Abstract: In this talk we will explore the current state of this famous question, first formulated this way by Marc Kac in 1966. The answer to this question is known to be "no, you can not," but that is not the end of it. We will discuss the mathematical formulation of the problem and the counter examples, followed by some, fairly recent, positive results and the tools, specifically the heat- and wave-trace, that can be used.

Abstract: We'll prove the Homological Sylow theorem using the elementary approach of Surowski. This involves poset theory, simplicial complexes and character theory of finite groups. The talk is quite elementary. The main highlights are Order complexes with a G-action, Quillen's fibre lemma and Lefschetz character. All of these interact in an interesting way to prove the required result.

Abstract: We give an introduction to A-infinity algebras in this talk, which is a generalisation of differential graded algebras. We show that for a graded algebra A, the Ext-algebra has an A-infinity structure that contains sufficient information to recover A. On the other hand, we will present an example where the usual associative algebra structure on the Ext-algebra cannot recover A. We also show that the A-infinity structure is closely related to Massey products.

Abstract: Tensor triangulated categories appear in algebraic geometry, in homotopy theory and in representation theory, and beyond. Once the naive idea of classifying all objects is abandoned, the natural question becomes to classify the so-called ``thick tensor-ideals". The latter classification can always be achieved, via the spectrum of prime ideals. We shall review these ideas and see what new results we can obtain in the equivariant stable homotopy category. -- This is joint work with Beren Sanders.