Past

I will discuss recent joint work with S. Galatius, in which we

generalise the Madsen--Weiss theorem from the case of surfaces to the

case of manifolds of higher even dimension (except 4). In the simplest

case, we study the topological group $\mathcal{D}_g$ of

diffeomorphisms of the manifold $\#^g S^n \times S^n$ which fix a

disc. We have two main results: firstly, a homology stability

theorem---analogous to Harer's stability theorem for the homology of

mapping class groups---which says that the homology groups

$H_i(B\mathcal{D}_g)$ are independent of $g$ for $2i \leq g-4$.

Secondly, an identification of the stable homology

$H_*(B\mathcal{D}_\infty)$ with the homology of a certain explicitly

described infinite loop space---analogous to the Madsen--Weiss

theorem. Together, these give an explicit calculation of the ring

$H^*(B\mathcal{D}_g;\mathbb{Q})$ in the stable range, as a polynomial

algebra on certain explicitly described generators.

In the absence of background fluxes and sources, the compactification of string theories on Calabi-Yau threefolds leads to supersymmetric solutions.Turning on fluxes, e.g. to lift the moduli of the compactification, generically forces the geometry to break the Calabi-Yau conditions, and to satisfy, instead, the weaker condition of reduced structure. In this talk I will discuss manifolds with SU(3) structure, and their relevance for heterotic string compacitications. I will focus on domain wall solutions and how explicit example geometries can be constructed.

We consider a class of stochastic control problems with fuel constraint that are closely connected to the problem of finding adaptive mean-variance-optimal portfolio liquidation strategies in the Almgren-Chriss framework. We give a closed-form solution to these control problems in terms of the log-Laplace transforms of certain J-functionals of Dawson-Watanabe superprocesses. This solution can be related heuristically to the superprocess solution of certain quasilinear parabolic PDEs with singular terminal condition as given by Dynkin (1992). It requires us to study in some detail the blow-up behavior of the log-Laplace functionals when approaching the singularity.

In this talk, I will show that Brauer classes of a locally solvable degree 4 del Pezzo surface X are vertical, that is, that every Brauer class is obtained by pullback from an element of Br k(P^1) for some rational map f : X ----> P^1. As a consequence, we see that a Brauer class does not obstruct the existence of a rational point if and only if there exists a fiber of f that is locally solvable. The proof is constructive and gives a simple and practical algorithm, distinct from that in [Bright,Bruin,Flynn,Logan (2007)], for computing all nonconstant classes in the Brauer group of X. This is joint work with Anthony V\'arilly-Alvarado.

We study a thin liquid film on a vertical fibre. Without gravity, there

is a Rayleigh-Plateau instability in which surface tension reduces the

surface area of the initially cylindrical film. Spherical drops cannot

form because of the fibre, and instead, the film forms bulges of

roughly twice the initial thickness. Large bulges then grow very slowly

through a ripening mechanism. A small non-dimensional gravity moves the

bulges. They leave behind a thinner film than that in front of them, and

so grow. As they grow into large drops, they move faster and grow

faster. When gravity is stronger, the bulges grow only to finite

amplitude solitary waves, with equal film thickness behind and in front.

We study these solitary waves, and the effect of shear-thinning and

shear-thickening of the fluid. In particular, we will be interested in

solitary waves of large amplitudes, which occur near the boundary

between large and small gravity. Frustratingly, the speed is only

determined at the third term in an asymptotic expansion. The case of

Newtonian fluids requires four terms.

I will give an introduction to how SU(3)-structures appear in heterotic string theory and string compactifications. I will start by considering the zeroth order SU(3)-holonomy Calabi-Yau scenario, and then see how this generalizes when higher order effects are considered. If time, I will discuss some of my own work.

A triangulated category admits a strong generator if, roughly speaking,

every object can be built in a globally bounded number of steps starting

from a single object and taking iterated cones. The importance of

strong generators was demonstrated by Bondal and van den Bergh, who

proved that the existence of such objects often gives you a

representability theorem for cohomological functors. The importance was

further emphasised by Rouquier, who introduced the dimension of

triangulated categories, and tied this numerical invariant to the

representation dimension. In this talk I will discuss the generation

time for strong generators (the least number of cones required to build

every object in the category) and a refinement of the dimension which is

due to Orlov: the set of all integers that occur as a generation time.

After introducing the necessary terminology, I will focus on categories

occurring in representation theory and explain how to compute this

invariant for the bounded derived category of the path algebras of type

A and D, as well as the corresponding cluster categories.

