Faculty Research Talks - Spring 2026

February 2

Professor Jonathan Jaquette

Location: CULM 611

Dynamics, Topology, and Computation

Understanding the long term behavior of solutions is a fundamental question in any time evolving system. This begins with questions of existence, and whether local existence can be extended globally. Of the solutions which exist globally, coherent structures (such as equilibria, traveling waves and periodic orbits) serve as emblematic examples of how solutions may behave. However, nonlinear differential equations are rarely solvable by hand. Instead of searching for arbitrary solutions, the dynamical systems viewpoint typically begins by identifying basic landmark invariant sets, and then studying connecting orbits between these. Often, it is just the topology of these landmarks which is robust; changes in topology signal important changes in the system.

With abstract theorems one can describe in great detail the dynamics on and around generic invariant sets. However, for a specific differential equation, verifying the hypotheses of such a theorem often requires hard quantitative analysis. In this talk I will discuss various ways one can develop a global understanding of how complex systems change over time, and bridge the gap between what can be proven mathematically and what can be computed numerically, with the aid of computer-assisted proofs.

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February 16

Professor Christina Frederick

Location: CULM 611

Multi-Agent Path-Planning in a Moving Medium via Wasserstein Hamiltonian Flow

I’ll discuss a finite dimensional variational model for multi-agent path-planning in which a group of agents traverses from initial positions to a target distribution in a moving medium. The model is derived using the agent-based formulation of the Wasserstein Hamiltonian flow that transports between probability distributions while optimizing a running cost. The objective is the mismatch between the agents' final positions and the target distribution. The constraints are a system of Hamiltonian equations that provide the trajectories of the agents. The free variables on which the optimization is defined form a finite vector of the initial velocities for the agents. The model is solved numerically by the L-BFGS method in conjunction with a shooting strategy. Several simulation examples, including a time-dependent moving medium, are presented to illustrate the performance of the model.

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March 2

Professor Travis Askham

Location: CULM 611

Integral equation methods and bulk-surface PDEs

I will outline some of the main considerations in using integral representations to solve partial differential equations (PDEs). I will then describe some recent work on developing integral representations for bulk-surface PDEs, which are partial differential equation models whose boundary conditions also take the form of a PDE. The motivating examples  include applications in geophysics and acoustics.

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March 30

Professor Shahriar Afkhami

Location: CULM 611

Numerical Methods for Viscoelastic Fluid Flows

Materials that show a viscoelastic behavior are ubiquitous; examples include mucus, polymers, egg white, dough, mayonnaise, clay, concrete, gels, detergent soap, and toothpaste. Viscoelastic fluids exhibit a combination of both fluid and solid characteristics, and require a constitutive model for relating the deformation to stresses. In this talk, I will give an overview of numerical methods for simulating viscoelastic fluids and will discuss data-driven approaches that can be leveraged to not only accelerate the simulations of viscoelastic flows, but also learn the constitutive relation of such systems. I will then present examples of simulations using spatial discretizations for both governing differential equations and constitutive models and will discuss the Physics-Informed Neural Network approach to infer the complex, functional form of the constitutive equation for simulating viscoelastic fluids.

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April 13

Professor Xinyu Zhao

Location: CULM 611

Start Time: 3:00 PM 

Conformal mapping method in the study of free-surface water waves

In this talk, we will demonstrate how conformal maps are used in the study of two-dimensional free-surface water waves, particularly for simulating waves over variable bottom topography (i.e., the riverbed or ocean floor). We will discuss two numerical examples: a periodic water wave propagating over periodic bottom topography whose period is irrationally related to that of the wave, and a steady solitary wave over bottom topography with a localized bump.

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April 27

Professor Amitabha Bose

Location: CULM 611

Title/Abstract Forthcoming

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Last Updated: April 7, 2026