Abstract: In tropical arithmetic, the sum of two numbers is their maximum and the product of two numbers is their usual sum. Many results familiar from algebra and geometry, including the Quadratic Formula, the Fundamental Theorem of Algebra, and Bezout's Theorem, continue to hold in the tropical world. In this lecture we learn how to draw tropical curves and why evolutionary biologists might care about this.
NB: A PDF version of this announcement (suitable for posting) is also available.
Abstract: This lecture concerns convex bodies with an interesting algebraic structure. A primary focus lies on the geometry of semideite optimization. Starting with elementary questions about ellipses in the plane, we move on to discuss the geometry of spectrahedra, orbitopes, and convex hulls of real varieties.
Abstract: The central curve of a linear program is the algebraic curve along which the interior point algorithms travel. We determine the degree, genus and defining ideal of this curve. These invariants, as well as the total curvature of the curve, are expressed in the combinatorial language of matroid theory. This is joint work with Jesus De Loera and Cynthia Vinzant.