OS Reelle Geometrie und Algebra: Carathéodory theorem and its influence in Mathematics
Freitag, 28. Juni 2019
13:30 – 14:30 Uhr
Jesus A. De Loera (University of California, Davis)
Diese Veranstaltung ist Teil der Veranstaltungsreihe „Oberseminar Reelle Geometrie und Algebra“.
Convex geometry has been an important tool in several areas of mathematics, e.g., convexity appears in the study of polynomials. My talk will show this continues to be the case today by focusing one influential theorem, Carathéodory's theorem from 1905. In its most basic form it describes the size of a minimal linear combination representing a vector in a cone as a sum of others, and it is among the most fundamental results in Convex Geometry and it has seen many variations and extensions. I will review some variations of Carathéodory's theorem that have interesting applications. In particular, I will talk about integer versions, given a system $Ax=b, x \geq 0$, what is the size of the sparsest solution integer? I will mention some open problems too. This talk is geared for non-experts, but all new results are joint work with Iskander Aliev, Gennadiy Averkov, Timm Oertel, and Chris O'Neill.