All Seminars
| Title: Scalable Bayesian optimal experimental design for efficient data acquisition |
|---|
| Seminar: CODES@Emory |
| Speaker: Peng Chen of Georgia Tech |
| Contact: Matthias Chung, matthias.chung@emory.edu |
| Date: 2022-11-10 at 10:00AM |
| Venue: MSC W301 |
| Download Flyer |
| Abstract: Bayesian optimal experimental design (OED) is a principled framework for maximizing information gained from limited data in Bayesian inverse problems. Unfortunately, conventional methods for OED are prohibitive when applied to expensive models with high-dimensional parameters. In this talk, I will present fast and scalable computational methods for large-scale Bayesian OED with infinite-dimensional parameters, including data-informed low-rank approximation, efficient offline-online decomposition, projected neural network approximation, and a new swapping greedy algorithm for combinatorial optimization. |
| Title: Athens-Atlanta joint Number Theory Seminar |
|---|
| Seminar: Algebra |
| Speaker: Alina Bucur and Samit Dasgupta of Duke and University of California, San Diego |
| Contact: David Zureick-Brown, david.zureick-brown@emory.edu |
| Date: 2022-11-07 at 4:00PM |
| Venue: MSC W301 |
| Download Flyer |
| Abstract: \textbf{Samit Dasgupta} (Duke University), 4:00 \\ Stark’s Conjectures and Hilbert’s 12th Problem \\ In this talk we will discuss two central problems in algebraic number theory and their interconnections: explicit class field theory (also known as Hilbert’s 12th Problem), and the special values of L-functions. The goal of explicit class field theory is to describe the abelian extensions of a ground number field via analytic means intrinsic to the ground field. Meanwhile, there is an abundance of conjectures on the special values of L-functions at certain integer points. Of these, Stark’s Conjecture has special relevance toward explicit class field theory. This conjecture states the existence of certain canonical elements in CM abelian extensions of totally real fields. I will state a conjectural exact formula for Brumer-Stark units that has been developed over the last 15 years. I will conclude with a description of my work with Mahesh Kakde that proves these conjectures away from $p = 2$, thereby giving an explicit class field theory for totally real fields. \\ \textbf{Alina Bucur, USCD} (University of California, San Diego), 5:15 \\ Counting $D_4$ fields \\ A guiding question in number theory, specifically in arithmetic statistics, is that of counting number fields of fixed degree whose normal closure has a given Galois group G as we let their discriminants grow to infinity. In this talk, we will discuss the history of this question and take a closer look at the story in the case that $n=4$, i.e. the counts of quartic fields. |
| Title: Regression with Tropical Rational Functions and Application to Neural Network Initialization |
|---|
| Seminar: CODES@emory |
| Speaker: Alex Dunbar of Emory University |
| Contact: Lars Ruthotto, lruthotto@emory.edu |
| Date: 2022-11-03 at 10:00AM |
| Venue: MSC W301 |
| Download Flyer |
| Abstract: The tropical semiring and its associated polynomial and rational functions provide an algebraic framework for understanding continuous piecewise linear functions. We propose an alternating minimization heuristic for regression over the space of tropical rational functions. The method alternates between fitting the numerator and denominator via tropical polynomial regression, which is known to admit a closed-form solution. Our work is motivated by applications to ReLU neural network training. ReLU neural networks are a popular class of network architectures in the machine learning community which have recently been connected to tropical rational functions. We present experiments demonstrating the behavior of the alternating minimization method. Additionally, we present preliminary experiments which leverage tropical rational regression to initialize weights in ReLU neural networks and discuss geometric aspects of the network initialization problem. |
| Title: (Lack of) Rank Growth of Elliptic Curves in Quartic Extensions |
|---|
| Seminar: Algebra |
| Speaker: Daniel Keliher of University of Georgia |
| Contact: David Zureick-Brown, david.zureick-brown@emory.edu |
| Date: 2022-11-01 at 4:00PM |
| Venue: MSC N304 |
| Download Flyer |
