All Seminars

Title: Triangular modular curves
Seminar: Algebra
Speaker: Juanita Duque-Rosero of Boston University
Contact: Andrew Kobin, ajkobin@emory.edu
Date: 2024-03-26 at 4:00PM
Venue: MSC W301
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Abstract:
Triangular modular curves are a generalization of modular curves and arise from quotients of the upper half-plane by congruence subgroups of hyperbolic triangle groups. These curves naturally parameterize hypergeometric abelian varieties, making them interesting arithmetic objects. In this talk we focus on the Borel-kind prime level triangular modular curves. We show that there are finitely many such curves of any given genus and present an algorithm to enumerate these curves. This is joint work with John Voight.
Title: Structure-conforming Operator Learning via Transformers
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: Nonlinear scientific computing in machine learning and applications
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
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
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.
Title: Ramsey and density results for approximate arithmetic progressions.
Seminar: Combinatorics
Speaker: Marcelo Sales of UC Irvine
Contact: Cosmin Pohoata, cosmin.pohoata@emory.edu
Date: 2024-02-23 at 4:00PM
Venue: MSC W201
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Abstract:
Let AP_k={a,a+d,\ldots,a+(k-1)d} be an arithmetic progression of length k. For a given epsilon>0, we call a set AP_k(epsilon)={x_0,…,x_{k-1}} an epsilon-approximate arithmetic progression of lenght k for some a and d, if the inequality |x_i-(a+id)|<\epsilon d holds for all i in {0,1,...,k-1}. In this talk we discuss numerical aspects of Van der Waerden and Szemeredi type of results in which arithmetic progressions are replaced by their epsilon-approximation. Joint work with Vojtech Rodl.
Title: On the evolution of structure in triangle-free graphs
Seminar: Discrete Analysis
Speaker: Will Perkins of Georgia Tech
Contact: Cosmin Pohoata, cosmin.pohoata@emory.edu
Date: 2024-02-19 at 5:30PM
Venue: MSC W301
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Abstract:
Erdos-Kleitman-Rothschild proved that the number of triangle-free graphs on n vertices is asymptotic to the number of bipartite graphs; or in other words, a typical triangle-free graph is a random subgraph of a nearly balanced complete bipartite graph. Osthus-Promel-Taraz extended this result to much lower densities: when m >(\sqrt{3}/4 +eps) n^{3/2} \sqrt{\log n}, a typical triangle-free graph with m edges is a random subgraph of size m from a nearly balanced complete bipartite graph (and this no longer holds below this threshold). What do typical triangle-free graphs at sparser densities look like and how many of them are there? We consider what we call the "ordered" regime, in which typical triangle-free graphs are not bipartite but do align closely with a nearly balanced bipartition. In this regime we prove asymptotic formulas for the number of triangle-free graphs and give a precise probabilistic description of their structure. Joint work with Matthew Jenssen and Aditya Potukuchi.
Title: Canonical colourings in random graphs
Seminar: Combinatorics
Speaker: Mathias Schact of University of Hamburg
Contact: Liana Yepremyan, liana.yepremyan@emory.edu
Date: 2024-02-16 at 4:00PM
Venue: MSC W201
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Abstract:
Rodl and Rucinski established Ramsey's theorem for random graphs. In particular, for fixed integers $r$, $\ell\geq 2$ they showed that $n^{-\frac{2}{\ell+1}}$ is a threshold for the Ramsey property that every $r$-colouring of the edges of the binomial random graph $G(n,p)$ yields a monochromatic copy of $K_\ell$. We investigate how this result extends to arbitrary colourings of $G(n,p)$ with an unbounded number of colours. In this situation Erd\H{o}s and Rado showed that \textit{canonically coloured} copies of~$K_\ell$ can be ensured in the deterministic setting. We transfer the Erd\H os--Rado theorem to the random environment and show that for $\ell\geq 4$ both thresholds coincide. As a consequence the proof yields $K_{\ell+1}$-free graphs~$G$ for which every edge colouring yields a canonically coloured $K_\ell$. This is joint work with Nina Kamev.
Title: A shifted convolution problem arising from physics
Seminar: Algebra
Speaker: Kim Klinger-Logan of Kansas State University/Rutgers University
Contact: Andrew Kobin, andrew.jon.kobin@emory.edu
Date: 2023-12-12 at 4:00PM
Venue: MSC W301
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Abstract:
Physicists Green, Russo, and Vanhove have discovered solution to differential equations involving automorphic forms appear at the coefficients to the $4$-graviton scattering amplitude in type IIB string theory. Specifically, for $\Delta$ the Laplace-Beltrami operator and $E_s(g)$ a Langlands Eisenstein series, solutions $f(g)$ of $(\Delta-\lambda) f(g) = E_a(g) E_b(g)$ for $a$ and $b$ half-integers on certain moduli spaces $G(Z)\backslash G(R)/K(R)$ of real Lie groups appear as coefficients to the analytic expansion of the scattering amplitude. We will briefly discuss different approaches to finding solutions to such equations and focus on a shifted convolution sum of divisor functions which appears as the Fourier modes associated to the homogeneous part of the solution. Initially, it was thought that, when summing over all Fourier modes, the homogeneous solution would vanish but recently we have found an exciting error term. This is joint work with Stephen D. Miller, Danylo Radchenko, and Ksenia Fedosova.
Title: Predicting Complex Spatiotemporal Cardiac Voltage Dynamics Using Reservoir Computing
Seminar: Numerical Analysis and Scientific Computing
Speaker: Elizabeth Cherry of Georgia Tech
Contact: Matthias Chung, matthias.chung@emory.edu
Date: 2023-12-05 at 10:00AM
Venue: MSC N306
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Abstract:
Disruptions to the electrical behavior of the heart caused by cardiac arrhythmias can result in complex dynamics, from period-2 rhythms in single cells to spatiotemporally complex spiral and scroll waves of electrical activity, which can inhibit contraction and may be lethal if untreated. Accurate forecasts of cardiac voltage behavior could allow new opportunities for intervention and control, but predicting complex nonlinear time series is a challenging task. In this talk, we discuss our recent work using machine-learning approaches based on reservoir computing to forecast cardiac voltage dynamics. First, we show that a novel method combining an echo state network with automated feature extraction via an autoencoder can successfully and efficiently predict time series of synthetic and experimental datasets of cardiac voltage in one cell with 20-30 action potentials in advance. Building on this work, we then demonstrate a novel method for predicting the complex spatiotemporal electrical dynamics of cardiac tissue using an echo state network integrated with a convolutional autoencoder. We show that our approach can forecast complex spiral-wave behavior, including breakup several periods in advance for time series ranging from model-derived synthetic datasets to optical-mapping recordings of explanted human hearts.