Oxford University

The Thue-Morse sequence is perhaps the simplest example of an automatic sequence. Various pseudorandomness properties of this sequence have long been studied. During the talk, I will discuss a new result in this direction, asserting that the Gowers uniformity norms of the Thue-Morse sequence are small in a quantitative sense. Similar results hold for the Rudin-Shapiro sequence, as well as for a much wider class of automatic sequences which will be introduced during the talk.

The talk is partially based on joint work with Jakub Byszewski.

Consider the following particle system. We are given a uniform random rooted tree on vertices labelled by $[n] = \{1,2,\ldots,n\}$, with edges directed towards the root. Each node of the tree has space for a single particle (we think of them as cars). A number $m \le n$ of cars arrive one by one, and car $i$ wishes to park at node $S_i$, $1 \le i \le m$, where $S_1, S_2, \ldots, S_m$ are i.i.d. uniform random variables on $[n]$. If a car wishes to park at a space which is already occupied, it follows the unique path oriented towards the root until it encounters an empty space, in which case it parks there; if there is no empty space, it leaves the tree. Let $A_{n,m}$ denote the event that all $m$ cars find spaces in the tree. Lackner and Panholzer proved (via analytic combinatorics methods) that there is a phase transition in this model. Set $m = \lfloor \alpha n \rfloor$. Then if $\alpha \le 1/2$, $\mathbb{P}(A_{n,\lfloor \alpha n \rfloor}) \to \frac{\sqrt{1-2\alpha}}{1-\alpha}$, whereas if $\alpha > 1/2$ we have $\mathbb{P}(A_{n,\lfloor \alpha n \rfloor}) \to 0$. In this talk, we will give a probabilistic explanation for this phenomenon, and an alternative proof via the objective method.

Joint work with Christina Goldschmidt.

This talk will introduce the notion of quasi-convex subgroups. As an application, we will prove that the intersection of two finitely generated subgroups of a free group is again finitely generated.
 

We pursue robust approach to pricing and hedging in mathematical
finance. We develop a general discrete time setting in which some
underlying assets and options are available for dynamic trading and a
further set of European options, possibly with varying maturities, is
available for static trading. We include in our setup modelling beliefs by
allowing to specify a set of paths to be considered, e.g.
super-replication of a contingent claim is required only for paths falling
in the given set. Our framework thus interpolates between
model-independent and model-specific settings and allows to quantify the
impact of making assumptions. We establish suitable FTAP and
Pricing-Hedging duality results which include as special cases previous
results of Acciaio et al. (2013), Burzoni et al. (2016) as well the
Dalang-Morton-Willinger theorem. Finally, we explain how to treat further
problems, such as insider trading (information quantification) or American
options pricing.
Based on joint works with Burzoni, Frittelli, Hou, Maggis; Aksamit, Deng and Tan.
 

Stallings' theorem states that a finitely generated group splits over a finite subgroup if and only if it has more than one end. As a consequence of this, group splittings over finite subgroups are invariant under quasi-isometry. I will discuss a generalisation of Stallings' theorem which shows that under suitable hypotheses, group splittings over classes of infinite groups, namely coarse $PD_n$ groups, are also invariant under quasi-isometry.

A Kähler group is a group which can be realised as fundamental group of a compact Kähler manifold. I shall begin by explaining why such groups are not arbitrary and then address Delzant-Gromov's question of which subgroups of direct products of surface groups are Kähler. Work of Bridson, Howie, Miller and Short reduces this to the case of subgroups which are not of type $\mathcal{F}_r$ for some $r$. We will give a new construction producing Kähler groups with exotic finiteness properties by mapping products of closed Riemann surfaces onto an elliptic curve. We will then explain how this construction can be generalised to higher dimensions. This talk is independent of last weeks talk on Kähler groups and all relevant notions will be explained.