# Winter 2018

## Nicholas Meadows, Local Higher Category Theory, Cocycles and Torsors (Tuesday Jan 16, 2018)

Motivated by geometric models of concurrency, we introduce the path category of a simplicial set, and the Joyal model structure on simplicial sets. We explain the sheaf theoretic extension of the Joyal model structure. We introduce cocycle categories as a way of talking about the homotopy category of a right proper model category, and produce a sheaf theoretic version of the Bergner model structure which is right proper. The Quillen equivalence of the Joyal and Bergner model structure extends to the sheaf theoretic level. If $X$ is a presheaf of Kan complexes, we describe the maps $[*, X]$, in the homotopy category of the Jardine model structure. This is a consequence of the cocycle theory. We interpret this as non-abelian $H^{1}$ with coefficients in a sheaf of $\infty$-groupoids, generalizing Giraud’s non-abelian $H^{1}$. This talk is based on original research by the speaker.

## Sergio Chaves, Freeness of Equivariant Cohomology as a Module over $H^*(BG)$ (Tuesday Feb 13, 2018)

The talk will start with a brief introduction to the classical Borel's construction of equivariant cohomology for topological spaces together with an action of a group $G$, and as an extension of the ordinary singular cohomology in algebraic topology. Such cohomology inherits naturally a module structure over the cohomology ring $H^*(BG)$ where $BG$ is a model for the classifying space of $G$; it is natural to ask whether the equivariant cohomology is a free module over $H^{*}(BG)$ or not. We will see that the fixed point subspace plays a crucial role in the core of this question, and the main results involving this problem which are due to Atiyah, Borel, Dieck, Seagal and recent progress made by Guillemin, Haussmann, Franz and myself will be discussed.

## Octavian Mitrea, Polynomial Convexity of Lagrangian Inclusions (Tuesday Mar 6, 2018)

Polynomial convexity is of key importance in the general theory of approximation of continuous functions, uncovering deep connections to topology, Banach algebras, symplectic geometry, and other areas of mathematics. If $S \subset \mathbb{C}^{2}$ is a compact real surface, a Lagrangian inclusion of $S$ is a map $S\rightarrow \mathbb{C}^2$ which is a local Lagrangian embedding, except for a finite number of singularities that are either transverse double points or open Whitney umbrellas. In 1986, Givental proved that any such surface $S$, orientable or not, admits a Lagrangian inclusion. In this talk we show that Lagrangian inclusions are locally polynomially convex at every point.