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Abstract: A wide variety of topics in pure and applied mathematics involve the problem of counting the number of lattice points inside a region in Euclidean space. Applications range from the very pure (Number theory, toric Hilbert functions, Kostant's partition function in representation theory) to the applied (cryptography, integer programming, contingency tables).
Perhaps the most basic case is when the region is a convex bounded polyhedron, for short called polytopes. This talk is a survey of this exciting and useful problem. Our goal will be to explain to an audience of non-experts the basic structure theorems about these counting problem. Perhaps the most famous special case are the so called, Ehrhart quasipolynomials, introduced in the 1960s by Eugene Ehrhart. Ehrhart quasipolynomials count the number of lattice points in the different integral dilations of a rational convex polytope. Toward the end of the talk, we present a new very general version of Ehrhart's theorem recently found by the speaker. We conclude with a look at what happens when counting lattice points in more complicated regions of space.
Colloquium - 16:00
Abstract: Let S be a planar point set. A decomposition of S into polygons is a set of internally disjoint polygons such that their union is the convex hull of S and all vertices are in S. In a convex-pseudo decompositions, all polygons are either convex polygons or pseudotriangles. We are interested in the minimal number of elements that a decomposition of an n-element point set can have. A minimal pseudotriangulation on n vertices has n-2 elements. For convex decompositions, some point sets require at least n+2 elements. For convex-pseudo decompositions, we have an upper bound of 0.7n.
Joint work with Bettina Speckmann and Oswin Aichholzer