The seminar will run on Fridays from 1:30 to 2:30 PM in MC 107.

Coffee and Snacks are usually provided.

A list of suggested topics for expository talks of MSc Students can be accessed here.

The PDF version of the abstracts can be accessed here.

## Organizational Meeting (Tuesday Sep 17, 2019)

3:30-4:30 PM, MC107

## Sergio Chaves, Equivariant Cohomology of $\mathbb{Z}/2$-manifolds (Friday Sep 27, 2019)

If $N$ is a compact manifold with boundary, then it can be realized as the orbit space of a $\mathbb{Z}/2$-manifold $M$; more precisely, as the quotient $M/\tau$ where $M$ is a manifold of the same dimension as $N$ and $\tau \colon M \rightarrow M$ is an involution. The manifold $M$ might not be unique and it depends on the principal $\mathbb{Z}/2$-bundles over $N$. However, the equivariant cohomology module-type of these manifolds is completely determined by the cohomology of $N$. Before discussing these results, the necessary background on principal bundles and equivariant cohomology will be firstly introduced; only basic notions about singular cohomology will be assumed.

## Félix Baril Boudreau, A twisted sheaf proof of Albert’s theorem (Friday Oct 4, 2019)

In 1939 A. A. Albert published in Structure of algebras a proof that given a central simple algebra $A$ over a field $K$, $A$ has an involution of the first kind if and only if its class in the Brauer group of $K$ has order $1$ or $2$. In this talk we will introduce, following Giraud (1971), a categorical analogue of a principal vector bundle, called $G$-gerbe, as well as the notion of twisted vector bundle as defined in the work of Lieblich (2004). These modern tools give us a completely different point of view on Albert's theorem and allow us to prove this result in an interesting new way. All the necessary background will be presented and some familiarity with the basic vocabulary of category theory will be assumed.

## Luuk Verhoeven, The Gelfand-Naimark Theorem and Noncommutative Geometry (Friday Oct 11, 2019)

In this talk I will talk about the Gelfand-Naimark Theorem, which provides a duality between locally compact Hausdorff spaces and commutative $C^*$-algebras. We will discuss the proof of this theorem, which has several interesting ingredients, as well as the conceptual implications which form, essentially, the basis of Noncommutative Geometry. If time permits, I will also discuss how my master's thesis fits in this conceptual framework. This talk is intended as a casual introduction, so basic knowledge of functional analysis (norms, duals), topology and complex analysis should suffice. Any extra familiarity with functional analysis (spectra of operators in particular) is beneficial but not necessary.

## Udit Ajit Mavinkurve, Sheaves on Posets (Friday Oct 18, 2019)

There is a particularly nice way to interpret posets as topological spaces (via the Alexandroff topology), which allows us to translate many tools from topology to the context of posets. Working over posets, several definitions simplify and become easier to introduce and work with - making posets an important pedagogical tool. In this expository talk, we will see (and hopefully prove) analogues of results such as Quillen's Theorem A. Passing familiarity with basic notions of category theory (e.g. adjunctions) and homological algebra (e.g. injective resolutions, right-derived functors) will be assumed, but not much more beyond that.

## Jarl Taxerås, Fibrations of Categories (Friday Oct 25, 2019)

By categorifying the homotopy lifting property of maps between topological spaces, one obtains discrete fibrations between categories. The classical correspondence between (isomorphism classes of) covers of a topological space $X$ and (conjugacy classes of) subgroups of the fundamental group $\pi(X)$, generalizes to an equivalence between the category $DFib(C)$ of discrete fibrations into a category $C$ and presheaves on $C$. We will see that under this equivalence, universal covers correspond to representable presheaves. Realizing this has some nice consequences, and will give us a curious perspective on some familiar constructions.

## Babak Beheshti Vadeqan, What is regularized determinant? (Friday Nov 1, 2019)

Taming infinities has its firm roots throughout the history of mathematics and physics. Euler’s work on zeta function and infinite products or Cantor’s stunning idea of counting infinities are just two classics among many others. In this talk we will see how one can use zeta function to define the determinant of certain operators -infinite matrices- such as differential operators. Then, if time permits, we will go through Quillen's discovery that connects the regularized determinant to the geometry of some determinant bundles.

## César Bardomiano Martínez , Sheaves as Étalé Spaces (Friday Nov 15, 2019)

Étale morphisms are certain kind of morphisms between schemes. In general case, a bundle over a topological space $X$ is said to be étale if it is a local homeomorphism. The goal of this talk is to present the equivalence between sheaves on a topological space $X$ and étalé spaces. We will give a fairly good amount of details of the ideas involved to prove such equivalence.

## James Leslie, Introduction to Toposes and Logic (Friday Nov 22, 2019)

Toposes can be seen as abstractions of the category of sets and functions. We will look at several examples of toposes and discuss some of the logical properties they can exhibit. We will conclude by sketching a topos theoretic proof of the independence of the axiom of choice from a slightly weakened version of Zermelo Frankel set theory.

## Marios Velivasakis, Schubert Varieties in Partial Flag Manifolds and Generalized Severi-Brauer Varieties (Friday Nov 29, 2019)

Schubert varieties form one of the most important classes of singular algebraic varieties. They are also a kind of moduli spaces. One problem is that these varieties are not easy to understand and manipulate using only their geometric nature. In this talk, we will discuss about Schubert varieties and present a way to characterize them combinatorially. In addition, we will discuss how they relate to Severi-Brauer varieties $SB(d,A)$ and how we can use their combinatorial description to answer questions about subvarieties of $SB(d,A)$.