Colloquium Series

Spring 2025

April 11, 2025

Amol Aggarwal, Clay Mathematics Institute and Columbia University

Topic: The Toda Lattice as a Soliton Gas
Time: 4:00-5:00 pm 
Location:  LL08 Barnum Complex
Reception:  JCC501
Abstract: A basic tenet of integrable systems is that, under sufficientl yirregular initial data, they can be thought of as dense collections of many solitons, or “soliton gases.” In this talk we focus on the Toda lattice, which is an archetypal example of an integrable Hamiltonian dynamical system. We explain how the system, under certain random initial data, can be interpreted through solitons, and provide a framework for studying how these solitons asymptotically evolve in time. The arguments use ideas from random matrix theory, particularly the analysis of Lyapunov exponents governing the decay rates of eigenvectors of random tridiagonal matrices.

March 7, 2025

Jen Hom, Georgia Institute of Technology

Topic: 3-manifolds, groups, and Heegaard Floer homology
Time: 4:00-5:00 pm 
Location: JCC 270
Reception: JCC 501
Abstract: We will consider various ways to build 3-manifolds. Under the operation of connected sum, the set of 3-manifolds forms a monoid, and modulo an appropriate equivalence relation, this monoid becomes a group. What is the structure of this group? What families of three-manifolds generate (or don’t generate) this group? We give some answers to these questions using Heegaard Floer homology. This is joint work with (various subsets of) I. Dai, K. Hendricks, M. Stoffregen, L. Truong, and I. Zemke.

February 7, 2025

Nick Trefethen, Harvard University

Topic: Rational Approximation and the AAA Algorithm
Time: 4:00-5:00 pm 
Location: JCC 270
Reception: JCC 501
Abstract: Approximation by rational functions used to be mainly a theoretical subject, but with the introduction of the AAA algorithm in 2018, it became computationally practical and indeed easy.  The implications for what we can do numerically are enormous.  This talk will outline the algorithm and demonstrate its application to a collection of problems.  We can also use it to demonstrate the potential theory that underlies the theory of rational approximation, a topic that goes back to Joseph Walsh at Harvard nearly a century ago.

Fall 2024

November 1, 2024

Lenore Cowen, Department of Computer Science and Department of Mathematics, Tufts University

Topic: Pathways for Learning from Structure and Organization of Protein Interaction Networks 

Time: 4:00-5:00 pm    
Location: 113 Breaker Hall   
Reception: JCC 501 5:00-6:00 pm    
Abstract: Biological networks are powerful resources for the discovery of genes and genetic modules that drive disease. Similar to in the social networks analysis domain, diffusion-based low-dimensional network embedding methods have proved quite powerful for biological networks. In particular, these methods have been highly successful in creating coherent local neighborhoods that correlate to gene function, enabling the downstream use of the entire machine learning toolbox to perform multiple inference tasks on these networks. We highlight the recent success of diffusion-based methods in biological networks for gene function prediction, link prediction, and for disease module prediction. We also discuss structural ways in which some types of biological networks seem to be organized differently than social networks, and show the advantage of customizing methods for particular types of biological network data. Also,  biological domain offers unique additional sources of principled structured data organization (such as evolution), that can be leveraged to improve inference, resulting in some interesting open problems in multiplex network inference.  If time permits, we will present some very recent work on how we are using these methods to understand coral reef resilience to heat stress in the Anthropocene.

October 18, 2024

Marc Hodes, Department of Mechanical Engineering and Department of Mathematics, Tufts University

Topic: Analytical Solutions Elucidating Transport Phenomena on Superhydrophobic Surfaces

Time: 4:00-5:00 pm    
Location: JCC 270    
Reception: JCC 501 5:00-6:00 pm    
Abstract: Superhydrophobic surfaces (SHs) are regularly microfabricated to mimic the lubricating properties of “self-cleaning” lotus leaves by trapping a gas layer beneath a portion of the liquid flowing over them. Technological applications exploiting them include overcoming diYusion- limited mixing within mm-scale droplets of liquid for chemical assays and pumping of liquid in microfluidic circuits. The analysis of momentum/heat/mass/charge transport phenomena on SHs is mathematically challenging because of mixed Dirichlet-Neumann boundary conditions as a consequence of (flat) liquid-solid interfaces adjacent to (curved) liquid-gas interfaces. Over the past decade we have applied mathematical techniques (asymptotics, conformal maps, boundary perturbations, reciprocity, etc.) to resolve a suite of problems. I will describe our solution approach to one such problem, viz., heat transfer to a liquid flowing over a SH textured with ridges (on which the liquid is suspended) that are oriented transverse to the flow. Then, I will discuss the ramifications of our results on the direct liquid cooling of microelectronics, whereby a liquid metal is pumped through the Silicon comprising a microprocessor and related physical experiments in our lab. Finally, I will, briefly, discuss the solutions we have obtained to other problems related to transport phenomena on SHs.