holomorphic strips and disk bubbles, and patch them together into a framed flow category. The inductive step relies on the vanishing of an obstruction class that takes values in a complex of positive domains with partitions. (This is joint work with Sucharit Sarkar.)

Mon, May 9

Algebra Seminar
Jie Ren, UB
Quivers and 2-Calabi-Yau categories
 
4:00PM

The framework of Calabi-Yau categories is appropriate forthe theory of motivic Donaldson-Thomas invariants. I will give an introductionto the Calabi-Yau categories associated to quivers, and the analyticity oftheir stability structures.

Mon, May 9

Special Event
Jie Ren, UB
Quivers and 2-Calabi-Yau categories
 
4:00PM

The framework of Calabi-Yau categories is appropriate forthe theory of motivic Donaldson-Thomas invariants. I will give an introductionto the Calabi-Yau categories associated to quivers, and the analyticity oftheir stability structures.

Tue, May 10

Applied Math Seminar
Panayotis Kevrekidis, U Mass
Some Vignettes of Nonlinear Waves in Granular Crystals: From Modeling and Analysis to Computations and Experiments
4:00PM, Zoom - contact sergeyd@buffalo.edu for link

In this talk, we will provide an overview of some results in the setting of granular crystals, consisting of beads interacting through Hertzian contacts. In 1d we show that there exist three prototypical types of coherent nonlinear waveforms: shock waves, traveling solitary waves and discrete breathers. The latter are time-periodic, spatially localized structures. For each one, we will discuss the existence theory, presenting connections to prototypical models of nonlinear wave theory, such as the Burgers equation, the Korteweg-de Vries equation and the nonlinear Schrodinger (NLS) equation, respectively. We will also explore the stability of such structures, presenting some explicit stability criteria analogous to the famous Vakhitov-Kolokolov criterion in the NLS model. Finally, for each one of these structures, we will complement the mathematical theory and numerical computations with recent experiments, allowing their quantitative identification and visualization. Finally, time permitting, ongoing extensions of these themes will be briefly touched upon, most notably in higher dimensions, in heterogeneous or disordered chains and in the presence of damping and driving; associated open questions will also be outlined.

Thu, May 26

Special Event
Tara Hudson
12:30PM

Tue, Aug 16

Special Event
103 Desk needs to be moved
12:30PM

Wed, Aug 31

Analysis Seminar
Yi Wang, Chongqing University
 Helton-Howe trace, Connes-Chern character and quantization

8:00PM, On Zoom - contact hfli@math.buffalo.edu for link

We study the Helton-Howe trace and the Connes-Chern character for Toeplitz operators on weighted Bergman spaces via the idea of quantization. We prove a local formula for the large t-limit of the Connes-Chern character as the weight goes to infinity. And we show that the Helton-Howe trace of Toeplitz operators is independent of the weight and obtain a local formula for the Helton-Howe trace for all weighted Bergman spaces. The proofs are based on an integration by parts formula and some harmonic analysis. This talk is based on joint work with Xiang Tang and Dechao Zheng. 

Tue, Sep 6

Applied Math Seminar
Maxim Bichuch, SUNY Buffalo
Introduction to Decentralized Finance

4:00PM, Math 250 and on Zoom - contact mbichuch@buffalo.edu for link

Decentralized Finance (DeFi) is an addition/alternative to the traditional financial system to handle financial transactions. The key difference between the traditional financial system and DeFi is the lack of central authority that establishes trust and regulates the financial system. In this talk we will explain the main concept of DeFi and consider three examples: Decentralized Payment Clearing using Blockchain and Optimal Bidding; Pricing by Staking; Axiomatic Automated Market Making.

Wed, Sep 7

Analysis Seminar
Mariusz Tobolski, University of Wroclaw
The Stone-von Neumann theorem for locally compact quantum groups
4:00PM, On Zoom - contact hfli@mah.buffalo.edu for link

The Stone-von Neumann theorem is a mathematical result that rigorously proves the equivalence between the two fundamental approaches to quantum mechanics, i.e. the matrix mechanics of Heisenberg and the wave mechanics of Schrodinger. It was then formulated by Mackey as a theorem about certain unitary representations of locally compact abelian groups. In my talk, based on yet another formulation due to Rieffel, I will present a Stone-von Neumann-type theorem in the setting of locally compact quantum groups introduced by Kustermans and Vaes and independently by Woronowicz.

