Georgia Institute of Technology College of Sciences

The uniformization theorem states that every Riemann surface is a quotient of some subset of the complex projective line by a group of Mobius transformations. However, a number of closely related questions regarding the structure of uniformization maps remain open. For example, it is unclear how one might associate a uniformizing map to a given Riemann surface. In this talk we will discuss an approach to this question due to Gunning by attaching a projective line bundle to a Riemann surface and studying its analytic properties.

 Nordhaus and Gaddum proved in 1956 that the sum of the chromatic number  of a graph G and its complement is at most |G|+1.  The Nordhaus-Gaddum graphs are the class of graphs satisfying this inequality with equality, and are well-understood. In this paper we consider a hereditary generalization: graphs G for which all induced subgraphs H of G satisfy that the sum of the chromatic numbers of H and its complement are at least |H|. We characterize the forbidden induced subgraphs of this class and find its intersection with a number of common classes, including line graphs. We also discuss chi-boundedness and algorithmic results.

The stein variational gradient descent (SVGD) algorithm is a deterministic particle method for sampling. However, a mean-field analysis reveals that the gradient flow corresponding to the SVGD algorithm (i.e., the Stein Variational Gradient Flow) only provides a constant-order approximation to the Wasserstein gradient flow corresponding to the KL-divergence minimization. In this work, we propose the Regularized Stein Variational Gradient Flow, which interpolates between the Stein Variational Gradient Flow and the Wasserstein gradient flow. We establish various theoretical properties of the Regularized Stein Variational Gradient Flow (and its time-discretization) including convergence to equilibrium, existence and uniqueness of weak solutions, and stability of the solutions. We provide preliminary numerical evidence of the improved performance offered by the regularization.

Ice sheets are fascinating dynamical systems that flow, fracture and melt on a wide range of time scales, presenting a range of challenging prediction problems with important implications for how coastal communities plan for sea level rise. In this talk, I will introduce a few outstanding problems concerning the evolution of Earth’s ice sheets under climate change. I will start by introduce the classical theory of “marine ice sheet instability” which describes how glacier ice flows from the land to ice which floats on the ocean, and leads to a saddle-node bifurcation in ice sheet size under climate change. Many contemporary predictions of ice sheet change hold that such a bifurcation is currently unfolding at a number of glaciers in Greenland and Antarctica and could lead to runaway ice sheet retreat even if global temperatures stop increasing in the future. I discuss our recent work on whether this bifurcation may actually play out as a sliding-crossing bifurcation, and the role of a stochastic climate system in driving the system through this bifurcation where nonlinearities cause evolution of the leading order moments of the distribution of glacier state.