Wed, Feb 11

Analysis Seminar

Didier Lesesvre, Université de Lille

A connection between zeros and central values of \(L\)-functions

4:00 PM, Mathematics Building

\(L\) -functions appear as generating functions encapsulating information about various objects such as Galois representations, elliptic curves, arithmetic functions, modular forms, Mass forms, etc. Studying  \(L\) -functions is therefore of utmost importance in number theory at large. Two of their attached data carry critical information: their zeros, which govern the distributional behavior of underlying objects; and their central values, which are related to invariants such as the class number of a field extension. We will discuss the important conjectures, one concerning the distribution of the zeros and one concerning the distribution of the central values, and explain a general principle that any restricted result towards the first conjecture can be refined to show that most corresponding central value shave the typical distribution predicted by the second conjecture. We will instantiate this general principle for a wide class of  \(L\)- functions, and provide a more detailed discussion in the case of  \(L\) -functions attached to modular forms.

Fri, Feb 13

Applied Mathematics Seminar

Di Qi (Purdue University)

Reduced-order data assimilation models for predicting probability distributions of multiscale turbulent systems

4:00 PM, MATH 250

A new strategy is presented for the statistical forecasts of multiscale nonlinear systems involving non-Gaussian probability distributions. The capability of using reduced-order models to capture key statistical features is investigated. A closed stochastic-statistical modeling framework is proposed using a high-order statistical closure enabling accurate prediction of leading-order statistical moments and probability density functions in multiscale complex turbulent systems. A new efficient ensemble forecast algorithm is developed dealing with the nonlinear multiscale coupling mechanism as a characteristic feature in high-dimensional turbulent systems. To address challenges associated with closely coupled spatio-temporal scales in turbulent states and expensive large ensemble simulation for high-dimensional complex systems, we introduce efficient computational strategies using the random batch method. Effective nonlinear ensemble filters are developed based on the nonlinear coupling structures of the explicit stochastic and statistical equations, which satisfy an infinite-dimensional Kalman-Bucy filter with conditional Gaussian dynamics. It is demonstrated that crucial principal statistical quantities in the most important large scales can be captured efficiently with accuracy using the new reduced-order model in various dynamical regimes of the flow field with distinct statistical structures.

Mon, Mar 9

Algebra Seminar

Michaela Vancliff, University of Texas at Arlington

Generalizing classical Clifford algebras, graded Clifford algebras and their associated geometry

4:00 PM, 250 Mathematics Building

Graded Clifford algebras are non-commutative graded algebras related to classical Clifford algebras, and certain properties of such an algebra can be deduced from certain commutative geometric data associated to it. In particular, a standard result is that a graded Clifford algebra \(C\) is quadratic and Artin-Schelter regular with Hilbert series equal to that of a polynomial ring if and only if a certain quadric system associated to \(C\) is base-point free. About two decades ago, T. Cassidy and the speaker introduced a generalization of such an algebra, called a graded skew Clifford algebra, and they found that many results concerning graded Clifford algebra shave analogues in the case of graded skew Clifford algebras, provided the appropriate non-commutative geometric data is defined. More recently, T.Cassidy and the speaker defined a "skew" version of classical Clifford algebras, and related such algebras to graded skew Clifford algebras. Indeed,just as (classical) Clifford algebras are the Poincaré-Birkhoff-Witt (PBW) deformations of exterior algebras, skew Clifford algebras may be viewed as \(\mathbb{Z}_2\)-graded PBW deformations of quantum exterior algebras.

Fri, Apr 10

Applied Mathematics Seminar

Yulong Lu (U Minnesota)

Title: TBD

4:00 PM, Room: TBD

Fri, Apr 10

Topology and Geometry Seminar

Roberta Shapiro (University of Michigan)

TBA

4:00 PM, 122 Mathematics Building

TBA