Faculty Research Talks - Spring 2025
Talks will be held at 2:30PM on every other Monday (M) at 2:30PM in CULM 611 unless otherwise noted.
February 17
Location: CULM 611
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).
February 24
Location: CULM 611
Title and Abstract Forthcoming
March 3
Location: CULM 611
Title and Abstract Forthcoming
March 31
Location: CULM 611
Integral Equation Methods for Flexural-Gravity Waves (or, a Song of Ice and Water)
I'll present some results from two recent pre-prints on flexural and flexural-gravity waves (https://arxiv.org/abs/2409.19160 and https://arxiv.org/abs/2501.00887). These models are commonly used to understand the waves that propagate along the interface between ice floes and sea water. Perhaps surprisingly, this problem can be reduced from three dimensions to a tractable, second-kind integral equation defined on the interface alone. I'll discuss the derivation, analysis, and numerical solution of this integral equation --- soup to nuts --- which touches on a lot of the mathematics from our first year courses. Finally, I'll present some of the interesting numerical experiments that these tools enable and some open projects. This is joint work with Manas Rachh (Flatiron Institute) and Jeremy Hoskins, Peter Nekrasov, and Tim Su (U.Chicago).
April 14
Location: CULM 611
Exploring Tree Balance Indices: Curious results, current developments, and future directions
Measures of tree balance play an important role in different research areas ranging from evolutionary biology to theoretical computer science. The balance of a tree is usually quantified in a single number, called a balance or imbalance index, and several such indices exist in the literature. Some of them are well-understood, while for others there are still open questions regarding their mathematical properties.
In this talk, I will introduce tree balance and discuss different tree balance indices. I will then focus on presenting some curious results related to tree balance indices, before describing recent advances to extending these concepts to phylogenetic networks and highlighting some open questions and directions for future research.
April 28
Location: CULM 611
Numerical methods for the magnetophoretic transport and assembly of paramagnetic particles
I will present recent numerical methods for the problem of the magnetophoretic transport of magnetic liquids (ferrofluids). I will discuss the extension of the numerical framework to include the fluid-structure interactions based on direct solutions of the fully three-dimensional governing (Navier-Stokes) equations, coupled with the solution of Maxwell equations for the magnetic field and the magnetic stress calculations. Complex geometries are treated with an adaptive Cartesian embedded boundary using the volume-of-fluid method for piecewise linear reconstruction of the particle geometries. I will then discuss the self-assembly of particles at fluid-fluid interfaces, which is of considerable interest for studying the collective behavior of colloidal suspensions at small scales, and then the future work to study the spatial arrangement and clustering of paramagnetic particles on fluid-fluid interfaces.