Fri, Sep 9

Geometry and Topology Seminar
Bill Menasco (UB)
Surface Embeddings in \(\mathbb{R}^2 \times \mathbb{R}\)
4:00PM, 122 Mathematics Building

In this joint work with Margaret Nichols, we consider \(\mathbb{R}^3\) as having the product structure \(\mathbb{R}^2 \times \mathbb{R}\) and let \(\pi : \mathbb{R}^2 \times \mathbb{R} \rightarrow \mathbb{R}^2\) be the natural projection map onto the Euclidean plane. Let \( \epsilon : S_g \rightarrow \mathbb{R}^2 \times \mathbb{R}\) be a smooth embedding of a closed oriented genus \(g\) surface such that the set of critical points for the map \(\pi \circ \epsilon\) is a piece-wise smooth (possibly multi-component) \(1\)-manifold, \(\mathcal{C} \subset S_g\). We say \(\mathcal{C}\) is the {\em crease set of \(\epsilon\)} and two embeddings are in the same {\em isotopy class} if there exists an isotopy between them that has \(\mathcal{C}\) being an invariant set. The case where \(\pi \circ \epsilon |_\mathcal{C}\) restricts to an immersion is readily accessible, since the turning number function of a smooth curve in \(\mathbb{R}^2\) supplies us with a natural map of components of \(\mathcal{C}\) into \(\mathbb{Z}\). The Gauss-Bonnet Theorem beautifully governs the behavior of \(\pi \circ \epsilon (\mathcal{C})\), as it implies \(\chi(S_g) = 2 \sum_{\gamma \in \mathcal{C}} t( \pi \circ \epsilon (\gamma))\), where \(t\) is the turning number function. Focusing on when \(S_g \cong S^2\), we give a necessary and sufficient condition for when a disjoint collection of curves \(\mathcal{C} \subset S^2\) can be realized as the crease set of an embedding \(\epsilon: S^2 \rightarrow \mathbb{R}^2 \times \mathbb{R}\).

Mon, Sep 12

Algebra Seminar
Shichen Tang
Arithmetic stability of higher rank Artin-Schreier-Witt towers
4:00PM, 250 Mathematics Building

Let \(q=p^a\), \(K\) the rational function field over the field of \(q\) elements, and \(G\) be the absolute Galois group of \(K\).For any continuous p-adic representation of \(G\), one can construct a tower of finite Galois extensions of \(K\). A conjecture of Daqing Wan states that if this representation "comes from algebraic geometry", then the slopes of the zeta functions of the fields in this tower have a stable behavior. In general, Wan's conjecture is wide open and already very hard when this tower is an Artin-Schreier-Witt tower. In this talk, we will discuss some recent progress related to Wan's conjecture for higher rank Artin-Schreier-Witt towers.

Wed, Sep 14

Analysis Seminar
Hanfeng Li, SUNY at Buffalo
Entropy and asymptotic pairs
4:00PM, 250 Math Building and on Zoom - contact hfli@buffalo.edu for Zoom link

Positive entropy and the existence of nontrivial asymptotic pairs are both kind of chaotic properties in topological dynamics. I will discuss the relation between these two properties for algebraic actions of amenable groups, and how this is related to the strong Atiyah conjecture in L2-invariants theory. This is joint work with Sebastian Barbieri and Felipe Garcia-Ramos.

Fri, Sep 16

Geometry and Topology Seminar
Yulan Qing (Fudan University/ University of Toronto)
Gromov boundary extended


4:00PM, 122 Mathematics Building

Gromov boundary provides a useful compactification for all infinite-diameter Gromov hyperbolic spaces. It consists of all geodesic rays starting at a given base-point and it has been an essential tool in the study of the coarse geometry of hyperbolic groups. In this study we introduce two topological spaces that are natural analogs of the Gromov boundary for a larger class of metric spaces. First we construct the sublinearly Morse boundaries and show that it is a QI-invariant topological space that can be associated to all finitely generated groups. Furthermore, for many groups, the sublinear boundary can be identified with the Poisson boundaries of the associated group, thus providing a QI-invariant model for Poisson boundaries. This result answers the open problems regarding QI-invariant models of CAT(0) groups and the mapping class group. Lastly,  for a subset of the metric spaces we define a compactification of the sublinearly Morse boundary and show that in these cases they are naturally identified with the Bowditch boundary. This is a series of joint work with Kasra Rafi and Giulio Tiozzo.

Mon, Sep 19

Algebra Seminar
Mariusz Tobolski, University of Wrocław
4:00PM, 250 Mathematics Building

Tue, Sep 20

Applied Math Seminar
Yangwen Zhang, CMU
A new reduced order model of linear parabolic PDEs.

