We are interested in classifying all finite groups which can act on a finite $CW$-complex homotopy equivalent to a sphere, such that all isotropy subgroups are rank one groups, meaning that they do not include $Z/p\times Z/p$ for any prime $p$. The similar question for free actions (all isotropy subgroups are trivial) has been answered completely by the works of P.A. Smith and R. Swan. There is a complete list of such groups (they are finite groups with periodic group cohomology) and we would like to obtain a similar list for actions with rank one isotropy. This is joint work with Ian Hambleton.

Let $f$ be a holomorphic function on the unit disc, and let $(S_{n_k})$ be a subsequence of its Taylor polynomials about 0. In this talk we will see that the nontangential limit of $f$ and $\lim_{k\rightarrow \infty} S_{n_{k}}$ agree at a.e. point of the unit circle where they simultaneously exist. In the first part of this talk we will focus on this result and its applications. In the second part of the talk we will discuss a convergence theorem of harmonic measures on domains in $\mathbb{R}^{N}$ which played a key role in the proof of the above result and it is of independent interest. (Joint work with Stephen Gardiner)

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.

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.

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.

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.

We consider the canonical spectral triple associated to a two-torus as a compact Riemannian spin manifold. As an example for the local index formula in the even case, we compute the index of the corresponding Dirac operator with coefficients in a rank one vector bundle given by a special class of projections.

In the early 90's, Bogomolov introduced a program whose ultimate goal is to reconstruct function fields of dimension $> 1$ over algebraically closed fields from their pro-$\ell$ 2-step nilpotent Galois groups. Although it is far from being resolved in full generality, this program has since been carried through for function fields over the algebraic closure of a prime field. After an introduction to birational anabelian geometry and Bogomolov's program, in this talk I will describe the possibility of a $Z/\ell$-analogue, including the inherent problems/difficulties, as well as some partial results in this direction.