Partial differential equations (PDEs) with random coefficients are used in computer simulations of physical processes in science, engineering and industry applications with uncertain data. The goal is to obtain quantitative statements on the effect of input data uncertainties for a comprehensive evaluation of simulation results. However, these equations are formulated in a physical domain coupled with a sample space generated by random parameters and are thus very computing-intensive.

We outline the key computational challenges by discussing a model elliptic PDE of single phase subsurface flow in random media. In this application the coefficients are often rough, highly variable and require a large number of random parameters which puts a limit on all existing discretisation methods. To overcome these limits we employ multilevel Monte Carlo (MLMC), a novel variance reduction technique which uses samples computed on a hierarchy of physical grids. In particular, we combine MLMC with mixed finite element discretisations to calculate travel times of particles in groundwater flows.

For coefficients which can be parameterised by a small number of random variables we employ spectral stochastic Galerkin (SG) methods which give rise to a coupled system of deterministic PDEs. Since the standard SG formulation of the model elliptic PDE requires expensive matrix-vector products we reformulate it as a convection-diffusion problem with random convective velocity. We construct and analyse block-diagonal preconditioners for the nonsymmetric Galerkin matrix for use with Krylov subspace methods such as GMRES.

We will discuss numerical solutions of Multi-objective Control problems governed by partial differential equations. More precisely, we will look for Nash Equilibria, which are solutions to non-cooperative differential games. First we will study the continuous case. Then, in order to compute solutions, we will combine finite difference schemes for the time discretization, finite element methods for the space discretization and a conjugate gradient algorithm (or other suitable alternative) for the iterative solution of the discrete differential game. Finally, we will apply this methodology to the solution of several test problems.

We investigate a mixed problem with Robin boundary conditions for a diffusion-reaction equation. We investigate the problem in the sublinear, linear and super linear cases, depending on the nonlinear part. We obtain relations between the parameters of the problem which are sufficient conditions for the existence of generalized solutions to the problem and, in a special case, for their uniqueness. The proof relies on a general existence theorem by Soltanov. Finally we investıgate the time-behaviour of solutions. We show that boundedness of solutions holds under some additional conditions as t is convergent to infinity. This study is joint work with Kamal Soltanov (Hacettepe University).

I'll sketch some context for future and past research around valued fields
and motivic integration, from a model theoretic viewpoint, leaving out technical details. 
The talk will be partly conjectural.

We give a brief overview of hyperbolic metric spaces and the relatively hyperbolic counterparts, with particular emphasis on the quasi-isometry class of trees. We then show that an understanding of the relative version of such spaces - quasi tree-graded spaces -  has strong consequences for mapping class groups. In particular, they are shown to embed into a finite product of (possibly infinite valence) simplicial trees. This uses and extends the work of Bestvina, Bromberg and Fujiwara.

I describe recent work with Pyber, Short and Szabo in which we study the `width' of a finite simple group. Given a group G and a subset A of G, the `width of G with respect to A' - w(G,A) - is the smallest number k such that G can be written as the product of k conjugates of A. If G is finite and simple, and A is a set of size at least 2, then w(G,A) is well-defined; what is more Liebeck, Nikolov and Shalev have conjectured that in this situation there exists an absolute constant c such that w(G,A)\leq c log|G|/log|A|. 
I will present a partial proof of this conjecture as well as describing some interesting, and unexpected, connections between this work and classical additive combinatorics. In particular the notion of a normal K-approximate group will be introduced.

Results that say local information is enough to guarantee global information provide the theoretical underpinnings of many reconstruction algorithms in evolutionary biology. Such results include Buneman's Splits-Equivalence Theorem and the Tree-Metric Theorem. The first result says that, for a collection $\mathcal C$ of binary characters, pairwise compatibility is enough to guarantee compatibility for $\mathcal C$, that is, there is a phylogenetic (evolutionary) tree that realises $\mathcal C$. The second result says that, for a distance matrix $D$, if every $4\times 4$ distance submatrix of $D$ is realisable by an edge-weighted phylogenetic tree, then $D$ itself is realisable by such a tree. In this talk, we investigate these and other results of this type. Furthermore, we explore the closely-related task of determining how much information is enough to reconstruct the correct phylogenetic tree.

I will explain recent results with Emanuele Macrì, in which we systematically study the birational geometry of moduli of sheaves on K3's via wall-crossing for

Bridgeland stability conditions. In particular, we obtain descriptions of their nef cones via the Mukai lattice of the K3, their moveable cones, their divisorial contractions, and obtain counter-examples to various conjectures in the literature. We also give a proof of the Lagrangian fibration conjecture (due to

Hassett-Tschinkel/Huybrechts/Sawon) via wall-crossing.