4:00PM, Math 250 and on Zoom - contact mbichuch@buffalo.edu for link

How to build an accurate reduced order model (ROM) for multidimensional time dependent partial differ- ential equations (PDEs) is quite open. In this paper, we propose a new ROM for linear parabolic PDEs. We prove that our new method can be orders of magnitude faster than standard solvers, and is also much less memory intensive. Under some assumptions on the problem data, we prove that the convergence rates of the new method is the same with standard solvers. Numerical experiments are presented to confirm our theoretical result.

Tue, Sep 20

Special Event
250 Sexual Harassment Prevention Training
5:15PM, 250 Mathematics Building

Thu, Sep 22

Colloquium
Abdul Zalloum (University of Toronto)
 
4:00PM, 250 Mathematics Building

The field of geometric group theory investigates theconnections between the algebraic structure of a group and the geometriesof the metric spaces on which that group acts. This modern approach to grouptheory has revolutionized the study of finitely generated groups andproduced deep applications in logic, topology, geometry, and dynamical systems.

Group actions on hyperbolic spaces tend to be particularlyinformative of the algebraic structure of the acting group.One very broad class of groups that can be studied using actions onhyperbolic spaces is the class of hierarchically hyperbolicgroups which includes free groups, surface groups, mapping class groups,fundamental groups of 3–manifolds without Nil or Sol components, and manyothers.

I will show how the introduction, formalism and study of such a class haveled to the resolution of many long-standing open problems in the fieldincluding Farb's quasi-flats conjecture and the semi-hyperbolicity conjectureof the mapping class group. Finally, I will discuss some of my recentwork with Sisto where we show that the geometric model introduced byHaettel, Hoda and Petyt is the first geometric model for mapping classgroups where pseudo-Anosov elements have stronglycontracting axes confirming Thurston's conjecture on genericity of theseelements in such a geometric model.

Fri, Sep 23

Geometry and Topology Seminar
Abdul Zalloum (University of Toronto)
Hyperbolic models for CAT(0) spaces

4:00PM, 122 Mathematics Building

Two of the most well-studied topics in geometric group theory are CAT(0) cube complexes and mapping class groups. This is in part because they both admit powerful combinatorial-like structures that encode their (coarse) geometry: hyperplanes for the former and curve graphs for the latter. In recent years, analogies between the two theories have become more apparent. For instance: there are counterparts of curve graphs for CAT(0) cube complexes and rigidity theorems for such counterparts that mirror the surface setting, and both can be studied using the machinery of hierarchical hyperbolicity. However, the considerably larger class of CAT(0) spaces is left out of this analogy, as the lack of a combinatorial-like structure presents a difficulty in importing techniques from those areas. In this talk, I will speak about recent work with Petyt and Spriano where we bring CAT(0) spaces into the picture by developing analogues of hyperplanes and curve graphs for them. The talk will be accessible to everyone, and all the aforementioned terms will be defined.

Mon, Oct 3

Algebra Seminar
Li Li, Oakland University
Cluster algebras and Nakajima's graded quivervarieties 
4:00PM, 250 Mathematics Building

Nakajima's graded quiver varieties are complex algebraicvarieties associated with quivers. They are introduced by Nakajima in the studyof representations of universal enveloping algebras of Kac-Moody Lie algebras,and can be used to study cluster algebras. In the talk, I will explain how toprecisely locate the supports of the triangular basis of skew-symmetric rank-2quantum cluster algebras by applying the decomposition theorem to variousmorphisms related to quiver varieties, thus prove a conjecture proposed byLee-Li-Rupel-Zelevinsky in 2014.

Wed, Oct 5

Special Event
2022 Myhill lecture Series: Gigliola Staffilani October 5-7
The study of wave interactions: where beautiful mathematical ideas come together.


4:00PM, 250 Mathematics Building,

  Phenomena involving interactions of waves happen at different scales and in different media: from gravitational waves to the waves on the surface of the ocean, from our milk and coffee in the morning to infinitesimal particles that behave like wave packets in quantum physics. These phenomena are difficult to study in a rigorous mathematical manner, but maybe because of this challenge mathematicians have developed interdisciplinary approaches that are powerful and beautiful. I will describe some of these approaches and show for example how the need to understand certain multilinear and periodic interactions gave also the tools to prove a famous conjecture in number theory or how classical tools in probability gave the right framework to still have viable theories behind certain deterministic counterexamples.

