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…

In this talk, I will discuss how incorporating geometric information into classical learning algorithms can improve their performance. The main focus will be on optimal mass transport (OMT), which has evolved as a major method to analyze distributional data.  In particular, I will show how embeddings can be used to build OMT-based classifiers, both in supervised and unsupervised learning settings. The proposed framework significantly…

Qualitatively, a tipping point in a dynamical system is when a small change in system inputs causes the system to move to a drastically different state. The discussion of tipping points in climate and related fields has become increasingly urgent as scientists are concerned that different aspects of Earth’s climate could tip to a qualitatively different state without…

In many recent works, analysis and number theory go beyond working side by side and team up in an interconnected back and forth interplay to become a powerful force. Here I describe two distinct meetings of the pair, which result in sharp counts for equilateral triangles in Euclidean space and statistics for how often a random polynomial has Galois group not isomorphic to the full symmetric group. https://ncsu.zoom.us/j/91896366693?pwd=YnFuZURGc1NNenRTQ3YrbjVTK0dQZz09 Meeting ID: 918 9636 6693 Passcode: 875811

Speaker: Lisa Fauci, Professor of Mathematics, Tulane University Abstract: In the past decade, the study of the fluid dynamics of swimming organisms has flourished. My research has been centered on answering questions about biophysics using computational models of fluid flow: How do mammalian sperm penetrate the outer layer of an egg for successful fertilization?  How…

The stochastic processes of evolution have generated DNA, RNA, and protein sequences. These sequences determine how these entities chemically interact with themselves and each other, form physical structures, and functionally behave as signals and/or machines within cells. My research involves reconstructing the history of the stochastic processes that led to the sequences we observe today…

We define vertex-colourings for edge-coloured digraphs, which unify the theory of P-partitions and proper vertex-colourings of graphs. Furthermore, we use our vertex-colourings to define generalized chromatic functions, which merge the chromatic symmetric and quasisymmetric functions and generating functions of P-partitions. We also discuss the relations between generalized chromatic functions, Schur functions in noncommuting variables, and the well-known Stanley-Stembridge (3+1)-free conjecture.

Stochastic fluctuations drive biological processes from particle diffusion to neuronal spike times. The goal of this talk is to use a variety of mathematical frameworks to understand such fluctuations and derive insight into the corresponding applications. We start by considering a novel stochastic process motivated by astrocytes, glial cells that ensheath neuronal synapses and can…

Machine learning (ML) has achieved unprecedented empirical success in diverse applications. It now has been applied to solve scientific problems, which has become an emerging field, Scientific Machine Learning (SciML). Many ML techniques, however, are very complex and sophisticated, commonly requiring many trial-and-error and tricks. These result in a lack of robustness and interpretability, which…

In this talk, we study an exploration version of continuous time expected utility maximization problem with reinforcement learning. It is shown that the optimal feedback policy is Gaussian. We then prove a policy improvement theorem. An implementable reinforcement learning algorithm is designed. Numerical examples are provided for illustrations. https://ncsu.zoom.us/j/95758380569?pwd=OFZKWnVQTkJVTTNPU1R2TkhXQzdPZz09 Meeting ID: 957 5838 0569 Passcode: 832132

The concept of multidegrees provides the right generalization of the degree of a projective variety to a multiprojective setting. The study of multidegrees goes back to seminal work by van der Waerden in 1929. We will slowly introduce the notion of multidegrees of a multiprojective variety. A complete characterization of the positivity of multidegrees will…

Recent years have witnessed tremendous progress in developing and analyzing quantum computing algorithms for quantum dynamics simulation of bounded operators (Hamiltonian simulation). However, many scientific and engineering problems require the efficient treatment of unbounded operators, which frequently arise due to the discretization of differential operators. Such applications include molecular dynamics, electronic structure theory, quantum control…

Many problems in scientific computing require minimizing nonsmooth optimization problems. In many applications, it is common to minimize the sum of a smooth nonconvex function and a nonsmooth convex function. For example, imaging  and data science applications require minimizing a measure of data misfit plus a sparsifying L1- or total-variation regularizer.  We develop a novel…

A hyperplane arrangement is a union of codimension one linear spaces.  These simple objects provide fertile ground for interactions between combinatorics, algebra, algebraic geometry, topology, and group actions.  The combinatorics of an arrangement is encoded by the pattern of intersections among the hyperplanes, called its intersection lattice.  On the other hand, a key algebraic object…