Georgia Institute of Technology College of Sciences

I will review a general framework for developing numerical methods working with non-parametrically defined surfaces for various problems. In this talk, I will focus on boundary integral equations. The main idea is to formulate appropriate extensions of a given problem defined on a surface to ones in the narrow band of the surface in the embedding space. The extensions are arranged so that the solutions to the extended problems are equivalent, in a strong sense, to the surface problems that we set out to solve. Such extension approaches allow us to analyze the well-posedness of the resulting system, develop, systematically and in a unified fashion, numerical schemes for treating a wide range of problems involving differential and integral operators, and deal with similar problems in which only point clouds sampling the surfaces are given. At the end of this talk, I will mention our work in developing multilevel neural network methods for inverting dense and large matrices that arise from boundary integral equations.

The capillary energy functional is used to model the equilibrium shape of a liquid drop meeting a substrate at a prescribed interior contact angle. We will discuss a rigidity theorem for volume-preserving critical points of the capillary energy in the half-space: among all sets of finite perimeter, every such critical configuration corresponding to a prescribed contact angle between 90 degrees and 120 degrees must be a finite union of spheres and spherical caps with the correct contact angle. Assuming that the tangential part of the capillary boundary is $\mathcal{H}^n$-null, this rigidity extends to the full hydrophobic range of contact angles between 90 degrees and 180 degrees. We will also present an anisotropic counterpart, establishing rigidity under suitable lower density assumptions. This talk is based on joint work with A. De Rosa and R. Resende.
 

A well known result in graph theory states that a graph is connected if and only if the second eigenvalue of its Laplacian matrix is positive. In fact, the larger the second eigenvalue, the more connected the graph is. By varying the weights on edges, one can in general increase the second eigenvalue which in turn affects many graph properties such as expansion, mixing times of random walks etc.

In this talk, I will introduce conformally rigid graphs, which are those unweighted undirected graphs in which one cannot increase the second eigenvalue or decrease the largest eigenvalue by changing
weights. This notion turns out to be deeply connected to graph embeddings, semidefinite programming and other ideas in geometry, optimization and combinatorics.

Joint work with Joao Gouveia and Stefan Steinerberger

Modern homology theories have given many knot invariants with the following useful properties: they are additive with respect to connected sum, they give a lower bound for a knot's slice genus, and this lower bound is equal to the slice genus for torus knots. These invariants, called slice-torus invariants, include the Ozsváth–Szabó $\tau$ and Rasmussen $s$ invariants. We discuss how, on a large class of knots, the value of a slice-torus invariant is fully determined by these properties, and can be computed without reference to the homology theory. We also discuss results that follow from the existence of slice-torus invariants, and a potential connection to the smooth 4-dimensional Poincaré conjecture.