Tue, Oct 11

Applied Math Seminar
Zechuan Zhang, SUNY Buffalo
Soliton resolution and asymptotic stability of N-soliton solutions for the defocusing mKdV equation with finite density type initial data

4:00PM, Math 250 and on Zoom - contact mbichuch@buffalo.edu for link

We consider the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation with finite density type initial data. According to the \(\bar\partial\) steepest descent method, we introduce a series of transformation to the original Riemann-Hilbert problem \(m(z)\) to extrapolate the leading order approximation to the solution of mKdV for large time in the solitonic space-time region \(|x/t + 4| < 2\), and we give bounds for the error which decay as \(t\to\infty\). Our results also provide a verification of the soliton resolution conjecture and asymptotic stability of N-soliton solutions for mKdV equation with finite density type initial data.

Wed, Oct 12

Analysis Seminar
Yuqing (Frank) Lin, Texas A&M University
Entropy for actions of free groups under bounded orbit equivalence

4:00PM, 250 math Building and on Zoom - contact hfli@math.buffalo.edu for link

Joint work with Lewis Bowen. The f-invariant is a notion of entropy for probability measure preserving (pmp) actions of free groups. It is invariant under measure conjugacy and is an extension of Kolmogorov-Sinai entropy for actions of the integers. Two pmp actions are orbit equivalent if their orbits can be matched almost everywhere in a measurable fashion. Although entropy is not invariant under orbit equivalence in general, work of Austin and Kerr-Li has shown in various settings that entropy is invariant under certain stronger notions of quantitative orbit equivalence. We add to these results by showing that the f-invariant is invariant under the assumption of bounded orbit equivalence. 

Fri, Oct 14

Geometry and Topology Seminar
Bojun Zhao (UB)
Left orderability and taut foliations with one-sided branching
4:00PM, 122 Mathematics Building

For a closed orientable irreducible 3-manifold \(M\) that admits a co-orientable taut foliation with one-sided branching, we show that \(\pi_1(M)\) is left orderable.

Mon, Oct 17

Algebra Seminar
Mihai Fulger, U of Connecticut
Positivity vs. semi-stability for bundles with vanishing discriminant
4:00PM, Zoom - contact achirvas@buffalo.edu for link

Positivity properties of vector bundles like ampleness or nefness are rich research topics, mostly studied in the rank 1 case where they have important applications to birational geometry. Semi-stability is another important property due to its application to the construction of moduli spaces. On curves there are connections between the two first observed in work of Hartshorne. In higher dimension a connection exists under the additional assumption that the discriminant of the bundle vanishes. We give algebraic proofs of this valid in arbitrary characteristic. This is in joint work with Adrian Langer.

Tue, Oct 18

Colloquium
Jie Shen, Purdue University
Efficient positivity/bound preserving schemes for complex nonlinear systems
4:00PM, Math Bldg Room 250

Solutions of a large class of partial differential equations (PDEs) arising from sciences and engineering
applications are required to be positive or within a specified bound, and also energy dissipative.
It is of critical importance that their numerical approximations preserve these structures at the discrete level, as violation of these structures may render the discrete problems ill posed or inaccurate.
I will review the existing approaches for constructing positivity/bound preserving schemes, and then present
several efficient and accurate approaches:
(i) through reformulation as Wasserstein gradient flows;
(ii) through a suitable functional transform
(iii) through a Lagrange multiplier.
These approaches have different advantages and limitations, are all relatively easy to implement and can be combined with most spatial discretizations.

Fri, Oct 21

Geometry and Topology Seminar
José Román Aranda Cuevas (Binghamton University)

4:00PM, 122 Mathematics Building

Take two 3-dimensional handlebodies with the same boundary surface. One can tell them apart by studying the curves on the boundary surface bounding disks on each handlebody. Hempel studied Heegaard splittings of closed 3-manifolds by comparing these disk sets in the curve complex. For trisections of 4-manifolds, one can measure the length of loops in some complex passing through the disk set of each 3-dimensional handlebody. Kirby and Thompson used cut systems this way to define the L-invariant of a trisection of a closed 4-manifold. Other authors extended this definition for relative trisections and bridge trisections. Naturally, L is hard to compute. We will discuss lower bounds for (b,c)-bridge trisections of closed surfaces. This is joint work with Taylor, Pongtanapaisan, and Zhang.

