Rothschild Distinguished Lectures in Mathematics and Computer Science

Lectures given by

Prof. Bernd Sturmfels

University of California, Berkeley

Monday , Jan 14, 2013 at 16:00

Lecture I: Convex Algebraic Geometry

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We introduce convex bodies with an interesting algebraic structure. A primary focus lies on the geometry of semidefinite 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.

This lecture has many beautiful pictures and can be enjoyed by undergraduates.

Tuesday, Jan 15, 2013 at 16:00

Lecture II:The Central Curve in Linear Programming

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Interior point methods in linear programming travel along the

central curve. We determine the degree, genus, and defining equations of this algebraic 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.

Wednesday, Jan 16, 2013 at 14:00

Lecture III:Maximum Likelihood for Matrices with Rank Constraints

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Maximum likelihood estimation is a fundamental computational task in statistics. We discuss this problem for manifolds of low rank matrices. These represent mixtures of independent distributions of two discrete random variables. This non-convex optimization leads to some beautiful geometry, topology, and combinatorics. We explain how numerical algebraic geometry is used to find the global maximum of the likelihood function.

This is joint work with Jon Hauenstein and Jose Rodriguez.