Past

We know since almost a century that a ball can be decomposed into five pieces and these pieces rearranged so as to produce two balls of the same size as the original. This apparent paradox has led von Neumann to the notion of amenability which is now much studied in many areas of mathematics. However, the initial paradox has remained tied down to an elementary property of free groups of rotations for most of the 20th century. I will describe recent progress leading to new paradoxical groups.

Traditional diffusion models for random phenomena have paths with Holder

regularity just greater than 1/2 almost surely but there are situations

arising in finance and telecommunications where it is natural to look

for models in which the Holder regularity of the paths can vary.

Such processes are called multifractal and we will construct a class of

such processes on R using ideas from branching processes.

Using connections with multitype branching random walk we will be able

to compute the multifractal spectrum which captures the variability in

the Holder regularity. In addition, if we observe one of our processes

at a fixed resolution then we obtain a finite Markov representation,

which allows efficient simulation.

As an application, we fit the model to some AUD-USD exchange rate data.

Joint work with Geoffrey Decrouez and Ben Hambly

In this talk, I will present our new local existence result to the shallow water equations describing the motions of vertically averaged flows, which are closely related to the $2$-D isentropic Navier-Stokes equations for compressible fluids with density-dependent viscosity coefficients. Via introducing the notion of regular solutions, the local existence of classical solutions is established for the case that the viscosity coefficients are degenerate and the initial data are arbitrarily large with vacuum appearing in the far field.

We shall talk our recent works on the well-posedness of the Prandtl boundary layer equations both in two and three space variables. For the two-dimensional problem, we obtain the well-posedness in the Sobolev spaces by using an energy method under the monotonicity assumption of tangential velocity, and for the three-dimensional Prandtl equations, we construct a special solution by using the Corocco transformation, and obtain it is linearly stable with respect to any three-dimensional perturbation. These works are collaborated with R. Alexandre, C. J. Liu, C. Xu and T. Yang.

Seismic exploration in the oil industry is one example where wave equations are used as models. When the wave speed is spatially varying one is naturally concerned with questions of homogenisation or upscaling, where one would like to calculate an effective or average wave speed. As a canonical problem this short workshop will introduce the one-dimensional acoustic wave equation with a rapidly varying wave speed, perhaps even a periodic variation. Three questions will be asked: (i) how do you calculate a sensible average wave speed (ii) does the wave equation suffice or is there a change of form after averaging and (iii) if one can induce any particular excitation at one end of a finite one-dimensional medium, and make any observations that we like at that end, what - if anything - can be inferred about the spatial variability of the wave speed?

Since their introduction in the context of symplectic geometry, moment maps and symplectic quotients have been generalized in many different directions. In this talk I plan to give an introduction to the notions of hyperkähler moment map and hyperkähler quotient through two examples, apparently very different, but related by the so called ADHM construction of instantons; the moduli space of instantons and a space of complex matrices arising from monads.

We extend Kisin's construction of integral canonical models of Hodge-type Shimura

varieties to p=2, using Dieudonné display theory.

Adaptive network models, in which node states and network topology coevolve, arise naturally in models of social dynamics that incorporate homophily and social influence. Homophily relates the similarity between pairs of agents' states to their network coupling strength, whilst social influence causes the convergence of coupled agents' states. In this talk, I will describe a deterministic adaptive network model of attitude formation in social groups that incorporates these effects, and in which the attitudinal dynamics are represented by an activator-inhibitor process. I will show that consensus, corresponding to all nodes adopting the same attitudinal state and being fully connected, may destabilise via Turing instability, giving rise to chaotic dynamics. For the case where there are just two agents, I will illustrate, using numerical continuation, how such chaotic dynamics arise.

We consider evaluation methods for payoffs with an inherent

financial risk as encountered for instance for portfolios held

by pension funds and insurance companies. Pricing such payoffs

in a way consistent to market prices typically involves

combining actuarial techniques with methods from mathematical

finance. We propose to extend standard actuarial principles by

a new market-consistent evaluation procedure which we call `two

step market evaluation.' This procedure preserves the structure

of standard evaluation techniques and has many other appealing

properties. We give a complete axiomatic characterization for

two step market evaluations. We show further that in a dynamic

setting with continuous stock prices every evaluation which is

time-consistent and market-consistent is a two step market

evaluation. We also give characterization results and examples

in terms of $g$-expectations in a Brownian-Poisson setting.

