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 observed transition paths from the stochastic optimal control viewpoint, which realizes the transitions almost surely. Then based on collected high dimensional point clouds and nonlinear…

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 of such patterns sheds valuable insight upon their origins, and from the viewpoint of applications, is necessary for their control. In this talk, after introducing a…

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 posets with good dynamical behavior. Here the order polynomial is a certain enumerative invariant of the poset. Meanwhile, the dynamics includes promotion of linear extensions,…

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 coexistence. This theory extends and makes rigorous Modern Coexistence Theory and leads to resolving a number of conjectures due to Chesson, Ellner, and Palis. I…

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 crowded room. Here I will focus on the specific example of pattern formation in zebrafish, which are named for the dark and light stripes that…

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 will present a generalization of the Robinson-Schensted-Knuth algorithm to the insertion of two-row arrays of multisets. This generalization leads to new enumerative results that have…

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 tensor product operation for higher representations as well as with new geometric constructions. https://sites.google.com/usc.edu/manion/home

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 interplay between many interconnected physical phenomena has been a major obstacle in understanding the physiology of the heart and integrating it in mathematical models. By…

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 and high-dimensional. Traditional topological data analysis considers the geometric features of a dataset, while in practice, there could be both geometric and non-geometric features. In…

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 of the geometry and topology of varieties and using this understanding to obtain practical and effective computational methods. Such methods may then in turn be…

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 in 3-manifold topology in the past decade. In this talk I will provide motivation and background on these virtual properties and discuss some recent results. Zoom…

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 novel algorithmic and theoretical results in two broad classes of optimization problems. The first class of problems are semidefinite programs (SDP). I will present the…

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 be recast as questions about the structure of cohomology rings of geometric spaces such as flag varieties.  Borel's isomorphism identifies the cohomology of the complete…

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 also brought about a great number of powerful mathematical tools.  The through line to the subject is the essential role played by probability theory, a…

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 arrays, which have received considerable attention due, in part, to their occurrence in integrable probability. Gibbsian line ensembles can be thought of as collections of…

In the study of disordered quantum systems, it is believed that a strong dichotomy should occur between two phases: a delocalized (or conducting) phase and a localized (or insulating) phase. While this is far from being proved in all generality, the study of large symmetric random matrices, which model simple systems, allows us to describe…

A classical result due to Dobrushin (1956) yields the central limit theorem for partial sums of functionals of inhomogeneous ("sufficiently contracting'') Markov chains. In the talk we will restrict to bounded functionals of uniformly elliptic inhomogeneous Markov chains, for which we can obtain: A Berry-Esseen theorem (optimal rates in the Central limit theorem); Correction terms of…

We will introduce both finite and infinite-type surfaces and study their collections of symmetries, known as mapping class groups. The study of the mapping class group of finite-type surfaces has played a central role in low-dimensional topology stretching back a hundred years to work of Max Dehn and Jakob Nielsen, and gaining momentum and significance through the…

I will present problems arising from biochemical reaction networks, optimization, computer science, and complexity theory. These problems share the following characteristics: 1) they can be modeled by multivariate polynomials, 2) they demand different theorems than the ones offered by the traditional theory of computation and state-of-the-art theory of polynomials. I will present recent results that blend…

The topology of smooth manifolds is governed largely by geometry in low dimensions and by algebraic topology in high dimensions. The phase transition occurs in dimension four, leading to "exotic" phenomena where continuous and differential topology diverge sharply. I will begin by surveying some ways that surfaces can be used to investigate this phase transition. Then I…