Properads are an extension of the notion of operad which allow one to model structures with many-to-many operations, such as various kinds of bialgebras. In this talk we will discuss up-to-homotopy versions of properads, as well as potential applications.

Necklace polynomials arise in different areas of mathematics: combinatorics, arithmetic and Lie theory. In my talk, I will discuss their significance in each of these areas with special emphasis on a generalised Witt formula that one may deduce for graded Lie algebras. This formula can be used to prove a Gromov-like theorem for graded Lie algebras of type FP. Necklaces, finite fields, and Lie algebras

The index of a transversally elliptic operator is not an integer. It is a character of an infinite dimensional representation. Such characters, when they can be defined, make sense only as distribution but under some mild conditions they can be shown to be representable by locally integrable functions. This talk is a general introduction to such characters mostly from a purely representation theoretic view and can be followed independently of index theory seminar talks.

Continuing from last week's lecture, I will discuss some of the functorial properties of the index of transversally elliptic operators, and give some basic examples.

HOTT is a new branch of mathematics arising from the unexpected encounter of logic with homotopy theory. It provides a new foundation of mathematics which can be implemented in a computerised proof assistant like Coq or Agda. I will briefly describe the history of the subject, from Martin-Löf, to Awodey, Warren and Voevodsky. I will describe HOTT in the language of category theory and discuss the geometric meaning of Voevodsky's univalence axiom.