Students will be able to build semidefinite programming hierarchies for a wide variety of problems, and apply tools from representation theory and (real) algebraic geometry in practice. 

Lecture

Semidefinite programming, a generalization of linear programming, is a surprisingly rich and quickly growing sub-field of optimization. Initially introduced for combinatoric questions about graphs, the method has since found applications in (real) algebraic geometry, PDEs, discrete geometry, quantum information theory and much more. In this lecture the focus will be on a particularly interesting subclass of problems: high dimensional problems exhibiting symmetries. To be able to attack these problems computationally, results from representation theory, algebraic combinatorics and harmonic analysis are required.Preliminary list of contents:Introduction to semidefinite programming and its duality theory: the theta-number of a graph.Goemanns-Williamson algorithm for Max-Cut.The Shannon-capacity of a graph: Bounding the independence number of an infinite graph.Error-correcting-codes: Symmetries and the Delsarte-LP bound.The regular representation of Matrix-*-Algebras and the crossing number of a graph.Energy-minimization and universal optimality, Bochner's theorem.Representation theory with focus on Artin-Wedderburn theory.Spherical harmonics: Bounds for the Kissing number (towards Maryna Viazovska's 2022 fields medal).Polynomial optimization and real algebraic geometry.Representation stability: Infinite dimensional and dimension-free optimization.

Good understanding of linear algebra and group theory. No knowledge of linear optimization or representation theory is required.

M. Laurent and F. Vallentin: Semidefinite OptimizationF. Vallentin: Semidefinite programs and harmonic analysisB. E. Sagan: The Symmetric GroupRelevant papers and additional references will be introduced in the lecture.

Oral exam.

Contents of the class.

Knowledge of the contents of the class.