Seminars and Colloquia by Series

Differential Equations for Continuous-Time Deep Learning

Series
PDE Seminar
Time
Friday, April 19, 2024 - 15:00 for 1 hour (actually 50 minutes)
Location
CSIP Library (Room 5126), 5th floor, Centergy one
Speaker
Dr.Lars RuthottoResearch Associate Professor in the Department of Mathematics and the Department of Computer Science at Emory University

In this talk, we introduce and survey continuous-time deep learning approaches based on neural ordinary differential equations (neural ODEs) arising in supervised learning, generative modeling, and numerical solution of high-dimensional optimal control problems. We will highlight theoretical advantages and numerical benefits of neural ODEs in deep learning and their use to solve otherwise intractable PDE problems.

Branching Brownian motion and the road-field model

Series
Stochastics Seminar
Time
Thursday, April 18, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Nick CookDuke University

The Fisher-KPP equation was introduced in 1937 to model the spread of an advantageous gene through a spatially distributed population. Remarkably precise information on the traveling front has been obtained via a connection with branching Brownian motion, beginning with works of McKean and Bramson in the 70s. I will discuss an extension of this probabilistic approach to the Road-Field Model: a reaction-diffusion PDE system introduced by H. Berestycki et al. to describe enhancement of biological invasions by a line of fast diffusion, such as a river or a road. Based on joint work with Amir Dembo.

 

Square Functions Controlling Smoothness with Applications to Higher-Order Rectifiability

Series
Analysis Seminar
Time
Wednesday, April 17, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
John HoffmanFlorida State University

We present new results concerning characterizations of the spaces $C^{1,\alpha}$ and “$LI_{\alpha+1}$” for $0<\alpha<1$.  The space $LI_{\alpha +1}$ is the space of Lipschitz functions with $\alpha$-order fractional derivative having bounded mean oscillation.  These characterizations involve geometric square functions which measure how well the graph of a function is approximated by a hyperplane at every point and scale.  We will also discuss applications of these results to higher-order rectifiability.

Conflict-free hypergraph matchings and generalized Ramsey numbers (Emily Heath, Iowa State University)

Series
Graph Theory Seminar
Time
Tuesday, April 16, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Emily HeathIowa State University

Given graphs G and H and a positive integer q, an (H,q)-coloring of G is an edge-coloring in which each copy of H receives at least q colors. Erdős and Shelah raised the question of determining the minimum number of colors, f(G,H,q), which are required for an (H,q)-coloring of G. Determining f(K_n,K_p,2) for all n and p is equivalent to determining the classical multicolor Ramsey numbers. Recently, Mubayi and Joos introduced the use of a new method for proving upper bounds on these generalized Ramsey numbers; by finding a “conflict-free" matching in an appropriate auxiliary hypergraph, they determined the values of f(K_{n,n},C_4,3) and f(K_n,K_4,5). In this talk, we will show how to generalize their approach to give bounds on the generalized Ramsey numbers for several families of graphs. This is joint work with Deepak Bal, Patrick Bennett, and Shira Zerbib.

Galois/Monodromy Groups in 3D Reconstruction

Series
Algebra Seminar
Time
Tuesday, April 16, 2024 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Tim DuffUniversity of Washington

Please Note: The seminar has been rescheduled from Monday to Tuesday.

Galois groups embody the structure of algebraic equations arising in both enumerative geometry and various scientific applications where such equations must be solved. I will describe a line of work that aims to elucidate the role of Galois groups in applications where data taken from multiple images are used to reconstruct a 3D scene. From this perspective, I will revisit two well-known solutions to camera pose estimation problems, which originate from classical photogrammetry and are still heavily used within modern 3D reconstruction systems. I will then discuss some less-classical problems, for which the insight we gleaned from computing Galois groups led to significant practical improvements over previous solutions. A key ingredient was the use of numerical homotopy continuation methods to (heuristically) compute monodromy permutations. Time-permitting, I will explain how such methods may also be used to automatically recover certain symmetries underlying enumerative problems. 

Monotone generative modeling via a geometry-preserving mapping

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 15, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/98355006347
Speaker
Wonjun LeeUniversity of Minnesota, Twin Cities

Generative Adversarial Networks (GANs) are powerful tools for creating new content, but they face challenges such as sensitivity to starting conditions and mode collapse. To address these issues, we propose a deep generative model that utilizes the Gromov-Monge embedding (GME). It helps identify the low-dimensional structure of the underlying measure of the data and then map it, while preserving its geometry, into a measure in a low-dimensional latent space, which is then optimally transported to the reference measure. We guarantee the preservation of the underlying geometry by the GME and c-cyclical monotonicity of the generative map, where c is an intrinsic embedding cost employed by the GME. The latter property is a first step in guaranteeing better robustness to initialization of parameters and mode collapse. Numerical experiments demonstrate the effectiveness of our approach in generating high-quality images, avoiding mode collapse, and exhibiting robustness to different starting conditions.

Eremenko’s Conjecture and Wandering Lakes of Wada

Series
CDSNS Colloquium
Time
Friday, April 12, 2024 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
James WatermanStonybrook University

In 1989, Eremenko investigated the set of points that escape to infinity under iteration of a transcendental entire function, the so-called escaping set. He proved that every component of the closure of the escaping set is unbounded and conjectured that all the components of the escaping set are unbounded. Much of the recent work on the iteration of entire functions is involved in investigating properties of the escaping set, motivated by Eremenko's conjecture. We will begin by introducing many of the basic dynamical properties of iterates of an analytic function, and finally discuss constructing a transcendental entire function with a point connected component of the escaping set, providing a counterexample to Eremenko's conjecture. This is joint work with David Martí-Pete and Lasse Rempe.

Enumeration of interval graphs and d-representable complexes (Amzi Jeffs, CMU)

Series
Combinatorics Seminar
Time
Friday, April 12, 2024 - 15:15 for 1 hour (actually 50 minutes)
Location
Speaker
Amzi JeffsCarnegie Mellon University

How many different ways can we arrange n convex sets in R^d? One answer is provided by counting the number of d-representable complexes on vertex set [n]. We show that there are exp(Theta(n^d log n))-many such complexes, and provide bounds on the constants involved. As a consequence, we show that d-representable complexes comprise a vanishingly small fraction of the class of d-collapsible complexes. In the case d = 1 our results are more precise, and improve the previous best estimate for the number of interval graphs.

Riemannian geometry and contact topology IV

Series
Geometry Topology Working Seminar
Time
Friday, April 12, 2024 - 14:00 for 2 hours
Location
Skiles 006
Speaker
John EtnyreGeorgia Tech

This series of talks will discuss connections between Riemannian geometry and contact topology. Both structures have deep connections to the topology of 3-manifolds, but there has been little study of the interactions between them (at least the implications in contact topology). We will see that there are interesting connections between curvature and properties of contact structures. The talks will give a brief review of both Riemannian geometry and contact topology and then discuss various was one might try to connect them. There will be many open problems discussed (probably later in the series). 

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