Title: Ehrhart polynomials with negative coefficients

Abstract: The Ehrhart polynomials of integral convex polytopes count integer points under dilations of the polytopes. In this talk, I will discuss the possible sign patterns of the coefficients of Ehrhart polynomials of integral convex polytopes. While the leading terms, the second leading terms and the constant of Ehrhart polynomials are always positive, the other terms aren't necessarily positive.  In fact, some examples of Ehrhart polynomials with negative coefficients were known before. For arbitrary dimension, I will describe a construction of Ehrhart polynomials with negative coefficients.