# The Vinberg Lecture

The Vinberg Distinguished Lecture Series aims at bringing together all mathematicians interested in wide dissemination of ideas from various domains of pure and applied mathematics.

The Vinberg Lecture Series: Lecture 5 (Dec 07, 2021) -------- Maryna Viazovska (EPF Lausanne, Switzerland) Title: The sphere packing problem Abstract: The sphere packing problem asks for the densest configuration of non-overlapping unit balls in space. In this talk I shall speak about the sphere packing problem in various spaces and its generalisations. The talk will focus on linear programming and semidefinite programming methods as powerful tools for analysing and, in some cases, completely solving geometric optimisation questions.

The Vinberg Lecture Series: Lecture 4 (Nov 30, 2021) -------- Peter Sarnak (IAS Princeton) Title: Reflection groups and monodromy Abstract: Vinberg's theory of reflection groups has wide applications. We discuss some of these, to monodromy groups of hypergeometric and Painlevé equations. The nonlinear case is intimately connected to affine Markoff surfaces and it is a central ingredient in the Diophantine analysis of these surfaces.

The Vinberg Lecture Series: Lecture 3 (Nov 23, 2021) -------- Dmitry Alekseevsky (IITP RAS, Moscow, Russia) Title: Special Vinberg cones and their application to Supergravity Abstract: In the early 1960s, Vinberg gave a description of homogeneous convex cones as cones of Hermitian positive definite matrices in a matrix T-algebra of (n \times n)-matrices whose diagonal entries are just real numbers, but off-diagonal elements belong to different vector spaces. It turns out that rank 3 special Vinberg cones (corresponding to Clifford algebras) have important applications to Supergravity. No special background is required. The talk is based on joint works with V. Cortes; and with A. Marrani and A. Spiro.

The Vinberg Lecture Series: Lecture 2 (Oct 19, 2021) -------- Alan Reid (Rice University, USA) Title: The geometry and topology of arithmetic hyperbolic manifolds of simplest type Abstract: This talk will survey as well as discuss geometric and topological properties of arithmetic hyperbolic manifolds of simplest type. These are precisely the class of arithmetic hyperbolic manifolds that contain an immersed co-dimension one totally geodesic submanifolds.