Many estimates in analysis have certain “common structures” which makes it possible to obtain them with what is now called Bellman function method. Originally the method appeared in control theory…
Free-boundary problems are partial differential equations in which the unknown function and its domain must be simultaneously determined. They arise ubiquitously as mathematical models for phenomena in many fields, most…
A central question in machine learning theory is whether an algorithm enjoys good generalization, which is the ability to correctly predict new examples from prior observations. While classical wisdom advocates…
Thin liquid films flowing down vertical fibers exhibit complex and interesting interfacial dynamics, including droplet formation and traveling wave patterns. Such dynamics play a crucial role in the design of…
Computerized tomography techniques, which are used for visualizing the interior structure of an object of interest in a non-invasive manner, have played a central role in medical imaging, industrial non-destructive…
We focus on analysis and data-driven algorithms for rare events such as essential conformational transitions in biochemical reactions which are modeled by Langevin dynamics on manifolds. We first reinterpret the…
Energy-driven pattern formation is ubiquitous in nature; the character and dynamics of such patterns is selected as local minimizers and gradient flows, respectively, of non-convex, and often, non-local energies with multiple spatio-temporal scales. Analysis…
Sam Hopkins will present a heuristic for finding special families of partially ordered sets. The heuristic says that the posets with order polynomial product formulas are the same as the…
A fundamental problem from population biology is finding conditions under which interacting species coexist or go extinct. I present results that lay the foundation for a general theory of stochastic…
Many natural and social phenomena involve individual agents coming together to create group dynamics, whether they are cells in a skin pattern, voters in an election, or pedestrians in a…
In algebraic combinatorics, the Robinson-Schensted-Knuth algorithm is a fundamental correspondence between words and pairs of semistandard tableaux illustrating identities of dimensions of irreducible representations of several groups. In this talk, I…
I will give a tour of the origins of Heegaard Floer homology and its applications in topology and representation theory, highlighting recent work that relates Heegaard Floer homology with a…
Cardiovascular diseases are a major health and economic concern both in the U.S. and worldwide. Although recent breakthroughs in medical treatments for heart diseases have improved patient outcomes, the complex…
Topological and geometric data analysis (TGDA) is a powerful framework for quantitative description and simplification of datasets' shapes. It is especially suitable for modern biological data that are intrinsically complex…
At its most basic, algebraic geometry studies algebraic varieties; that is, the solution sets of systems of polynomial equations. In this talk our focus is on developing a concrete understanding…
A virtual property of a 3-manifold is a property satisfied by a finite cover of the 3-manifold. The study of such properties has been at the heart of several major developments…
Optimization is a broad area with ramifications in many disciplines, including machine learning, control theory, signal processing, robotics, computer vision, power systems, and quantum information. I will talk about some…
Schubert calculus has its origins in enumerative questions asked by the geometers of the 19th century, such as "how many lines meet four fixed lines in three-space?" These problems can…
Disordered systems are mathematical models (typically of the physical world) that are governed by random variables. These models have offered insights into a diverse array of research problems, and have…
Gibbs measures are ubiquitous in statistical mechanics and probability theory. In this talk I will discuss two types of classes of Gibbs measures – random line ensembles and triangular particle…