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Abstract: The Ehrhart polynomial of a convex lattice polytope P counts the number of integer points in integral dilates of P. We will discuss what information about P can be obtained by expanding the Ehrhart polynomial in different bases, and give bounds on the possible location of the roots of Ehrhart polynomials in the complex plane and on the real line. We close by discussing the Ehrhart polynomials of lattice tetrahedra with few interior points.
Colloquium - 16:00
Abstract: Recently Bokowski and Gevay presented a new family Emn of self-dual 3-spheres based on a construction of Gevay. This family contains in particular the 24-cell.
In my talk I will show that this family of spheres is a very special case of the ``E-construction'' invented by Eppstein, Kuperberg and Ziegler for the construction of 2-simple and 2-simplicial 4-polytopes. The family of spheres Emn can be derived from products Cm x Cn of two polygons Cm and Cn with m and n vertices. We prove that all the Emn are polytopal with flexible geometric realizations. E44 is the 24-cell. We look at the projective realisation spaces of the two cases E33 and E44 in detail. We show that their respective dimensions are at least nine for E33 and four in the case of the 24-cell.