2012-2013 Milliman Lectures
April 16 - 18, 2013
Smith Hall, Room 120
Bio: Bernd Sturmfels received doctoral degrees in Mathematics in 1987 from the University of Washington, Seattle, and the Technical University Darmstadt, Germany. After postdoctoral years in Minneapolis and Linz, Austria, he taught at Cornell University, before joining UC Berkeley in 1995, where he is Professor of Mathematics, Statistics and Computer Science. His honors include a National Young Investigator Fellowship, a Sloan Fellowship, and a David and Lucile Packard Fellowship, a Clay Senior Scholarship, an Alexander von Humboldt Senior Research Prize, the SIAM von Neumann Lecturership, and a Sarlo Distinguished Mentoring Award. Recently, he served as Vice President of the American Mathematical Society. A leading experimentalist among mathematicians, Sturmfels has authored ten books and over 200 research articles, in the areas of combinatorics, algebraic geometry, symbolic computation and their applications. He has mentored 35 doctoral students and numerous postdocs. His current research focuses on algebraic statistics and computational algebraic geometry.
Tuesday: Lecture I-Convex Algebraic Geometry
Abstract: 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 undergraduate students.
Wednesday: Lecture II-Maximum Likelihood for Matrices with Rank Constraint
Abstract: 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 problems 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, and we present a remarkable duality theorem due to Draisma and Rodriguez.
Thursday: Lecture III-Tropicalization of Classical Moduli Spaces
Abstract: Algebraic geometry is the study of solutions sets to polynomial equations. Solutions that depend on an infinitesimal parameter are studied combinatorially by tropical geometry. Tropicalization works especially well for varieties that are parametrized by monomials in linear forms. Many classical moduli spaces (for curves of low genus and few points in the plane) admit such a representation, and we here explore their tropical geometry. Examples to be discussed include the Segre cubic, the Igusa quartic, the Burkhardt quartic, and moduli of marked del Pezzo surfaces. Matroids, hyperplane arrangements, and Weyl groups play a prominent role. Our favorites are E6, E7 and G32.
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