Georgia Institute of Technology College of Sciences

The problem in the talk is motivated by the following problem. Suppose we need to place sprinklers on a field to ensure that every point of the field gets certain minimal amount of water. We would like to find optimal places for these sprinklers, if we know which amount of water a point $y$ receives from a sprinkler placed at a point $x$; i.e., we know the potential $K(x,y)$. This problem is also known as finding the $N$-th Chebyshev constant of a compact set $A$. We study how the distribution of $N$ optimal points (sprinklers) looks when $N$ is large. Solving such a problem also provides an algorithm to approximate certain given distributions with discrete ones. We discuss connections of this problem to minimal discrete energy and to potential theory.

For every surface (sphere, torus, etc.) there is an associated graph called the curve graph. The vertices are curves in the surface and two vertices are connected by an edge if the curves are disjoint. The curve graph turns out to be very important in the study of surfaces. Even though it is well-studied, it is quite mysterious. Here are two sample problems: If you draw two curves on a surface, how far apart are they as edges of the curve graph? If I hand you a surface, can you draw two curves that have distance bigger than three? We'll start from the beginning and discuss these problems and some related computational problems on surfaces.

In this lecture series, held jointly (via video conference) with the University of Buffalo and the University of Arkansas, we aim to understand the lecture notes by Vincent Guirardel on geometric small cancellation. The lecture notes can be found here: https://perso.univ-rennes1.fr/vincent.guirardel/papiers/lecture_notes_pcmi.pdf This week we will begin Lecture 4.

The compact transverse cross-sections of a cylinder over a central ovaloid in Rn, n ≥ 3, with hyperplanes are central ovaloids. A similar result holds for quadrics (level sets of quadratic polynomials in Rn, n ≥ 3). Their compact transverse cross-sections with hyperplanes are ellipsoids, which are central ovaloids. In R3, Blaschke, Brunn, and Olovjanischnikoff found results for compact convex surfaces that motivated B. Solomon to prove that these two kinds of examples provide the only complete, connected, smooth surfaces in R3, whose ovaloid cross sections are central. We generalize that result to all higher dimensions, proving: If M^(n-1), n >= 4, is a complete, connected smooth hypersurface of R^n, which intersects at least one hyperplane transversally along an ovaloid, and every such ovaloid on M is central, then M is either a cylinder over a central ovaloid or a quadric.