Please come join us for coffee, tea, snacks, and pleasant conversation!

Please come join us for coffee, tea, snacks, and pleasant conversation!

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. 

Wesley Hough will present the documentary "In the Footsteps of Newton", which he worked on while he was a undergraduate at Hanover College.  For more details on the documentary visit the following link.  Come and hangout with other mathematically minded students at the first math club meeting of the semester. There will be pizza.

Title: The Waldschmidt constant

Abstract:  A (projective) variety V is a set of common zeros of the polynomials in an ideal I that is generated by homogenous polynomials. Given the generators of the ideal I,  one would like to know the minimum degree of a polynomial F such that each point of V is a root of f of a given multiplicity, say k. As this is often a difficult problem one studies first the corresponding question for large k. This leads to the Waldschmidt constant, which gives an asymptotic answer to the problem.

If I is a an ideal that is generated by squarefree monomials, then the Waldschmidt constant can be expressed as the optimal solution to a linear program or as a fractional chromatic number. This leads to the new bounds and computations of the Waldschmidt constant.

No prior knowledge of monomial ideals or graph theory is assumed. All concepts will be explained in the talk.