Mon, Oct 24

Algebra Seminar
Doyon Kim, Rutgers University
The existence and uniqueness of Whittaker functionals for \(GL(n,R)\): an algebraic-geometric proof
4:00PM, Zoom; please email achirvas@buffalo.edu for meeting info

The "multiplicity one theorem," proved by Piatetski-Shapiro and Shalika, asserts that the space of Whittaker functionals on unitary irreducible representations of \(GL(n,R)\) is at most one-dimensional. In this talk, we discuss a new, algebraic-geometric proof that the space of Whittaker functionals on principal series representations of \(GL(n,R)\) is exactly one-dimensional. Additionally, we discuss its possible applications on Jacquet integrals.

Tue, Oct 25

Applied Math Seminar
Naoki Masuda, SUNY Buffalo
Core-periphery structure in networks.

4:00PM, Math 250 and on Zoom - contact mbichuch@buffalo.edu for link

Core-periphery structure is a mesoscale structure of networks, with which nodes in the core set are densely interconnected, peripheral nodes are connected to core nodes to different extents, and peripheral nodes are sparsely interconnected. I will introduce new scalable and principled algorithms to find core-periphery structure in networks. Core-periphery structure composed of a single core and a single periphery has been observed for various networks. In contrast, our algorithm aims to find multiple non-overlapping groups of core-periphery structure in a network. We also argue that, relative to a standard random graph model, core-periphery structure in the given network is mathematically possible only when we allow at least three blocks of nodes, thereby excluding conventional core-periphery structure composed of two blocks, i.e., a core block and a peripheral block. We illustrate our algorithms with empirical networks with applications.

Mon, Oct 31

Special Event
Jesse Huang, University of Alberta
Some attempts to build NCCRs for higher dimensional toric Gorenstein rings 
4:00PM, Zoom; please email achirvas@buffalo.edu for meeting info

A noncommutative crepant resolution (NCCR) is a nice endomorphism algebra of a sum of modules that ``resolves'' a normal Gorenstein ring. In the toric context, mirror symmetry suggests that questions surrounding the existence of NCCRs and derived equivalences among them could have geometric answers. In this talk, I will discuss some speculations on a geometric method to construct NCCRs as a quiver algebra for certain toric Calabi-Yau singularities ,potentially generalizing results of Mozgovoy and Bocklandt in dimension 3.

Tue, Nov 1

Applied Math Seminar
Scott Rich, Krembil Brain Institute
Resilience through diversity: Reduced heterogeneity in human epilepsy destabilizes neuronal circuits and promotes seizure-like transitions.

4:00PM, Math 250 and on Zoom - contact mbichuch@buffalo.edu for link

 
Epilepsy is the most common serious neurological disorder in the world, characterized by sudden transitions between sparse and asynchronous neuronal activity into hyper-synchronous and hyper-active oscillatory dynamics that envelop the entire brain. This phenomenon echoes effects of bifurcations in dynamical systems, supporting a robust subfield of computational neuroscience focused on modeling seizure dynamics. This pursuit is made more challenging considering the multitude of etiologies that can yield recurrent seizure, and in turn be classified under the broad banner of epilepsy. However, many of these etiologies can be understood as reduced variability in the properties of neuronal circuits, yielding the hypothesis that epileptogenesis can be recontextualized as a progressive loss of biophysical heterogeneity.
In this talk, I will present interdisciplinary results from my recently published work supporting this novel perspective on epilepsy. Neurons from human cortical tissue exhibit decreased heterogeneity when taken from brain regions that initiate seizure. When implemented in computational neuronal circuits, these reduced heterogeneities yield sudden transitions into synchronous oscillatory dynamics that are absent with physiological levels of heterogeneity. Perhaps most interestingly, this phenomenon is explained at a fundamental level by the differing stability structures of these systems with low and high heterogeneities. Alongside further research in preparation, these results showcase a fundamental role for biophysical heterogeneity in the brain in imparting resilience to pathological dynamica such as seizure.

Wed, Nov 9

Analysis Seminar
Hongming Nie, SUNY at Stony Brook
A metric on hyperbolic components

4:00PM, 250 Math Building and on Zoom - contact hfli@math.buffalo.edu for Zoom link

In this talk, under a mild condition, I will introduce a metric on hyperbolic components of rational maps. This metric is constructed by considering the measure-theoretic entropy with respect to some equilibrium state. Moreover, this metric is conformal equivalent to the pressure from the thermodynamics. It is a joint work with Y.M. He.

