You are cordially invited to our next Monday Lecture &
Colloquium on July 5th at 14:15 h & 16:00 h, online via Zoom. Invitation link: Zoom - Invitation Time: Monday, July 5th - 14:15 h Lecture: Papa Sissokho (Illinois State University) Title: Geometry of the minimal solutions of a linear Diophantine Equation Abstract:Let a_{1},...,a_{n} and b_{1},...,b_{m} be fixed positive integers, and let S denote the set of all nonnegative integer solutions of the equation x_{1}a_{1}+...+x_{n}a_{n}=y_{1}b_{1}+...+y_{m}b_{m}. A solution (x_{1},...,x_{n},y_{1},...,y_{m}) in S is called minimal if it cannot be expressed as the sum of two nonzero solutions in S. For each pair (i,j), with 1 ≤ i ≤ n and 1 ≤ j ≤ m, the solution whose only nonzero coordinates are x_{i} = b_{j} and y_{j} = a_{i} is called a generator. We show that every minimal solution is a convex combination of the generators and the zero-solution. This proves a conjecture of Henk-Weismantel and, independently, Hosten-Sturmfels. Break! Time: Monday, July 5th - 16:00 h s.t. Colloquium: Ansgar Freyer (Technische Universität Berlin) Title: Shaking a convex body in order to count its lattice points Abstract:A key step in the proof is a technique from convex geometry known as Blascke's shaking procedure by which the problem can be reduced to anti-blocking bodies, i.e., convex bodies that are "located in the corner of the positive orthant". As a corollary of
our result, we will obtain an upper bound on the number of lattice
points in K in terms of the successive minima, which is equivalent
to Minkowski's Second Theorem, giving a partial answer to a
conjecture by Betke et al. from 1993.
This is a joint work with Eduardo Lucas Marín. |