# Faculty Research Talks - Spring 2021

Talks will be held at 2:30PM on every other Monday (M) at 2:30PM via WebEx unless otherwise noted. Please see more information below:

Talks will be held at 2:30PM on every other Monday (M) at 2:30PM via WebEx unless otherwise noted. Please see more information below:

**An Emergent Autonomous Flow for Spin Glass Dynamics**

We study the large size limiting dynamics of symmetric and asymmetric spin-glass models of size N. These are stochastic interacting particle systems with random connections, of zero mean, and relatively high variance. They are generally considered to be prototypes of high dimensional stochastic systems navigating a complex disordered environment, with diverse applications, including magnetism, colloids, neuroscience, and deep-learning algorithms. Existing work on the large size limiting dynamics has established delayed integro-differential equations that are very difficult to analyze rigorously. The content of this work is to derive an emergent flow operator that is autonomous, and hence more amenable to standard PDE techniques. It is proved that the same flow operator holds for a wide range of initial conditions, including when the distribution of the initial values of the spins depend on the realization of the connections. In particular, the flow operator is accurate for chaotic initial conditions (a `deep quench'), and initial conditions given by the long-time equilibrium Gibbs measure. The analysis is in terms of the double empirical process: this contains both the spins, and the field felt by each spin, at a particular time (without any knowledge of the correlation history), spanning M `replicas' with identical connections and independent stochasticity. Preliminary numerical results suggest that, as the temperature is lowered, a `dynamical phase transition' occurs when a stable fixed point of this flow destabilizes (as long as there are more than two replicas).

**Finite-Time Singularity Formation in the Generalized Constantin-Lax-Majda Equation**

This talk has two parts. In the first part, I will provide some backgound on the problem of singularity formation in the 3D Euer equations, which is considered one of the most outstanding open problems in mathematics and physics. In the second part, I will investigate the possibility of finite-time singularity formation in the generalized Constantin-Lax-Majda (gCLM) equation. This equation was originally introduced by Constantin, Lax and Majda as a 1D model to study singularity formation in the 3D Euler equations, and was later generalized by Okamoto, Sakajo, and Wensch to include a term that models the advection of vorticity. The gCLM equation has generated intense interest, but relatively little is known about singularity formation. I will discuss recent analytical and numerical results, obtained in collaboration with Pavel Lushnikov (U. New Mexico) and Denis Silantyev (NYU), which give a nearly complete description of singularity formation over a wide range of parameter values.

**Dynamical Systems Analysis of Walking Droplet Models**

After a brief description of past and current development and analysis of mathematical models for walking droplet phenomena, we adumbrate our recent and envisaged research in these areas. We outline our plan to conduct a detailed, dynamical systems-based investigation of walking droplet models, both continuous and discrete, ranging from the fundamentals of well-posedness, to bifurcation mechanisms, transitions to chaotic dynamics and possible existence of chaotic

strange attractors, most of which have been observed in experiments and simulations of extant models. There are also areas of walking droplet research where reliable dynamical models are still needed, such as in environments with various types of obstacles, in which we intend to try our hand at model formulation and testing.

**Using Applied Math to Study Interesting Problems in Math Biology**

In this talk I will use two examples from my current research projects to illustrate how applied math can be used to study interesting problems in math biology. I will focus on (1) centrosome oscillation in cell division, and (2) training a swimmer in a viscous fluid using re-enforcement Q-learning. In the first example I aim to show you how results from classical dynamical system may be useful for understanding the role of large noises during cell division. In the second example, I aim to illustrate how to use re-enforcement Q-learning to train a simple swimmer immersed in a viscous solvent.

**Title: TBA**

*Abstract: TBA*

*Updated: March 16, 2021*