Algebra, Geometry, and Combinatorics Seminar

Title: Grassmann and flag varieties in linear algebra, optimization, and statistics: an algebraic perspective

Abstract: Grassmann and flag varieties lead many lives in pure and applied mathematics. In this talk,  I will focus on the algebraic complexity of solving various problems in linear algebra and statistics as optimization problems over these varieties. The measure of the algebraic complexity is the amount of complex critical points of the corresponding optimization problem. After an exposition of different realizations of these manifolds as algebraic varieties I will present a sample of optimization problems over them and compute their algebraic complexity. This is based on joint work with Hannah Friedman. 

Title: Curves, surfaces, and 3-manifolds

Abstract: In this talk, I will give a gentle introduction to the beautiful world of geometry and topology in low dimensions. We will start with some historical work and perspectives of Thurston and end with some more modern results and research directions (in the same vein). My goal is to make this talk accessible to a broad audience so I will include many pictures/cartoons and aim to focus on building intuition rather than fleshing out technicalities.

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Title: What is an abstract commensurator?

Abstract: Groups are the symmetries of things, but a group is also a highly symmetric object itself. The automorphism group of a group consists of all isomorphisms f : G –> G and there is a rich theory of automorphism groups. Groups are richer still: containing many hidden symmetries, isomorphisms between finite-index subgroups. These hidden symmetries also form a group, the abstract commensurator of the group. The goal of this talk is to introduce the abstract commensurator from both algebraic and geometric points of view, illustrated with the central examples of the integers and the fundamental group of a torus. We will conclude with some open questions in current research on commensurators. 

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