Simplicial localization is a localization of a category at a class of morphisms yielding a higher category that captures homotopy theory of the original category. I will begin with a review of motivations behind the simplicial localization and some of its classical constructions. In the main part of the talk, I will discuss a recent result, joint with Chris Kapulkin, about a new way of constructing simplicial localizations.

In this lecture, we continue with the following topics: ---Clifford algebras, Clifford modules, spin structures, Dirac operators, Weitzenbock formula. ---Heat kernel and its asymptotic expansion, Gilkey's formula, McKean-Singer formula.

I will discuss the following result, which is joint work with A. Sukhov: on any compact real surface $S$, except the sphere and $RP^2$, there exist two smooth functions with the property that their rational combinations are dense in the space of continuous functions on $S$. The proof relies on many different results in several complex variables, symplectic geometry, and approximation theory. I will give a general overview of this.

In the last few years, there has been a wave of interest in thermodynamics of very small or strongly correlated systems. This research, rooted in quantum information theory, has given rise to several interesting mathematical questions on majorization and its behavior under tensor products. In the talk, I will present one theorem that concerns majorization of probability distributions under the creation of correlations in auxiliary systems. As a by-product, we obtain a characterization of Shannon entropy as the essentially unique real function on probability distributions which is Schur-concave, additive, and subadditive.

Given a number field, we form the Hilbert p-class field tower by repeatedly iterating the Hilbert p-class field construction. Work of Golod and Shafarevich demonstrated that this construction can fail to stabilize after finitely many steps, and the slew of natural questions that this observation unleashed remain largely unanswered. In particular, there is no known definitive criterion for deciding if a number field has an infinite such tower. In the case of quadratic imaginary number fields, however, we are close to honing in on a set of invariants which completely resolve the question. This talk will cover both historical such invariants and modern ones, giving the current status of our ability to determine the infinitude of such towers.