Georgia Institute of Technology College of Sciences

We study the global existence of classical solutions to the incompressible viscous MHD system without magnetic diffusion in 2D and 3D. The lack of resistivity or magnetic diffusion poses a major challenge to a global regularity theory even for small smooth initial data. However, the interesting nonlinear structure of the system not only leads to some significant challenges, but some interesting stabilization properties, that leads to the possibility of the theory of global existence of classical and/or strong solutions. This talk is based on joint works with Yi Zhou, Yi Zhu, Shijin Ding, Xiaoying Zeng, and Jingchi Huang.

Ion channels are a class of proteins embedded in biological membranes, acting as biological devices or 'valves' for cells and playing a critical role in controlling various biological functions. To compute macroscopic ion channel kinetics, such as Gibbs free energy, electric currents, transport fluxes, membrane potential, and electrochemical potential, Poisson-Nernst-Planck ion channel (PNPIC) models have been developed as systems of nonlinear partial differential equations. However, they are difficult to solve numerically due to solution singularities, exponential nonlinearities, multiple physical domain issues, and the requirement of ionic concentration positivity. In this talk, I will present the recent progress we made in the development of finite element methods for solving PNPIC models. Specifically, I will introduce our improved PNPIC models and describe the mathematical and numerical techniques we utilized to develop efficient finite element iterative methods. Additionally, I will introduce the related software packages we developed for a voltage-dependent anion-channel protein and a mixture solution of multiple ionic species. Finally, I will present numerical results to demonstrate the fast convergence of our iterative methods and the high performance of our software package. This work was partially supported by the National Science Foundation through award number DMS-2153376 and the Simons Foundation through research award 711776.

This talk will be an introduction to the theory of surfaces, some tools we use to study surfaces, and some uses of surfaces in "real life". In particular, we will discuss the mapping class group and the curve complex. This talk will be aimed at an audience with a minimal background in low-dimensional topology. 

Contact 3-manifolds arise organically as boundaries of symplectic 4-manifolds, so it’s natural to ask: Given a contact 3-manifold Y, does there exist a symplectic 4-manifold X filling Y in a compatible way? Stein fillability is one such notion of compatibility that can be explored via open books: representations of a 3-manifold by means of a surface with boundary and its self-diffeomorphism, called a monodromy. I will discuss joint work with Andy Wand in which we exhibit first known Stein-fillable contact manifolds whose supporting open books of genus one have non-positive monodromies. This settles the question of correspondence between Stein fillings and positive monodromies for open books of all genera. Our methods rely on a combination of results of J. Conway, Lecuona and Lisca, and some observations about lantern relations in the mapping class group of the twice-punctured torus.