Seminars and Colloquia by Series

Interpolating between the optimal transport problems of Monge and Kantorovich

Series
Math Physics Seminar
Time
Friday, February 21, 2025 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Brendan Pass – University of Alberta –

I will present joint work in progress with Gero Friesecke.  We introduce a two parameter family of variational problems; varying the first parameter interpolates between a regularized version of Monge's optimal transport (OT) problem and Kantorovich's relaxed version.  The first limit problem has the advantage over Monge's original problem of always admitting a solution.  In cases where a (sufficiently regular) Monge map exists, the solution will be of such a form; if not, the limit problem essentially minimizes the transportation cost among all best approximations of the target measure by  pushforwards of the source.  When the source measure is discrete, we show that this is equivalent to the optimal quantization of the target measure, with the additional constraint that the weights of the approximating discrete masses are prescribed.  The second parameter controls the regularity of the pseudo-Monge map. In both the high and low regularity limits, the problem converges to the classical Kantorovich problem, under mild assumptions.

 

Part of the motivation for this problem is to understand whether the strictly correlated electron ansatz is valid in the semi-classical limit of density functional theory (DFT).  We will briefly discuss the corresponding application of OT to DFT, and outline what is known about the existence of Monge solutions (or, equivalently, the validity of the strictly correlated electron ansatz).

Approximate Messaging Passing Algorithms for High-dimensional Estimation and Inference

Series
Stochastics Seminar
Time
Thursday, February 20, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Cynthia Rush – Columbia University –

In this talk, I discuss how one can use approximate message passing (AMP) algorithms — a class of efficient, iterative algorithms that have been successfully employed in many statistical learning tasks like high-dimensional linear regression and low-rank matrix estimation — for characterizing exact statistical properties of estimators in a high-dimensional asymptotic regime, where the sample size of the data is proportional to the number of parameters in the problem. As a running example, we will study sorted L1 penalization (SLOPE) for linear regression and show how AMP theory can be used to give insights on the variable selection properties of this estimator by characterizing the optimal trade-off between measures of type I and type II error. Collaborators on this work include Zhiqi Bu, Oliver Feng, Jason Klusowski, Richard Samworth, Weijie Su, Ramji Venkataramanan, and Ruijia Wu.

Hamilton-Jacobi equations on Wasserstein spaces

Series
PDE Seminar
Time
Wednesday, February 19, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ben Seeger – University of North Carolina at Chapel Hill –

Please Note: Please note the unusual time and place.

The study of differential equations on infinite spaces of probability measures has become an active field of research in recent years. Such equations arise when describing nonlinear effects in systems with a large number of interacting particles or agents. A rigorous well-posedness theory for such equations then leads to results about large deviations for interacting particle systems, limiting statements about free energies in mean field spin glasses, and mean field descriptions in high-dimensional optimization and game theory, among many other examples.
 
This talk will give an overview of some recent well-posedness results for Hamilton-Jacobi equations on probablity spaces. The nonlinear and infinite-dimensional nature of the underlying space necessitates a mix of techniques from functional analysis and probability theory. We also discuss how these PDE techniques can be used to deduce qualitative and quantitative convergence results for stochastic control and differential games for a large system of interacting agents. This talk is based on joint work with Joe Jackson (University of Chicago) and Samuel Daudin (Université Paris Cité, Laboratoire Jacques Louis Lions).

Cylindrical Martingale-Valued Measures, Stochastic Integration and SPDEs

Series
Analysis Seminar
Time
Wednesday, February 19, 2025 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dario Mena – University of Costa Rica –

We develop a theory of Hilbert-space valued stochastic integration with respect to cylindrical martingale-valued measures. As part of our construction, we expand the concept of quadratic variation, to the case of cylindrical martingale-valued measures that are allowed to have discontinuous paths; this is carried out within the context of separable Banach spaces. Our theory of stochastic integration is applied to address the existence and uniqueness of solutions to stochastic partial differential equations in Hilbert spaces. 

