C6

We show that the group cohomology of simple connected Lie groups, and consequently their lattices and symmetric spaces, with L^p-coefficients vanishes below the rank. This confirms a conjecture of Gromov in the special case of the coefficient module L^p(G). We relate the vanishing of L^1-cohomology to a fixed-point theorem and confirm a conjecture of Farb. Farb conjectured a fixed point property for actions of lattices in such groups on CAT(0) cell complexes of dimension lower than the rank.
In this talk, I will give a short introduction to group cohomology and discuss ideas from the proofs of both results. This talk is based on joint work with Saar Bader, Uri Bader, and Roman Sauer.

A natural energy arising in the modelling of thin ferromagnetic films and liquid crystals is the Dirichlet energy, along with a penalisation term which favours a particular configuration. In this talk, I present recent work which establishes a quantitative analysis for sequences of minimisers to both the Baby-Skyrme model and two regularisation arising from geometry and physics respectively.

Riemannian manifolds whose holonomy group lies inside the exceptional Lie group G_2 are called G_2-manifolds. These manifolds have several interesting properties (they are Ricci-flat) and are of interest in geometric analysis as well as in mathematical physics and other fields. In this seminar, we will give an introduction to the theory of G_2-manifolds and discuss an associated non-linear Laplacian-type operator whose kernel essentially determines whether a compact manifold is G_2. We will present uniqueness and existence results for the Poisson’s equation of this operator on homogeneous and cohomogeneity-one manifolds.

We show that if a group admits an action with no algebraicity, then all of its mixed identities are singular. This generalises a theorem of Abért showing such groups are lawless. Our result confirms a conjecture of Bodirsky, Schneider, and Thom for a large class of oligomorphic permutation groups, and it also applies to several groups arising in topological contexts. I'll also discuss how ideas from independence relations in model theory arise in the proof and yield more general results, also applying to infinite-dimensional general (and projective) linear groups over arbitrary fields, recovering results of Bradford, Schneider, and Thom.

This is joint work with Michael Pinsker.

The classical Leopoldt conjecture predicts that the global units of a number field (tensored with Qp) inject into the local units at p. In this talk, I'll discuss some non-abelian generalisations of this in the setting of Galois representations.

Recurrence is central in ergodic Ramsey theory, and its multiplicative analogue is only now emerging. In this talk, I will define multiplicative recurrence, give illustrative examples, and explain how pretentious number theory is applied to establish it.