All Seminars
| Title: Probabilistic Bezout Over Finite Fields, and Some Applications |
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| Seminar: Math Colloquia |
| Speaker: Bhargav Narayanan of Rutgers University |
| Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
| Date: 2022-02-23 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: What is the distribution of the number of distinct roots of k random polynomials (of some fixed degree) in k variables? I will talk about a recently proved Bezout-like theorem that gives us a satisfactory answer over (large) finite fields. This result can be used to construct several interesting families of “extremal graphs”. I shall illustrate this method by 1) discussing the easiest applications in detail, reproving some well-known lower bounds in extremal graph theory, and 2) outlining how this method has recently found applications in establishing hardness results for a few basic computational problems. |
| Title: Harmonic Analysis on Cosphere Bundle |
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| Seminar: Analysis Reading |
| Speaker: Guangqiu Liang of Emory University |
| Contact: Yiran Wang, yiran.wang@EMORY.EDU |
| Date: 2022-02-18 at 3:00PM |
| Venue: MSC E406 |
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| Abstract: |
| Title: Structured Low-Rank Approximation and the Proxy Point Method |
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| Seminar: Computational Math |
| Speaker: Mikhail Lepilov of Purdue University |
| Contact: Lars Ruthotto, lruthotto@emory.edu |
| Date: 2022-02-17 at 1:00PM |
| Venue: Online |
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| Abstract: Structured algorithms for large, dense matrices require efficient low-rank approximation methods to obtain their computational cost savings. There are many ways of obtaining such approximations depending on the type of matrix involved. For kernel matrices, analytic approximation methods such as truncated Taylor expansions or the proxy point method have been used in the Fast Multipole Method and other structured matrix algorithms. In this talk, we focus on the proxy point method, in which pairwise interactions between two separated clusters are approximated using the interactions of each cluster with a smaller chosen set of "proxy" points that separate the clusters. We perform a new accuracy analysis of this method when applied to 1D analytic kernels, and we then use it to devise a sublinear-time algorithm for constructing the HSS approximation of certain Cauchy and Toeplitz matrices. Finally, we extend this method its analysis to analytic kernels in several complex variables. |
| Title: Euler’s Polyhedron Formula and The Euler Characteristic |
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| Seminar: Mathematics |
| Speaker: Daniel Hess of University of Chicago |
| Contact: Bree Ettinger, betting@emory.edu |
| Date: 2022-02-04 at 10:00AM |
| Venue: MSC W201 and Zoom |
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| Abstract: A standard soccer ball is constructed using 12 regular pentagons and 20 regular hexagons. Is it possible to build one using only pentagons? How about only hexagons? It turns out that one of these is possible and one is not! The key to answering these questions is Euler’s Polyhedron Formula, which expresses a certain relationship between the number of vertices, edges, and faces in any convex polyhedron. In this talk, we will discuss this formula, the more general Euler characteristic, and applications such as the classification of the Platonic solids and triangulations of surfaces. |
| Title: Hamiltonian system and spectral inverse problems |
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| Seminar: Analysis Reading Seminar |
| Speaker: Guangqiu Liang of Emory University |
| Contact: Yiran Wang, yiran.wang@emory.edu |
| Date: 2022-02-04 at 3:00PM |
| Venue: MSC E406 |
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| Abstract: |
| Title: Geometric Group Theory and Untangling Earphones |
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| Seminar: Mathematics |
| Speaker: Neha Gupta of Georgia Institute of Technology |
| Contact: Bree Ettinger, betting@emory.edu |
| Date: 2022-02-02 at 10:00AM |
| Venue: MSC W201 and Zoom |
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| Abstract: Suppose you get your earphones entangled around a doughnut with two holes... an entirely probable scenario, right?! Then how "big" does your doughnut need to be, for you to successfully untangle your earphones? This is going to be a gentle introduction to groups, and how they connect to geometry. We will slowly build up to answering our original question. No prior experience with groups, geometry, or topology is required or assumed. This is joint work with Ilya Kapovich. |
| Title: Riemannian Geometry and Biomedical Data |
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| Seminar: Mathematics |
| Speaker: Sima Ahsani of Emory University |
| Contact: Bree Ettinger, betting@emory.edu |
| Date: 2022-01-28 at 10:00AM |
| Venue: MSC W201 and Zoom |
