All Seminars
| 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. |
| Title: Bounds on the Torsion Subgroups of Second Cohomology |
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| Seminar: Algebra |
| Speaker: Hyuk Jun Kweon of University of Georgia |
| Contact: Andrew Kobin, ajkobin@emory.edu |
| Date: 2024-02-27 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: Let $X \hookrightarrow \mathbb{P}^r$ be a smooth projective variety defined by homogeneous polynomials of degree $\leq d$ over an algebraically closed field $k$. Let $\mathbf{Pic}\, X$ be the Picard scheme of $X$, and $\mathbf{Pic}\, ^0 X$ be the identity component of $\mathbf{Pic}\, X$. The N\'eron--Severi group scheme of $X$ is defined by $\mathbf{NS} X = (\mathbf{Pic}\, X)/(\mathbf{Pic}\, ^0 X)_{\mathrm{red}}$, and the N\'eron--Severi group of $X$ is defined by $\mathrm{NS}\, X = (\mathbf{NS} X)(k)$. We give an explicit upper bound on the order of the finite group $(\mathrm{NS}\, X)_{{\mathrm{tor}}}$ and the finite group scheme $(\mathbf{NS} X)_{{\mathrm{tor}}}$ in terms of $d$ and $r$. As a corollary, we give an upper bound on the order of the torsion subgroup of second cohomology groups of $X$ and the finite group $\pi^1_\mathrm{et}(X,x_0)^{\mathrm{ab}}_{\mathrm{tor}}$. We also show that $(\mathrm{NS}\, X)_{\mathrm{tor}}$ is generated by $(\deg X -1)(\deg X - 2)$ elements in various situations. |
| Title: Off-diagonal Weyl Laws for Commuting Selfadjoint Operators |
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| Seminar: Analysis and Differential Geometry |
| Speaker: Suresh Eswarathasan of Dalhousie University |
| Contact: David Borthwick, dborthw@emory.edu |
| Date: 2024-02-23 at 10:00AM |
| Venue: MSC W301 |
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| Abstract: The Weyl Law concerns the asymptotics of the eigenvalue counting function for, amongst other operators, Laplacians on compact manifolds. In this talk, we focus on the joint spectrum for commuting selfadjoint operators on compact manifolds (a special case being the joint spectrum for the Laplacian and the generator for rotations on a surface of revolution). In joint work with Blake Keeler (CRM Montréal and AARMS Halifax), we prove a corresponding "off-diagonal" Weyl asymptotic in this setting. Such an asymptotic describes the covariance function for certain types of "random waves" and gives a complementary eigenvalue counting result to that of Colin de Verdière from 1979. |