All Seminars
| Title: Homogeneous Substructures in Ordered Matchings |
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| Seminar: Combinatorics |
| Speaker: Andrzej Rucinski of Adam Mickiewicz University, Poznan |
| Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
| Date: 2024-03-29 at 4:00PM |
| Venue: MSC W201 |
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| Abstract: An ordered matching M_n is a partition of a linearly ordered set of size 2n into n pairs (called edges). Taking the linear ordering into account, every pair of edges forms one of three patterns: AABB, ABBA, or ABAB. A submatching with all pairs of edges forming the same pattern is called a clique. In my talk, I will first show an Erdos-Szekeres type result guaranteeing a large clique in every matching M_n. Then I will move on to a random (uniform) setting and investigate the largest size of a clique of a given type (pattern) present in almost all matchings. Finally, I will attempt to generalize these results to r-uniform hypermatchings, that is, partitions of a linearly ordered set of size rn into n r-element subsets. This is joint work with Andrzej Dudek and Jarek Grytczuk. |
| Title: Degeneracy of eigenvalues and singular values of parameter dependent matrices |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Alessandro Pugliese of Georgia Tech/University of Bary |
| Contact: Manuela Manetta, manuela.manetta@emory.edu |
| Date: 2024-03-28 at 10:00AM |
| Venue: MSC W201 |
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| Abstract: Hermitian matrices have real eigenvalues and an orthonormal set of eigenvectors. Do smooth Hermitian matrix valued functions have smooth eigenvalues and eigenvectors? Starting from such question, we will first review known results on the smooth eigenvalue and singular values decompositions of matrices that depend on one or several parameters, and then focus on our contribution, which has been that of devising topological tools to detect and approximate parameters' values where eigenvalues or singular values of a matrix valued function are degenerate (i.e. repeated or zero). The talk will be based on joint work with Luca Dieci (Georgia Tech) and Alessandra Papini (Univ. of Florence). |
| Title: Collective migration model on a viscoelastic collagen network |
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| Seminar: Analysis and Differential Geometry |
| Speaker: Andrei Tarfulea of Louisiana State University |
| Contact: Maja Taskovic, maja.taskovic@emory.edu |
| Date: 2024-03-27 at 10:00AM |
| Venue: White Hall 110 |
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| Abstract: We explore a model of self-generated directional cell migration on viscoelastic substrates in the absence of apparent intrinsic polarity. Mathematically, this takes the form of a reaction-diffusion equation for the network deformation, along with a moving cell-cluster source term which itself moves according to the local network deformation. This creates a strange form of nonlinear interaction. We show global well-posedness, conditional existence/absence of traveling waves, and address the stability of traveling waves. |
| Title: A few steps towards the Erdos–Hajnal conjecture |
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| Seminar: Combinatorics |
| Speaker: Tung Nguyen of Princeton University |
| Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
| Date: 2024-03-26 at 10:00AM |
| Venue: MSC W201 |
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| Abstract: A cornerstone of Ramsey theory says that every graph contains a clique or independent set of logarithmic size, which is asymptotically optimal for almost all graphs. The Erd?s–Hajnal conjecture from 1977 predicts a very different situation in graphs with forbidden induced subgraphs; more precisely, the conjecture asserts that for every graph $H$, there exists $c=c(H)>0$ such that every $n$-vertex graph with no induced copy of $H$ has a clique or independent set of size at least $n^c$. This conjecture remains open, and we will discuss recent progress on it in the talk. |
| Title: Local heights on hyperelliptic curves for quadratic Chabauty |
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| Seminar: Algebra |
| Speaker: Juanita Duque-Rosero of Boston University |
| Contact: Andrew Kobin, ajkobin@emory.edu |
| Date: 2024-03-26 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: The method of quadratic Chabauty is a powerful tool to determine the set of rational points on curves. A key input for this method is the values of local height functions. In this talk, we will discuss an algorithm to compute these local heights at odd primes v not equal to p for hyperelliptic curves. Through applications, we will see how this work extends the reach of quadratic Chabauty to curves previously deemed inaccessible. This is joint work with Alexander Betts, Sachi Hashimoto, and Pim Spelier. |
| Title: Integer distance sets |
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| Seminar: Discrete Analysis |
| Speaker: Rachel Greenfeld of Institute for Advanced Study |
| Contact: Cosmin Pohoata, cosmin.pohoata@emory.edu |
| Date: 2024-03-25 at 5:30PM |
| Venue: MSC W301 |
