Association for Women in Mathematics

As a mathematician at the National Security Agency, I get to do a lot of interesting math. I work in an office called cryptography in systems which develops new cryptography. Although I can't share any of the details of my work, I'm instead going to give an overview of public key cryptography and what it looks like in the past, present and future. I will also talk a little about NSA, what it's like to work there and some job opportunities for mathematicians.

A group of scientists make a new discovery. We can probably all agree that the world should hear about their work, and we can probably also agree that the scientists deserve some recognition. But how does that science make it from the lab bench to the Twitter feed? And what is the likelihood that the information you eventually consume will be intact, interesting, and accurate? As a mathematician with a background in radio, blogging, social media, and podcasting, I will attempt to answer these questions, and more. I will discuss the evolution of a science story from beginning to end, and make a case for quantitative literacy in journalism. And like much click-bait, this talk might not actually blow your mind…but then again, maybe it will. There’s really only one way to find out.

Graphical models have proven to be a valuable tool for connecting genotypes and phenotypes. Structural learning of phenotype-genotype networks has received considerable attention in the post-genome era. In recent years, a dozen different methods have emerged for network inference, which leverage natural variation that arises in certain genetic populations. The structure of the network itself can be used to form hypotheses based on the inferred direct and indirect network relationships, but represents a premature endpoint to the graphical analyses. In this work, we extend this endpoint. We examine the unexplored problem of perturbing a given network structure, and quantifying the system-wide effects on the network in a node-wise manner. We leverage belief propagation methods in Conditional Gaussian Bayesian Networks (CG-BNs), in order to absorb and propagate phenotypic evidence through the network. We show that the modeling assumptions adopted for genotype-phenotype networks represent an important sub-class of CGBNs, which possess properties that ensure exact inference in the propagation scheme. Applications to kidney and skin cancer expression Quantitative Trait Loci (eQTL) data from different musculus populations are presented. We demonstrate how these predicted system-wide effects can be examined in connection with estimated class probabilities for covariates of interest, e.g., cancer status. Despite the uncertainty in the network structure, we demonstrate the system-wide predictions are stable across an ensemble of highly likely networks. A software package, geneNetBP, which implements our approach, was developed in the R programming language.

When working with data, it is imperative to have a definition of a distance in order to rigorously define concepts such as noise and approximations. The interleaving distance was first defined in the context of generalizing the bottleneck distance for persistence diagrams, a common tool from topological data analysis (TDA) by Chazal et al. It was then shown to fit easily into the categorified framework of persistence modules provided by Bubenik and Scott. Category theory allows the idea to be fluidly moved and redefined to work on many other constructions. In particular, some standard metrics such as L∞ and Hausdorff distance, can also be viewed as special cases of the interleaving distance. In this talk, we will discuss the basic ideas for the interleaving, its generalized definition in the category theory language, and recent work extending this idea to give interleavings on Reeb graphs and Merge trees. This work is joint with Vin de Silva, Amit Patel, and Anastasios Stefanou.

In the past decade, the field of neuroscience has benefited from myriad experimental advances, allowing researchers to explore the brain at across multiple spatial and temporal scales with ever increasing resolution and sensitivity. Along with this wave of new information has come a need to develop novel methods and techniques for visualizing, quantifying, and comparing data across scales and modalities. Network theory – the art of mapping physical systems to mathematical graphs – provides an attractive methodology to study the brain, but also requires the development of new techniques to (i) identify network nodes and connections from large data sets and (ii) measure and interpret subtle differences in topological features of brain networks. In this talk, I’ll discuss my journey from math to physics to neuroscience (and back to math!) research and how quantitative researchers are helping to drive new areas of brain science. Finally, I will highlight some of the different types of data sets that are studied in network neuroscience and describe specific projects my group is working on to extract and measure network topology from diverse modalities ranging from single neuron imaging to whole brain fMRI datasets.