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
Navigating complex environments: from microbes to fish to robots
Active or motile agents and their collective motion are ubiquitous in nature and can be found at every scale, from microbial colonies to fish schools, bird flocks, and human crowds. The collective behavior emerges due to each agent interacting with others and the surrounding environment, but even if each interaction is simple per se, their sum can be non-trivial and difficult to predict. The interactions vary from mechanical, to chemical, visual or auditory cues, and even conscious decisions. Better understanding of these interactions and how their interplay gives rise to the observed dynamics can help us develop ways to control and direct these dynamics in many applications. Moreover, these discoveries are inspiring the design and development of synthetic active agents with desirable functionalities, e.g. micro-agents that can self-assemble or robots that can collectively complete a given task.
I will give a broad overview of the field and will focus on the examples of micro-swimmer and fish, how they interact with others and boundaries, and how they individually and collectively move in complex terrains. In particular, I will describe the mathematical modeling approaches, the analysis or the models, the computational challenges that arise, and list a few open problems. Lastly, I will discuss how micro-swimmers and fish inspired us to design simple robots that are able to reliably navigate a complex terrain, and I will do some table-top demonstrations of the concepts with a few of such robots.
March 3
Location: CULM 611
Space-Filling Designs for Computer Experiments and Their Application to Big Data Research
Computer models are essential for exploring complex systems across nearly all natural and social science disciplines. Yet, their complexity often leads to high computational costs, prompting the use of computer experiments that develop statistical surrogate models from simulation-generated data. Among these techniques, space-filling designs have emerged as the most widely accepted approach, and I will provide a comprehensive introduction to them.
Moreover, standard statistical analyses of large-scale datasets frequently encounter chal- lenges like limited storage and computational expense. One practical solution is to perform these analyses on a carefully selected subset of the full dataset, making the method of subdata selection crucial. In the second part, I will discuss how the ideas behind space-filling designs can be applied to big data research, with a focus on choosing representative subdata for robust analysis.
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.