# Fall 2018

## Béatrice Chetard, Graded Character Rings (Thursday Oct 4, 2018)

The characters (say, over the complex numbers) of a group $G$ form a ring $R(G)$. Exterior powers of representations induce a filtration on this ring, defined by Grothendieck. Can we compute the associated ring? What information about the group does it remember?

## Marco Vergura, Localization Theory in an Infinity-topos (Wednesday Oct 10, 2018)

What happens if we take spaces and kill a sphere? What if we kill other spaces? In this talk, I will introduce nullification of spaces and use it as a guiding example to gently lead into my research on localization theory in an infinity-topos.

## Dinesh Valluri, Essential Dimension of Stacks (Wednesday Oct 17, 2018)

In this talk we will introduce the notion of essential dimension. We are interested in finding essential dimension of a coherent $G$-sheaf on a non-singular algebraic curve (think Riemann surface) with a finite group $G$ acting on it. One of the tools we use to do this is the (equivariant) Riemann-Roch theorem. In particular, we compute (usual) dimensions of several classes of smooth stacks using (equivariant) Riemann-Roch. This helps us give an upper bound for the essential dimension of a coherent $G$-sheaf.

## Udit Mavinkurve, Hopf Objects in the Category of Combinatorial Species (Wednesday Oct 24, 2018)

Studying a graded object is often easier than studying the same thing as an ungraded object. Similarly, studying Hopf objects in the category of combinatorial species is often easier than studying classical Hopf algebras. Braid arrangements happen to be central to the study of such combinatorial Hopf algebras, and suggest an extension of the theory to more general contexts: e.g. hyperplane arrangements. In this talk, we begin by defining what a hyperplane arrangement is, and explore some basic aspects of the (commutative) theory.

## Chris Hellmann, Von Neumann Algebras (Wednesday Oct 31, 2018)

A von Neumann algebra is an algebra of Hilbert space operators which is "nice" in a certain way. These objects have connections with other areas of mathematics (for instance, knot theory and representation theory), but also possess a rather beautiful theory of their own. I'll talk about a few highlights of this theory, such as spectral theory in the abelian case, the phenomenon of continuous dimension, and type classification, with the goal of giving a sense of why these things might be interesting to study.

## Chandrahas Piduri, Morse Theory (Wednesday Nov 7, 2018)

We will define a Morse function on a smooth manifold. A Morse function can be used to obtain a cellular decomposition of a manifold. We will state the required results and obtain a cellular decomposition of the complex projective space. If time permits, we will characterize compact manifolds with a Morse function having only two critical points.

## Félix Baril Boudreau, Stacks and Gerbes (Wednesday Nov 28, 2018)

This talk aims to introduce the concepts of fibered category, stack and gerbe. We will discuss in some details the examples of the gerbe $B\mathbb{G}$ of a sheaf of abelian groups $\mathbb{G}$ and the gerbe of trivializations of an Azumaya algebra over a scheme. In some sense this can talk be seen as an (incomplete) background session for my comprehensive exam in December. Therefore the audience is invited to ask questions during the presentation. Algebraic Geometry will be there in the background but should be understandable with mild exposure to category theory.