Convexity over lattices and discrete sets: new theorems on Minkowski's Geometry of Numbers.

School of Mathematics Colloquium
Monday, February 8, 2016 - 4:00pm for 1 hour (actually 50 minutes)
Skiles 005
Jesus De Loera – University of California, Davis
Greg Blekherman
Convex analysis and geometry are tools fundamental to the foundations of several applied areas (e.g., optimization, control theory, probability and statistics), but at the same time convexity intersects in lovely ways with topics considered pure (e.g., algebraic geometry, representation theory and of course number theory). For several years I have been interested interested on how convexity relates to lattices and discrete subsets of Euclidean space. This is part of mathematics H. Minkowski named in 1910 "Geometrie der Zahlen''. In this talk I will use two well-known results, Caratheodory's & Helly's theorems, to explain my most recent work on lattice points on convex sets. The talk is for everyone! It is designed for non-experts and grad students should understand the key ideas. All new theorems are joint work with subsets of the following mathematicians I. Aliev, C. O'Neill, R. La Haye, D. Rolnick, and P. Soberon.