| Abstract: Let $E$ be an elliptic curve over the rationals, and let $E(K)$ be the group of K-rational points of E over a number field K. The Mordell-Weil Theorem says that $E(K)$ factors as a finitely generated abelian group; the rank of the free abelian part, $rk(E/K)$, is the rank of E over K. We will consider the related notion of rank growth. That is, for an extension of number fields $F/K$, we will consider the quantity $rk(E/F)-rk(E/K)$. We will survey some results, conjectures, and hopes in this area and outline an approach to produce infinitely many $S_4$ and $A_4$ quartic extensions of the rationals for which an elliptic curve E does not gain rank. The approach adapts methods of Mazur, Rubin, and others to manipulate the 2-Selmer group of a thin family of quadratic twists of our starting E in a prescribed fashion. In doing so, we’ll construct quadratic extensions for which we can understand the local conditions of the corresponding twist and such that the quadratic extensions parameterize the quartic extensions of interest. |
| Title: Algorithmic and combinatorial applications of the cluster expansion |
|---|
| Seminar: Combinatorics |
| Speaker: Matthew Jenssen of University of Birmingham, UK |
| Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
| Date: 2022-10-28 at 3:30PM |
| Venue: MSC E406 |
| Download Flyer |
| Abstract: The cluster expansion is a classical tool from statistical physics traditionally used to study the phase diagram of lattice spin models. Recently, the cluster expansion has enjoyed a number of applications in two new contexts: i) the design of efficient approximate counting and sampling algorithms for spin models on graphs and ii) classical enumeration problems in combinatorics. In this talk, I’ll give an introduction to the cluster expansion and discuss some of these recent developments. |
| Title: Neural networks investigation of bifurcating phenomena in fluid-dynamics |
|---|
| Seminar: CODES@Emory |
| Speaker: Federico Pichi of EPFL Lausanne |
| Contact: Alessandro Veneziani, ale@mathcs.emory.edu |
| Date: 2022-10-27 at 10:00AM |
| Venue: MSC W301 |
| Download Flyer |
| Abstract: Bifurcating phenomena, i.e. sudden changes in the qualitative behavior of the system linked to the non-uniqueness of the solution naturally arise in several fields. Since the reconstruction of bifurcation diagrams requires a many-query context, which is usually unaffordable using high-fidelity simulations, we propose a combination of Reduced Order Models (ROMs) and Machine Learning techniques to reduce the computational burden associated with the investigation of such complex phenomena.This work aims to show the applicability of the Reduced Basis (RB) model reduction and Artificial Neural Network (ANN), utilizing the POD-NN approach and its physics-informed variant [2, 1], to analyze multi-parameter bifurcating applications in fluid-dynamics. We considered the Navier-Stokes equations for a viscous, steady, and incompressible flow: (i) in a planar straight channel with a narrow inlet of varying width and (ii) in a triangular parametrized lid-driven cavity. Within this context, we present a new empirical strategy to employ the RB and ANN coefficients for a non-intrusive detection of the bifurcation points [3]. Finally, we introduce a newly developed ROM methodology based on Graph Neural Network, with powerful applications to general parametrized PDEs and branches classification when dealing with bifurcating phenomena [4]. References: [1] W. Chen, Q. Wang, J. S. Hesthaven, and C. Zhang. Physics-informed machine learning for reduced-order modeling of nonlinear problems. Journal of Computational Physics, 446:110666, 2021. [2] J. S. Hesthaven and S. Ubbiali. Non-intrusive reduced order modeling of nonlinear problems using neural networks. Journal of Computational Physics, 363:55–78, 2018. [3] F. Pichi, F. Ballarin, G. Rozza, and J. S. Hesthaven. Artificial neural network for bifurcating phenomena modelled by nonlinear parametrized PDEs. arXiv:2109.10765, 2021. [4] F. Pichi, B. Moya, and J. S. Hesthaven. A convolutional graph neural network approach to model order reduction for nonlinear parametrized PDEs. In preparation, 2022. |
| Title: Overconvergent differential operators acting on Hilbert modular forms |
|---|
| Seminar: Algebra |
| Speaker: Jon Aycock of University of California, San Diego |
| Contact: David Zureick-Brown, david.zureick-brown@emory.edu |
| Date: 2022-10-25 at 4:00PM |
| Venue: MSC N304 |
| Download Flyer |
| Abstract: In 1978, Katz gave a construction of the $p$-adic $L$-function of a CM field by using a $p$-adic analog of the Maass--Shimura operators acting on $p$-adic Hilbert modular forms. However, this $p$-adic Maass--Shimura operator is only defined over the ordinary locus, which restricted Katz's choice of $p$ to one that splits in the CM field. In 2021, Andreatta and Iovita extended Katz's construction to all $p$ for quadratic imaginary fields using overconvergent differential operators constructed by Harron--Xiao and Urban, which act on elliptic modular forms. Here we give a construction of such overconvergent differential operators which act on Hilbert modular forms. |
