We would like to get a Lie algebra functor a la SGA3 for infinity-group sheaves $G$. There, this process is based on the semi-direct product $1 \to Lie(G) \to T(G) \to G \to 1$. Therefore, we classify semi-direct products of infinity-group sheaves.

Semialgebraic and subanalytic sets have become ubiquitous in mathematics since their introduction by Lojasiewicz in the 1960s, following the famous Tarski-Seidenberg theorem on quantifier elimination. I will discuss two long-standing questions in real-analytic geometry, on global smoothing of a subanalytic set (an analogue of resolution of singularities), and on transformation of a proper real-analytic mapping to a mapping with equidimensional fibres by global blowings-up of the target (a classical result in the complex-analytic case).

These questions are related: a positive answer to the second can be used to reduce the first to the simpler semianalytic case. It turns out that the second question has a negative answer, in general, and the first nevertheless has a positive solution.

Speaker's web page: http://www.math.toronto.edu/bierston/

We will introduce Complete Segal spaces and prove they describe an equivalent homotopy theory to the one of quasi-categories.

Massey products are, originally, operations defined in the context of algebraic topology, on cohomology rings. However, when one specializes to group cohomology, work of Dwyer shows that the study of these operations amounts to that of elementary extension problems, involving the group of unipotent matrices over a finite field. When we specialize further, and consider Galois groups in particular, an ambitious conjecture predicts that Massey products always vanish. This has been actually proved for triple Massey products, in complete generality. In this talk, I will describe joint work with Minac, Tan, Topaz and Wittenberg, showing that the conjecture is true for fourfold Massey products in the cohomology of number fields.