Oxford

The lecture will introduce the notion of a folded 4-dimensional hyperkähler manifold, give examples and prove a local existence theorem from boundary data using twistor methods, following an idea of Biquard.  

we discuss various conjectures about the absolute Galois group G_Q  of the field Q of rational numbers and to what extent it encodes the elementary theory of Q.

Even a cursory inspection of the Hodge plot associated with Calabi-Yau threefolds that are hypersurfaces in toric varieties reveals striking structures. These patterns correspond to webs of elliptic K3 fibrations whose mirror images are also elliptic K3 fibrations. Such manifolds arise from reflexive polytopes that can be cut into two parts along slices corresponding to the K3 fibers. Any two half-polytopes over a given slice can be combined into a reflexive polytope. This fact, together with a remarkable relation on the additivity of Hodge numbers, explains much of the structure of the observed patterns.

Groups which act on rooted trees, and branch groups in particular, have provided examples of groups with exotic properties for the last three decades. This and their links to other areas of mathematics such as dynamical systems has made them the object of intense research.
One of their more useful properties is that of having a "tree-like" subgroup structure, in several senses. 
I shall explain what this means in the talk and give some applications.