In 2001, Andrei Bulatov generalized the binary commutator operation on a general algebra to an n-ary commutator operation. In a simple algebra higher commutators may only be the congruence 0 or 1. I have shown that for any natural number, n>1, we can construct an algebra with the n-ary commutator [1, ..., 1] = 1 while the (n+1)-ary commutator [1, ..., 1, 1] = 0. In this talk I will present the construction in the case where n=2.

I will give a short overview of the technique of "formal geometry", with its applications to deformation quantization of algebraic varieties and to characteristic classes. Then I will show how the moduli space of quantizations can be described in terms of certain characteristic classes of symplectic algebraic varieties introduced by Rozansky and Witten (joint work in progress with V. Baranovsky).

I will talk about a recent construction of Rips, Segev and Tent, which solves a longstanding problem.

It is a classical result of Beauville and Donagi that Fano varieties of lines on cubic fourfolds are hyper-Kahler. More recently, Lehn, Lehn, Sorger and van Straten constructed a hyper-Kahler eightfold out of twisted cubics on cubic fourfolds. In this talk, I will explain a new approach to these hyper-Kahler varieties via moduli of stable objects on the Kuznetsov components, and further generalizations. An application towards the study of 0-cycles on these hyper-Kahler varieties will also be discussed. This is based on a joint work with Chunyi Li and Laura Pertusi.

Now that the classification of infinite dimensional simple separable unital C*-algebras with finite nuclear dimension and which satisfy the UCT is complete, it becomes an interesting question to determine the range of the invariant for certain naturally-occurring subclasses. I will discuss joint work in progress with Robin Deeley and Ian Putnam that considers this question for the C*-algebras arising from groupoids associated to minimal dynamical systems. In particular, I will discuss dynamical  constructions which realize certain prescribed Elliott invariants.