Oxford

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.

We saw earlier that a subquadratic isoperimetric inequality implies a linear one. I will give examples of groups, due to Brady and Bridson, which prove that this is the only gap in the isoperimetric spectrum. 

The commuting probability of a finite group is defined to be the probability that two randomly chosen group elements commute. Not all rationals between 0 and 1 occur as commuting probabilities. In fact Keith Joseph conjectured in 1977 that all limit points of the set of commuting probabilities are rational, and moreover that these limit points can only be approached from above. In this talk we'll discuss a structure theorem for commuting probabilities which roughly asserts that commuting probabilities are nearly Egyptian fractions of bounded complexity. Joseph's conjectures are corollaries.

"We consider certain distinguished extensions of the field F_p((t)) of formal Laurent series over F_p, and look at questions about their model theory and Galois theory, with a particular focus on decidability."

In 1983 Kerckhoff settled a long standing conjecture by Nielsen proving that every finite subgroup of the mapping class group of a compact surface can be realized as a group of diffeomorphisms. An important consequence of this theorem is that one can now try to study subgroups of the mapping class group taking the quotient of the surface by these groups of diffeomorphisms. In this talk we will study quotients of surfaces under the action of a finite group to find bounds on the cardinality of such a group.

The Dehn function of a group measures the complexity of the group's word problem, being the upper bound on the number of relations from a group presentation required to prove that a word in the generators represents the identity element. The Filling Theorem which was first stated by Gromov connects this to the isoperimetric functions of Riemannian manifolds. In this talk, we will see the classification of hyperbolic groups as those with a linear Dehn function, and give Bowditch's proof that a subquadratic isoperimetric inequality implies a linear one (which gives the only gap in the "isoperimetric spectrum" of exponents of polynomial Dehn functions).

We will give an outline of the proof by Kahn and Markovic who showed that a closed hyperbolic 3-manifold $\textbf{M}$ contains a closed $\pi_1$-injective surface. This is done using exponential mixing to find many pairs of pants in $\textbf{M}$, which can then be glued together to form a suitable surface. This answers a long standing conjecture of Waldhausen and is a key ingredient in the proof of the Virtual Haken Theorem.

Dinits, Karzanov and Lomonosov showed that the minimal edge cuts of a finite graph have the structure of a cactus, a tree-like graph constructed from cycles. Evangelidou and Papasoglu extended this to minimal cuts separating the ends of an infinite graph. In this talk we will discuss a similar structure theorem for minimal vertex cuts separating the ends of a graph; these can be encoded by a succulent, a mild generalization of a cactus that is still tree-like.