In this talk, we will present the nature of inverse problems in ocean acoustics. We will focus on localizing a sound-transmitting source and on estimating properties related to the propagation medium. New work in our group features denoising acoustic signals for better inversion results, implementing optimal transport methods to improve the convexity of the implicit misfit minimization, and developing Machine Learning algorithms for seabed classification. We will describe open problems in the field and seek solutions.
Phylogenetic trees and networks provide mathematical representations of evolutionary history while raising a wide range of mathematical questions. In this talk, I will introduce phylogenetics through three problems that highlight different mathematical perspectives.
First, I will discuss probabilistic models of lateral gene transfer and the resulting questions of identifiability and statistical inference. Second, I will consider the combinatorics of phylogenetic networks, focusing on how many evolutionary trees a network can display and the resulting extremal questions. Finally, I will describe a divide-and-conquer approach to reconstructing evolutionary networks from data on small subsets of taxa.
No background in phylogenetics is assumed. The goal is to illustrate how questions arising from evolutionary biology can lead to interesting problems in several areas of mathematics.
Sampling by Transport and the Approximation of Measures
Transportation of measure underlies many contemporary methods in machine learning and statistics. Sampling, which is a fundamental building block in computational science, can be done efficiently given an appropriate measure-transport map. We ask: what is the effect of using approximate maps in such algorithms? We propose a new framework to analyze the approximation power of measure transport. This framework applies to existing algorithms, but also suggests new ones. At the core of our analysis is the theory of optimal transport regularity, approximation theory, and an emerging class of inequalities, previously studied in the context of uncertainty quantification (UQ).