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