Mathematics

Tilings in vertex ordered graphs

Lina Li (University of Waterloo)

Abstract: Over recent years, there has been much interest in both Turan and Ramsey properties of vertex ordered graphs. In this talk, we initiate the study of embedding spanning structures into vertex ordered graphs. In particular, we introduce a general framework for approaching the problem of determining the minimum degree threshold for forcing a perfect $H$-tiling in an ordered graph. (In the unordered graph setting, this problem was resolved by Kuhn and Osthus in 2009.) We use our general framework to resolve the perfect $H$-tiling problem for all ordered graphs $H$ of interval chromatic number $2$. Already in this restricted setting the class of extremal examples is richer than in the unordered graph problem. This is joint work with Jozsef Balogh and Andrew Treglown.

Please contact Sean at SEnglish (at) illinois (dot) edu for the Zoom information.

Empirical measures, geodesic lengths, and a variational formula in first-passage percolation 

Erik Bates (University of Wisconsin-Madison)

Abstract: We consider the standard first-passage percolation model on Z^d, in which each edge is assigned an i.i.d. nonnegative weight, and the passage time between any two points is the smallest total weight of a nearest-neighbor path between them.  Our primary interest is in the empirical measures of edge-weights observed along geodesics from 0 to ne_1.  For various dense families of edge-weight distributions, we prove that these measures converge weakly to a deterministic limit as n tends to infinity.  The key tool is a new variational formula for the time constant.  In this talk, I will derive this formula and discuss its implications for the convergence of both empirical measures and lengths of geodesics.