## Dmitry Vagner : Higher Dimensional Algebra in Topology

- Graduate/Faculty Seminar ( 241 Views )In his letter, "Pursuing Stacks," Grothendieck advocated to Quillen for the use of "higher" categories to encode the higher homotopy of spaces. In particular, Grothendieck dreamt of realizing homotopy n-types as n-groupoids. This powerful idea both opened the field of higher dimensional algebra but also informed a paradigm in which the distinction between topology and algebra is blurred. Since then, work by Baez and Dolan among others further surveyed the landscape of higher categories and their relationship to topology. In this talk, we will explore this story, beginning with some definitions and examples of higher categories. We will then proceed to explain "the periodic table of higher categories" and the four central hypotheses of higher category theory. In particular, these give purely algebraic characterizations of homotopy types, manifolds, and generalized knots; and account for the general phenomena of stabilization in topology. No prerequisites beyond basic ideas in algebraic topology will be expected.

## Dmitry Vagner : Introduction to Diagrammatic Algebra

- Graduate/Faculty Seminar ( 234 Views )We show how algebraic relations can be encoded in suggestive topological diagrams and use this to prove various algebraic equations in a purely pictorial way. We will first go over a few canonical examples: monoids, self-dual objects, Frobenius algebras, and monads. Then we will briefly discuss the underlying theory that makes this miracle rigorous.

## Jeff Jauregui : Geometry and topology in low dimensions: an introduction togeometric flows

- Graduate/Faculty Seminar ( 136 Views )This talk will be geared toward first and second year grad students and/or anyone with limited geometry experience. We will discuss the idea of curvature for curves and surfaces and the notion of "best metrics." The classical Uniformization Theorem will be introduced from a modern angle: Ricci flow. This will motivate studying the Ricci flow in dimension 3 as a tool to understand topology in terms of geometry. Time permitting, we will finish by discussing the Geometrization Theorem.