I will explain how to construct the symmetric monoidal infinity-category of complexes of quasi-coherent sheaves on a stack using higher algebra. Afterwards, I will talk about how this might be used to compactify moduli stacks of torsors.

We continue the lectures with : ---Clifford algebras, Clifford modules, spin structures, Dirac operators, Weitzenbock formula. ---Heat kernel and its asymptotic expansion, Gilkey's formula, McKean-Singer formula.

The talk will focus on the classical problem of traces of functions from Sobolev spaces, which had originated in connection with some specific problems in PDEs and then mushroomed into a separate field of research in functional analysis and the function spaces theory. One important property enjoyed by functions from the Sobolev space $W^ {m,p}(\mathbb{R}^ n )$, where $m\in \mathbb{N}$ and $p\in[1,\infty]$, is that their restrictions, called traces, to lower dimensional spaces $\mathbb{R}^ d$ can be properly defined, provided that the dimension $d$ of the relevant subspaces is not too small, depending on the values of $n$, $m$ and $p$. In such case one can ask whether some properties such as a certain degree of integrability of a trace can be expected, and, naturally, which of these properties are the best possible. We shall survey both classical and recent results concerning traces of Sobolev functions. We shall consider basic questions concerning the very existence of trace as well as deeper problems such as optimal trace embeddings involving specific function spaces.

We investigate equivalence relations for quadratic forms that can be expressed in terms of algebro-geometric properties of their associated quadrics, more precisely, birational, stably birational and motivic equivalence, and isomorphism of quadrics. We provide some examples and counterexamples and highlight some important open problems.