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| Abstract: Statistics is a science for everyone who wishes to collect, analyze, interpret, and understand data. Over the last few years, due to rapid technological developments, we handle increasingly large and complex data. For example, understanding biomedical data, that lie on matrix manifolds, in order to early detection of disease to prevent, control, or provide improved health care with low costs. Therefore, one of the most important steps in analyzing this type of data is understanding the structure of the surfaces where data live on them. To this end, differential geometry allows us to develop local methods to understand the global properties of surfaces that data lie on them. In this talk, after giving some examples of datasets that lie on curved spaces, I will provide an intuitive definition of Riemannian manifolds and their basic properties. Then, I will describe matrix manifolds and explain how to measure distances between two points and challenges that may arise when we want to calculate the mean of datasets and mention techniques that can be used to tackle these challenges. In the end, some resources and content will be provided to engage undergraduate students to know more about this research area to find the right way to develop their ideas and interest and take steps to build their future research areas. |
| Title: Game Theory, AI, and Optimal Transportation |
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| Seminar: Computational Math |
| Speaker: Levon Nurbekyan of University of California, Los Angeles |
| Contact: Lars Ruthotto, lruthotto@emory.edu |
| Date: 2022-01-24 at 10:00AM |
| Venue: MSC W201 |
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| Abstract: The modern world constitutes a network of complex socioeconomic interactions leading to increasingly challenging decision-making problems for participating agents such as households, governments, firms, and autonomous systems. We consequently need refined mathematical models and solution techniques for addressing these difficulties. In this talk, I will demonstrate how to apply mathematical game theory, optimal control, and statistical physics to model large systems of interacting agents and discuss novel dimension reduction and machine learning techniques for their solution. An intriguing aspect of this research is that the mathematics of interacting agent systems provides a foundation for fast and robust core machine learning algorithms and their analysis. For example, I will demonstrate how to solve the regularity problem in normalizing flows based on their "crowd-motion" or optimal transportation interpretation. Yet another essential utility of optimal transportation in data science is that it provides a metric in the space of probability measures. I will briefly discuss the application of this metric for robust solution methods in inverse problems appearing in physical applications. I will conclude by discussing future research towards socioeconomic applications, data science, and intelligent autonomous systems. |
| Title: Zarankiewicz problem, VC-dimension, and incidence geometry |
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| Seminar: Discrete Mathematics |
| Speaker: Cosmin Pohoata of Yale University |
| Contact: Dwight Duffus, dwightduffus@emory.edu |
| Date: 2022-01-21 at 10:00AM |
| Venue: Zoom and W201 |
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| Abstract: The Zarankiewicz problem is a central problem in extremal graph theory, which lies at the intersection of several areas of mathematics. It asks for the maximum number of edges in a bipartite graph on $2n$ vertices, where each side of the bipartition contains $n$ vertices, and which does not contain the complete bipartite graph $K_{s,t}$ as a subgraph. One of the many reasons this problem is rather special among Tur\'an-type problems is that the extremal graphs in question, whenever available, always seem to have to be of an algebraic nature, in particular witnesses to basic intersection theory phenomena. The most tantalizing case is by far the diagonal problem, for which the answer is unknown for most values of $s = t$, and where it is a complete mystery what the extremal graphs could look like. In this talk, we will discuss a new phenomenon related to an important variant of this problem, which is the analogous question in bipartite graphs with bounded VC-dimension. We will present several new consequences in incidence geometry, which improve upon classical results. \\ \\ This is based on joint work with Oliver Janzer. |
| Title: Coloring hypergraphs of small codegree, and a proof of the Erdős–Faber–Lovász conjecture |
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| Seminar: Discrete Mathematics |
| Speaker: Tom Kelly of The University of Birmingham |
| Contact: Dwight Duffus, dwightduffus@emory.edu |
| Date: 2022-01-19 at 10:00AM |
| Venue: Zoom |
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| Abstract: A long-standing problem in the field of graph coloring is the Erd$\ddot{\mathrm{o}}$s–Faber–Lovász conjecture (posed in 1972), which states that the chromatic index of any linear hypergraph on n vertices is at most n, or equivalently, that a nearly disjoint union of n complete graphs on at most n vertices has chromatic number at most n. In joint work with Dong Yeap Kang, Daniela Kühn, Abhishek Methuku, and Deryk Osthus, we proved this conjecture for every sufficiently large n. Recently, we also solved a related problem of Erd$\ddot{\mathrm{o}}$s from 1977 on the chromatic index of hypergraphs of small codegree. In this talk, I will survey the history behind these results and discuss some aspects of the proofs. |