University of Oxford

Let X_1 and X_2 be finite graphs with isomorphic universal covers.

Leighton's graph covering theorem states that X_1 and X_2 have a common finite cover.

I will discuss recent work generalizing this theorem and how myself and Sam Shepherd have been applying it to rigidity questions in geometric group theory.

We discuss hyperkahler implosion spaces, their relevance to group actions and why they should fit into the symplectic duality picture. For certain groups we present candidates for the symplectic duals of the associated implosion spaces and provide computational evidence. This is joint work with Amihay Hanany and Frances Kirwan.
 

Abstract: The signature of a path is a sequence of iterated coordinate integrals along the path. We aim at reconstructing a path from its signature. In the special case of lattice paths, one can obtain exact recovery based on a simple algebraic observation. For general continuously differentiable curves, we develop an explicit procedure that allows to reconstruct the path via piecewise linear approximations. The errors in the approximation can be quantified in terms of the level of signature used and modulus of continuity of the derivative of the path. The main idea is philosophically close to that for the lattice paths, and this procedure could be viewed as a significant generalisation. A key ingredient is the use of a symmetrisation procedure that separates the behaviour of the path at small and large scales.We will also discuss possible simplifications and improvements that may be potentially significant. Based on joint works with Terry Lyons, and also with Jiawei Chang, Nick Duffield and Hao Ni.

The second fundamental form of an embedded manifold must satisfy a set of constraint equations known as the Gauß-Codazzi equations. Since work of Chen-Slemrod-Wang, these equations are known to satisfy a particular div-curl structure: under suitable L^p bound on the second fundamental form, the curvatures are weakly continuous. In this talk we explore generalisations of this original result under weaker assumptions. We show how techniques from fluid dynamics can yield interesting insight into the weak continuity properties of isometric embeddings.

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Teichmüller harmonic map flow is a geometric flow designed to evolve combinations of maps and metrics on a surface into minimal surfaces in a Riemannian manifold. I will introduce the flow and describe known existence results, and discuss recent joint work with M. Rupflin that demonstrates how singularities can develop in the metric component in finite time.

Limit groups are a powerful tool in the study of free and hyperbolic groups (and even broader classes of groups). I will define limit groups in various ways: algebraic, logical and topological, and draw connections between the different definitions. We will also see how one can equip a limit group with an action on a real tree, and analyze this action using the Rips machine, a generalization of Bass-Serre theory to real trees. As a conclusion, we will obtain that hyperbolic groups whose outer automorphism group is infinite, split non-trivially as graphs of groups.