Instructor:
Alexander Barvinok
1858 East Hall, tel.: 763-0294;
e-mail: barvinok@umich.edu;
http://www.math.lsa.umich.edu/~barvinok
Time and Place: Tuesday and Thursday 1:00 p.m. -- 2:30 p.m., 3096 East Hall
Office hours: Wednesday 2-3 p.m., Thursday 3-4 p.m. and Friday 1-2 p.m. or by appointment or by chance in 1858 East Hall
Prerequisite: good knowledge of linear algebra
Convexity is very easy to define formally and to understand intuitively. It plays an important role in number theory, algebraic geometry, analysis, optimization, control theory, and mathematical economics. The goal of this course is to demonstrate the unifying power of convexity on a variety of applications, chosen from the following topics.
1. Classical structural results: Theorems of Caratheodory and Helly, their generalizations, separation theorems and the Krein-Milman Theorem, their applications in analysis, combinatorics, and optimization. Theorems of Birkhoff - von Neumann and Schur - Horn.
2. Positive semidefinite convexity. The structure of the cone of positive semidefinite matrices, applications to metric embedding problems and to quadratic convexity theorems of Brickmann and Toeplitz - Hausdorff. Applications of semidefinite programming to codes, sphere packing, graph coloring, connections to Lovasz' theta-function and Grothendieck's inequality.
3. Hyperbolic polynomials and associated convex cones. The recent proof of Gurvits of a far-reaching generalization of the van der Waerden conjecture for permanents.
Some of the more classical material can be found in "A Course in Convexity" by A. Barvinok. For newer developments, lecture notes will be posted.
Grading: we will have a number of homework problem sets
Text: There is no required text. Recommended text: A. Barvinok, "A Course in Convexity"