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| Abstract: A set S in the Euclidean plane is an integer distance set if the distance between any pair of its points is an integer. Interestingly, all so-far-known integer distance sets have all but up to four of their points on a single line or circle. And it had long been suspected, going back to Erd?s, that any integer distance set must be of this special form. In a recent work, joint with Marina Iliopoulou and Sarah Peluse, we developed a new approach to the problem, which enabled us to make the first progress towards confirming this suspicion. In the talk, I will discuss the study of integer distance sets, its connections to other problems, and our new developments. |
| Title: Structure-conforming Operator Learning via Transformers |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Shuhao Cao of University of Missouri-Kansas City |
| Contact: Yuanzhe Xi, yuanzhe.xi@emory.edu |
| Date: 2024-03-21 at 10:00AM |
| Venue: MSC W201 |
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| Abstract: GPT, Stable Diffusion, AlphaFold 2, etc., all these state-of-the-art deep learning models use a neural architecture called "Transformer". Since the emergence of "Attention Is All You Need" paper by Google, Transformer is now the ubiquitous architecture in deep learning. At Transformer's heart and soul is the "attention mechanism". In this talk, we shall dissect the "attention mechanism" through the lens of traditional numerical methods, such as Galerkin methods, and hierarchical matrix decomposition. We will report some numerical results on designing attention-based neural networks according to the structure of a problem in traditional scientific computing, such as inverse problems for Neumann-to-Dirichlet operator (EIT) or multiscale elliptic problems. Progresses within different communities will be briefed to answer some open problems on the mathematical properties of the attention mechanism in Transformers. |
| Title: The Hassett-Keel program in genus 4 |
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| Seminar: Algebra |
| Speaker: Kristin DeVleming of University of Massachusetts Amherst |
| Contact: Andrew Kobin, ajkobin@emory.edu |
| Date: 2024-03-19 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: Studying the minimal model program with scaling on the moduli space of genus g curves and interpreting the steps in a modular way is known as the Hassett-Keel program. The first few steps are well-understood yet the program remains incomplete in general. We complete the Hassett-Keel program in genus 4 using wall-crossing in K-moduli and modular interpretations. This is joint work with Kenneth Ascher, Yuchen Liu, and Xiaowei Wang. |
| Title: The Schrödinger equations as inspiration of beautiful mathematics. |
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| Colloquium: N/A |
| Speaker: Gigliola Staffilani of Massachusetts Institute of Technology |
| Contact: Maja Taskovic, maja.taskovic@emory.edu |
| Date: 2024-03-08 at 10:00AM |
| Venue: MSC W301 |
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| Abstract: In the last two decades great progress has been made in the study of dispersive and wave equations. Over the years the toolbox used in order to attack highly nontrivial problems related to these equations has developed to include a collection of techniques: Fourier and harmonic analysis, analytic number theory, math physics, dynamical systems, probability and symplectic geometry. In this talk I will introduce a variety of results using as model problem mainly the periodic 2D cubic nonlinear Schrödinger equation. I will start by giving a physical derivation of the equation from a quantum many-particles system, I will introduce periodic Strichartz estimates along with some remarkable connections to analytic number theory, I will move on the concept of energy transfer and its connection to dynamical systems, and I will end with some results following from viewing the periodic nonlinear Schrödinger equation as an infinite dimensional Hamiltonian system. |
| Title: Nonlinear scientific computing in machine learning and applications |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Wenrui Hao of Pennsylvania State University |
| Contact: Yuanzhe Xi, yuanzhe.xi@emory.edu |
| Date: 2024-02-29 at 1:00PM |
| Venue: MSC E300 |
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| Abstract: Machine learning has seen remarkable success in various fields such as image classification, speech recognition, and medical diagnosis. However, this success has also raised intriguing mathematical questions about optimizing algorithms more efficiently and applying machine-learning techniques to address complex mathematical problems. In this talk, I will discuss the neural network model from a nonlinear scientific computing perspective and present recent work on developing a homotopy training algorithm to train neural networks layer-by-layer and node-by-node. I will also showcase the use of neural network discretization for solving nonlinear partial differential equations. Finally, I will demonstrate how machine learning can be used to learn a mathematical model from clinical data in cases where the pathophysiology of a disease, such as Alzheimer's, is not well understood. |