Georgia Institute of Technology College of Sciences

In first-passage percolation (FPP), one assigns i.i.d. weights to the edges of the cubic lattice Z^d and analyzes the induced weighted graph metric. If T(x,y) is the distance between vertices x and y, then a primary question in the model is: what is the order of the fluctuations of T(0,x)? It is expected that the variance of T(0,x) grows like the norm of x to a power strictly less than 1, but the best lower bounds available are (only in two dimensions) of order \log |x|. This result was found in the '90s and there has not been any improvement since. In this talk, we discuss the problem of getting stronger fluctuation bounds: to show that T(0,x) is with high probability not contained in an interval of size o(\log |x|)^{1/2}, and similar statements for FPP in thin cylinders. Such a statement has been proved for special edge-weight distributions by Pemantle-Peres ('95) and Chatterjee ('17). In work with J. Hanson, C. Houdré, and C. Xu, we extend these bounds to general edge-weight distributions. I will explain some of the methods we use, including an old and elementary "small ball" probability result for functions on the hypercube.

The celebrated Hadwiger's theorem says that linear combinations of intrinsic volumes on convex sets are the only isometry invariant continuous valuations(i.e. finitely additive measures). On the other hand H. Weyl has extended intrinsic volumes beyond convexity, to Riemannian manifolds. We try to understand the continuity properties of this extension under theGromov-Hausdorff convergence (literally, there is no such continuityin general). First, we describe a new conjectural compactification of the set of all closed Riemannian manifolds with given upper bounds on dimensionand diameter and lower bound on sectional curvature. Points of thiscompactification are pairs: an Alexandrov space and a constructible(in the Perelman-Petrunin sense) function on it. Second, conjecturally all intrinsic volumes extend by continuity to this compactification. No preliminary knowledge of Alexandrov spaces will be assumed, though it will be useful.

Valuations are finitely additive measures on convex compact subsets of a finite dimensional vector space. The theory of valuations originates in convex geometry. Valuations continuous in the Hausdorff metric play a special role, and we will concentrate in the talk on this class of valuations. In recent years there was a considerable progress in the theory and its applications. We will describe some of the progress with particular focus on the multiplicative structure on valuations and its applications to kinematic formulas of integral geometry.