ε-series by Caleb McFarland, Richter Jordaan, Owen Huang

Series
Graph Theory Seminar
Time
Tuesday, February 18, 2025 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker

 

Caleb McFarland: We prove a structure theorem for Γ-labeled graphs G which forbid a fixed Γ-labeled graph H as an immersion in the case that Γ is a finite abelian group. Joint work with Rose McCarty and Paul Wollan.
Richter Jordaan: In this expository talk I will give introduce an approach to the cycle double cover based on the more general problem of finding specific cycle covers of cubic graphs. After stating the basics of the cycle double cover conjecture and structure of a minimal counterexample, I'll try to describe the setup and basic intuition behind how the general cyle cover problem could be used to approach the cycle double cover conjecture.
Owen Huang: We will discuss some recent work with Rose McCarty concerning the product structure of Cayley graphs. We also introduce an integer-valued invariant of finitely generated groups and note its relevance in geometric group theory. 

Recent progress on completely integrable systems

Series
Job Candidate Talk
Time
Tuesday, February 18, 2025 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Monica Visan – UCLA –

 We will survey a number of recent developments in the theory of completely integrable nonlinear dispersive PDE.  These include a priori bounds, orbital stability of multisolitons, well-posedness at optimal regularity, and the existence of dynamics for Gibbs distributed initial data. I will describe the basic objects that tie together these disparate results, as well as the diverse ideas required for each problem.

 

 

From Optics to the Deift Conjecture

Series
Job Candidate Talk
Time
Monday, February 17, 2025 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Rowan Killip – UCLA –

After providing a mathematical background for some curious optical experiments in the 19th century, I will then describe how these ideas inform our understanding of the Deift conjecture for the Korteweg--de Vries equation. Specifically, in joint work with Chapouto and Visan, we showed that the evolution of almost-periodic initial data need not remain almost periodic.

 

Uniform set systems with small VC-dimension and the Erdős--Ko--Rado theorem

Series
Combinatorics Seminar
Time
Friday, February 14, 2025 - 15:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Chi Hoi (Kyle) Yip – Georgia Institute of Technology –

Let $d\geq 2$, and $n\geq 2d+2$. Frankl and Pach initiated the study of the maximum size of a $(d+1)$-uniform set system in $[n]$ with VC-dimension at most $d$. The best-known upper bound is essentially $\binom{n}{d}$, and the best-known lower bound is $\binom{n-1}{d} + \binom{n-4}{d-2}$. In this talk, I will discuss some recent improvements on the upper bound and some interesting connections between this problem and the celebrated Erdős--Ko--Rado theorem. In particular, I will discuss our conjecture, which can be viewed as a generalization of the EKR as well as an "uniform version" of the disproved Erdős--Frankl--Pach conjecture, and highlight some of our partial progress. Joint work with Ting-Wei Chao, Zixiang Xu, and Shengtong Zhang.

Trivialized Momentum Facilitates Diffusion Generative Modeling on Lie Groups

Series
SIAM Student Seminar
Time
Friday, February 14, 2025 - 11:00 for
Location
Skiles 006
Speaker
Yuchen Zhu – Georgia Tech

The generative modeling of data on manifolds is an important task, for which diffusion models in flat spaces typically need nontrivial adaptations. This article demonstrates how a technique called `trivialization' can transfer the effectiveness of diffusion models in Euclidean spaces to Lie groups. In particular, an auxiliary momentum variable was algorithmically introduced to help transport the position variable between data distribution and a fixed, easy-to-sample distribution. Normally, this would incur further difficulty for manifold data because momentum lives in a space that changes with the position. However, our trivialization technique creates a new momentum variable that stays in a simple fixed vector space. This design, together with a manifold preserving integrator, simplifies implementation and avoids inaccuracies created by approximations such as projections to tangent space and manifold, which were typically used in prior work, hence facilitating generation with high-fidelity and efficiency. The resulting method achieves state-of-the-art performance on protein and RNA torsion angle generation and sophisticated torus datasets. We also, arguably for the first time, tackle the generation of data on high-dimensional Special Orthogonal and Unitary groups, the latter essential for quantum problems.

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