Georgia Institute of Technology College of Sciences

For some nonlocal PDEs, its steady states can be seen as critical points of an associated energy functional. Therefore, if one can construct perturbations around a function such that the energy decreases to first order along the perturbation, this function cannot be a steady state. In this talk, I will discuss how this simple variational approach has led to some recent progresses in the following equations, where the key is to carefully construct a suitable perturbation.

I will start with the aggregation-diffusion equation, which is a nonlocal PDE driven by two competing effects: nonlinear diffusion and long-range attraction. We show that all steady states are radially symmetric up to a translation (joint with Carrillo, Hittmeir and Volzone), and give some criteria on the uniqueness/non-uniqueness of steady states within the radial class (joint with Delgadino and Yan).

I will also discuss the 2D Euler equation, where we aim to understand under what condition must a stationary/uniformly-rotating solution be radially symmetric. Using a variational approach, we settle some open questions on the radial symmetry of rotating patches, and also show that any smooth stationary solution with compactly supported and nonnegative vorticity must be radial (joint with Gómez-Serrano, Park and Shi).

Numerical integration of given dynamic systems can be viewed as a forward problem with the learning of unknown dynamics from available state observations as an inverse problem. The latter has been around in various settings such as the model reduction of multiscale processes. It has received particular attention recently in the data-driven modeling via deep/machine learning. Indeed, solving both forward and inverse problems forms the loop of informative and intelligent scientific computing. A natural question is whether a good numerical integrator for discretizing prescribed dynamics is also good for discovering unknown dynamics. This lecture presents a study in the context of Linear multistep methods (LMMs).

In this talk we will discuss epidemic modeling in the context of COVID-19.  We will review the basics of classical epidemic models, and present joint work with Maria Chikina on the use of age-targeted strategies in the context of a COVID-19-like epidemic.  We will also discuss the broader roles epidemic modeling has played over the past year, and the limitations it as presented as a primary lens through which to understand the pandemic.

For a manifold M, the (generalized) Nielsen realization problem asks if the surjection Diff(M) → π_0 Diff(M) is split, where Diff(M) is the diffeomorphism group. When M is a surface, this question was posed by Thurston in Kirby's problem list and was addressed by Morita. I will discuss some more recent work on Nielsen realization problems with connections to flat fiber bundles, K3 surfaces, and smooth structures on hyperbolic manifolds.