| Title: Stability, Optimality, and Fairness in Federated learning |
|---|
| Seminar: CODES@Emory |
| Speaker: Kate Donahue of Cornell University |
| Contact: Elizabeth Newman, elizabeth.newman@emory.edu |
| Date: 2022-10-20 at 10:00AM |
| Venue: MSC W301 |
| Download Flyer |
| Abstract: Federated learning is a distributed learning paradigm where multiple agents, each only with access to local data, jointly learn a global model. There has recently been an explosion of research aiming not only to improve the accuracy rates of federated learning, but also provide certain guarantees around social good properties such as total error or fairness. In this talk, I describe two papers analyzing federated learning through the lens of cooperative game theory (both joint with Jon Kleinberg). In the first paper, we discuss fairness in federated learning, which relates to how error rates differ between federating agents. In this work, we consider two notions of fairness: egalitarian fairness (which aims to bound how dissimilar error rates can be) and proportional fairness (which aims to reward players for contributing more data). For egalitarian fairness, we obtain a tight multiplicative bound on how widely error rates can diverge between agents federating together. For proportional fairness, we show that sub-proportional error (relative to the number of data points contributed) is guaranteed for any individually rational federating coalition. The second paper explores optimality in federated learning with respect to an objective of minimizing the average error rate among federating agents. In this work, we provide and prove the correctness of an efficient algorithm to calculate an optimal (error minimizing) arrangement of players. Building on this, we give the first constant-factor bound on the performance gap between stability and optimality, proving that the total error of the worst stable solution can be no higher than 9 times the total error of an optimal solution (Price of Anarchy bound of 9). Relevant Links: https://arxiv.org/abs/2010.00753, https://arxiv.org/abs/2106.09580, https://arxiv.org/abs/2112.00818 |
| Title: Subspace configurations and low degree points on curves |
|---|
| Seminar: Algebra |
| Speaker: Borys Kadets of University of Georgia |
| Contact: David Zureick-Brown, david.zureick-brown@emory.edu |
| Date: 2022-10-18 at 4:00PM |
| Venue: MSC N304 |
| Download Flyer |
| Abstract: The hyperelliptic curve given by the equation $y^2=f(x)$ with coefficients in $\mathbf{Q}$ has an unusual arithmetic property: it admits infinitely many points with coordinates in quadratic extensions of $\mathbf{Q}$ (namely $(a, \sqrt{f(a)})$). Hindry, motivated by arithmetic questions about modular curves, asked if the only curves that possess infinite collections of quadratic points are hyperelliptic and bielliptic; this conjecture was confirmed by Harris and Silverman. I will talk about the general problem of classifying curves that possess infinite collections of degree $d$ points. I will explain how to reduce this classification problem to a study of curves of low genus, and use this reduction to obtain a classification for $d \leq 5$. This relies on analyzing a discrete-geometric object -- the subspace configuration -- attached to curves with infinitely many degree $d$ points. This talk is based on joint work with Isabel Vogt (arXiv:2208.01067). |
| Title: Model order reduction for parametrized optimal control problems: from time-dependency to nonlinearity. |
|---|
| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Maria Strazzullo of Politecnico di Torino, ITALY |
| Contact: Alessandro Veneziani, ale@mathcs.emory.edu |
| Date: 2022-10-17 at 10:00AM |
| Venue: Atwood 360 |
| Download Flyer |
| Abstract: Parametrized optimal control problems can represent an asset to fill the gap between collected data and partial differential equations in many scientific and industrial applications. Despite their indisputable usefulness, their computational complexity still limits their applicability in many-query and real-time parametric settings, most of all when the problem is time-dependent or nonlinear.\\ \\ We propose reduced order methods as a valid strategy to deal with this issue. The talk focuses on the approaches that provide a low-dimensional framework to accelerate the simulations of the system, maintaining a fair degree of accuracy.\\ \\ The first part of the talk is about the numerical algorithms used to reach this goal. The second part is more related to the applied viewpoint, analyzing the potential of reduced optimal control in many fields, such as bifurcating phenomena and numerical stabilization. |