Title: Furstenberg correspondence and invariants of simplices
Abstract: Given a simplex with vertices in Z^r (i.e. the convex hull of r+1 many points), there is an associated function which maps a positive integer n to the number of Z^r-lattice points in the dilated simplex nP. A classical result tells us that this function is in fact a polynomial called the Ehrhart polynomial, and its coefficients carry rich information about the simplex, such as its volume and the surface area of the boundary. I will review the main results of the recent preprint “Ehrhart spectra of large subsets of Z^r” by Michael Björklund, Rickard Cullman and Alexander Fish (and perhaps a related paper), where it is demonstrated using the Furstenberg correspondence principle that to a large subset E of Z^r we can associate a positive integer n, such that every simplex with vertices in nZ^r has the same Ehrhart polynomial as some simplex with vertices in E. No prior knowledge in combinatorics will be assumed.