Tue, Nov 15

Applied Math Seminar
Anita Layton, Waterloo
TBA

4:00PM, Zoom - contact mbichuch@buffalo.edu for link

TBA

Wed, Nov 16

Analysis Seminar
Sagun Chanillo, Rutgers University
Local Version of Courant's Nodal Domain Theorem
4:00PM, On Zoom - conact hfli@math.buffalo.edu for link

Let \((M^n, g)\) denote a smooth, compact Riemannian manifold with no boundary. A fundamental object on this manifold is the Laplace-Beltrami operator which has a discrete spectrum. If we arrange the eigenvalues of the Laplacian in increasing order (for the negative of the Laplacian) with multiplicity, Courant's theorem states, that the number of nodal components for the k-th eigenfunction is at most k. A nodal component of an eigenfunction u is a connected component of the set where u does not vanish. In this talk we study a local version of the global result of Courant. The local question was raised by Donnelley and C.Fefferman in the late1980s. Our theorems are joint work with A. Logunov, E. Mallinikova and D. Mangoubi.

Thu, Nov 17

Colloquium
Colloquium Hossein Shahmohamad
Graphs & Their potent Energy Drinks 
4:00PM

Speaker: Hossein Shahmohamad, RIT

Title: Graphs & Their potent Energy Drinks

Fri, Nov 18

Geometry and Topology Seminar
Yvon Verberne (University of Toronto)
Postponed to Spring 2023 due to Storm
Automorphisms of the fine curve graph

4:00PM, 122 Mathematics Building

The fine curve graph of a surface was introduced by Bowden, Hensel and Webb. It is defined as the simplicial complex where vertices are essential simple closed curves in the surface and the edges are pairs of disjoint curves. We show that the group of automorphisms of the fine curve graph is isomorphic to the group of homeomorphisms of the surface, which shows that the fine curve graph is a combinatorial tool for studying the group of homeomorphisms of a surface. This work is joint with Adele Long, Dan Margalit, Anna Pham, and Claudia Yao.

Wed, Nov 30

Analysis Seminar
Joseph Hundley, SUNY at Buffalo
Functorial Descent in the Exceptional Groups
4:00PM, Zoom - contact hfli@math.buffalo.edu for link

In this talk, I will discuss the method of functorial descent. This method, which was discovered by Ginzburg, Rallis and Soudry, uses Fourier coefficients of residues of Eisenstein series to attack the problem of characterizing the image of Langlands functorial liftings. We'll discuss the general structure, the results of Ginzburg, Rallis and Soudry in the classical groups, some recent attempts to extend the same ideas to the exceptional groups, and challenges and new phenomena which merge in these attempts. The new work discussed will mainly be joint with Baiying Liu. Time permitting I may also comment on unpublished work of Ginzburg and joint work with Ginzburg.

Tue, Dec 6

Applied Math Seminar
Weiqi Chu, UCLA
Non-Markovian opinion models inspired by random processes on networks

4:00PM, Math 250 and on Zoom - contact mbichuch@buffalo.edu for link

The study of opinion dynamics models opinion evolution as dynamical processes on social networks. For social networks, nodes encode social entities (such as people and twitter accounts), while edges encode relationship or events between entities. Traditional models of opinion dynamics consider how opinions evolve either on time-independent networks or on temporal networks with edges that follow Poisson statistics. However, in many real-life networks, interactions between individuals (and hence the edges in a network) follow non-Poisson processes, which leads to dynamics on networks with memory-dependent effects (such as stereotypes). In this talk, we model social interactions as random processes on temporal networks and derive the opinion model that is governed by an arbitrary waiting-time distribution (WTD). When random processes have non-Poisson interevent statistics, the corresponding opinion models yield non-Markovian dynamics naturally. We analyze the convergence to consensus of these models and illustrate a variety of induced opinion models from common WTDs (including Dirac delta, exponential, and heavy-tailed distributions). When the opinion model does not have an explicit form (such as models induced by heavy-tailed WTDs), we provide a discrete-time approximation method and derive an associate set of discrete-time opinion-dynamics models.

Sat, Dec 17

Special Event
150 KIm Javor
10:00AM

Mon, Dec 19

Special Event
Liviu Paunescu, Simion Stoilow Institute of Mathematics

4:00PM, : Zoom; please email achirvas@buffalo.edu for meeting info

Two permutations that almost commute are close to two commuting permutations. The same question can be asked for other relations, not only the commutant. Moreover, the answer to this question depends only on the group that the equations describe. We then survey some recent results where this question is answered affirmatively or negatively, depending on the group, and study the connections to the theory of sofic groups.