Introduction to Research Seminar

4/15/20 - Speaker: Tsao-Hsien Chen

Title: On the Beilinson-Bernstein localization theorem

Abstract: I will give an introduction to the celebrated Beilinson-Bernstein localization theorem relating representations of Lie algebras (such as sl₂) and algebraic geometry.

2/26/20 - Overview of the PDE and Dynamical Systems Groups

Description: Harini and Laurel will be giving an overview of what it is like to work in the PDE and dynamical systems research groups! Come ask questions and learn about the research that goes on in these groups!

2/19/20 - Gilad Lerman

Title: On the problem of group synchronization and on a solution via message passing

Abstract: The problem of group synchronization requires estimating the states of objects, which are represented by group elements, from the relative state measurements between pairs of objects. One example is rotation synchronization (or rotation averaging), which in practice aims to recover rotations of cameras (with respect to a fixed frame) from the relative rotations between pairs of cameras. In this case, the group is SO(3). The general problem can be difficult as the given measurements are often erroneous or missing. Also, direct optimization formulations take place on a non-convex product of groups with non-trivial energy landscapes. We focus on a new general framework of Yunpeng Shi and the speaker for solving a certain mathematical setting of the problem. It applies a novel message passing procedure that uses the "cycle consistency" of the underlying group in order to estimate the corruption levels of group ratios (i.e., relative measurements). We first present an explanation for the physicist (as message passing has gained popularity among statistical physicists). That is, we motivate the ideas of the framework by assuming a probabilistic model. We then establish exact recovery and linear convergence guarantees for the proposed message passing procedure under a deterministic setting with adversarial corruption (while neglecting the former probabilistic model). These guarantees hold as long as the ratio of corrupted graph cycles per edge is bounded by a reasonable constant. We further show that under a uniform corruption model, the recovery results are sharp in terms of an information-theoretic bound. The talk aims to convey general ideas and experiences of the speaker, in particular, discussion of related, but different, problems. It may evolve according to the interests of the audience and will encourage discussion.

2/12/20 - Ben Brubaker

Title: Special functions in automorphic forms

Abstract: I'll begin with a very down-to-earth description of how questions in number theory might be solved by methods in automorphic forms (and I'll define automorphic forms in the process). Then we'll discuss how certain special functions arise in the description of automorphic forms, and how to compute them using various constructions from combinatorial representation theory. I won't assume any prior acquaintance with number theory, automorphic forms, or much combinatorial representation theory.

1/29/20 - Max Engelstein

Title: How to figure out what shape is best?

Abstract: In calculus, our students learn how to maximize and minimize functions of several variables. There are theorems about when extremal points exist and how to detect them using derivatives. It turns out that a lot of the same results are true with functions of infinitely many variables, and functions of functions and even functions of sets. This study of these minimization problems is called the calculus of variations and, over the last few centuries, has become a central field in math, with connections to topology, physics and machine learning (amongst other areas).

12/11/19 - Mitch Luskin

Title: Twistronics: manipulating the spectrum of Hamiltonian operators for two-dimensional layered structures through their twist angle

12/04/19 - Christine Berkesch

Title: Virtual resolutions: homological algebra wielded at toric varieties

Abstract: I will discuss - through examples - a current research project that aims to use homological algebra to better understand toric varieties in algebraic geometry. The project seeks to generalize a classical theory for projective space that dates back to work of Hilbert in the late 1800's.

11/20/19 - Sarah Brauner

Title: What people mean when they say Type A: an introduction to reflection groups

Abstract: If you’ve ever gone to the Combinatorics Seminar (or the Student Algebra and Combinatorics Seminar, or the Representation Theory Seminar, or the Student Number Theory Seminar, and probably other seminars too..) you’ve likely heard someone mention Type A and “other types.” In this talk, I will explain what these cryptic comments mean. Along the way, I will present a broad research program that views the symmetric group as one example in a wide-ranging class of groups called reflection groups; the aim of such a program is to generalize the many nice properties of the symmetric group to all reflection groups.

11/13/19 - "Ask a Post-Doc Panel" featuring Patricia Klein, Chris Fraser, Nadejda Drenska, Mykhailo Kuian

10/30/19 - Weiyan Chen

Title: Obstructions to choosing points on cubic curves

Abstract: I will tell you a new theorem about some classical objects (such as inflection points in calculus) dating back to Maclaurin and Cayley.

10/23/19 - Paul Carter

Title: The geometry of localized roll patterns

Abstract: In the study of PDEs, techniques from dynamical systems can be powerful tools in analyzing the structure and behavior of solutions. I will talk about these ideas in the context of localized patterns, which appear ubiquitously in nature, e.g. in chemical reactions, vegetation patterns, and buckling of metal shells. I will focus on a particular example: the Swift-Hohenberg PDE, an equation frequently used in the study of pattern formation. This equation admits localized pattern solutions which display unusual "snaking" behavior as parameters in the equation are varied. By viewing this PDE as a dynamical system in an appropriate sense, the underlying geometry provides insight into the structure of these patterns and the nature of snaking.

10/06/19- Jasmine Foo

Title: Evolutionary Models of Cancer

Abstract: In this talk I will present a few examples of questions in cancer biology than can be addressed using mathematical modeling. These questions will cover various aspects of cancer evolution (e.g. initiation, treatment/personalized medicine and drug resistance), and we will discuss some probabilistic models (e.g. branching processes, interacting particle systems) to describe these processes.

9/25/19- Peter Webb

Title: The use of categories in representation theory

Abstract: When regarded as algebraic structures in their own right, categories include groups, posets and quivers, for example and, because they are so general, their appearance is widespread throughout mathematics. In my own work I have made use of categories constructed from groups that enable us to compute group cohomology, that allow reformulations of a major conjecture in group representation theory, and that permit the definition of so-called Mackey functors that encode most of the structure in group representation theory. I will describe a selection of constructions, show their significance, and indicate some of my work in this area.

9/11/19 - Richard McGehee

Title: Saving the Planet with MATLAB

Abstract: Overwhelming scientific evidence has accumulated to the point where there is no doubt that the Earth’s climate is changing at a rate faster than any event that has been seen in the geologic record. The predictions about the future climate are made with huge computer programs running on the largest supercomputers. I will explore some very simple mathematical models than can be used not only to understand the climate processes but also to make predictions that can be compared with those of the large computer models. The models are so simple that they do not challenge the capabilities of a laptop running MATLAB.