Infinity categories simultaneously generalize topological spaces and categories. As a result, their study benefits from a combination of techniques from homotopy theory and category theory. While the theory of ordinary categories provides a suitable context to analyze objects up to isomorphism (e.g. abelian groups), the theory of infinity categories provides a reasonable framework to study objects up to a weaker concept of identification (e.g. complexes of abelian groups). In the talk, we will introduce infinity categories from scratch, mention some of the fundamental results, and try to illustrate some features in concrete examples.

PLEASE NOTE: THIS EVENT HAS BEEN CANCELLED

Brady showed that there are hyperbolic groups with non-hyperbolic finitely presented subgroups. I will present a new construction of this kind using Bestvina-Brady Morse theory.

An important moral truth about the mapping class group of a closed orientable surface is the following: a generic mapping class has no power fixing a finite family of simple closed curves on the surface. Such "generic" elements are called pseudo-Anosov. In this talk I will discuss one instantiation of this principle, namely that the probability of a simple random walk on the mapping class group returning a non-pseudo Anosov element decays exponentially quickly.

A finitely generated group has the Haagerup property if it admits a proper isometric action on a Hilbert space. It was a long open question whether Haagerup property is a quasi-isometry invariant. The negative answer was recently given by Mathieu Carette, who constructed two quasi-isometric generalized Baumslag-Solitar groups, one with the Haagerup property, the other not. Elaborating on these examples, we proved (jointly with S. Arnt and T. Pillon) that the equivariant Hilbert compression is not a quasi-isometry invariant. The talk will be devoted to describing Carette's examples.

Given a model dynamical system, a model of any measuring apparatus relating states to observations, and a prior assessment of uncertainty, the probability density of subsequent system states, conditioned upon the history of the observations, is of some practical interest.

When observations are made at discrete times, it is known that the evolving probability density is a solution of the Bayesian filtering equations. This talk will describe the difficulties in approximating the evolving probability density using a Gaussian mixture (i.e. a sum of Gaussian densities). In general this leads to a sequence of optimisation problems and related high-dimensional integrals. There are other problems too, related to the necessity of using a small number of densities in the mixture, the requirement to maintain sparsity of any matrices and the need to compute first and, somewhat disturbingly, second derivatives of the misfit between predictions and observations. Adjoint methods, Taylor expansions, Gaussian random fields and Newton’s method can be combined to, possibly, provide a solution. The approach is essentially a combination of filtering methods and '4-D Var’ methods and some recent progress will be described.

Erdos, Faudree, Gould, Gyarfas, Rousseau and Schelp, conjectured that
whenever the edges of a complete graph are coloured using three colours
there always exists a set of at most three vertices which have at least
two-thirds of their neighbours in one of the colours.  We will describe a
proof of this conjecture. This is joint work with Rahil Baber

Everybody has heard of the Faraday cage effect, in which a wire mesh does a good job of blocking electric fields and electromagnetic waves. For example, the screen on the front of your microwave oven keeps the microwaves from getting out, while light with its smaller wavelength escapes so you can see your burrito.  Surely the mathematics of such a famous and useful phenomenon has been long ago worked out and written up in the physics books, right?

Well, maybe.   Dave Hewett and I have communicated with dozens of mathematicians, physicists, and engineers on this subject so far, and we've turned up amazingly little.   Everybody has a view of why the Faraday cage mathematics is obvious, and most of their views are different.  Feynman discusses the matter in his Lectures on Physics, but so far as we can tell, he gets it wrong. 

For the static case at least (the Laplace equation), Hewett and I have made good progress with numerical explorations based on  Mikhlin's method backed up by a theorem.   The effect seems to much weaker than we had imagined -- are we missing something?  For time-harmonic waves (the Helmholtz equation), our simulations lead to further puzzles.  We need advice!  Where in the world is the literature on this problem? 

Monotonicity formulas play a pervasive role in the study of variational inequalities and free boundary problems. In this talk we will describe a new approach to a classical problem, namely the thin obstacle (or Signorini) problem, based on monotonicity properties for a family of so-called frequency functions.

Let $F \in \mathbb{Z}[x_1,\ldots,x_n]$. Suppose $F(\mathbf{x})=0$ has infinitely many integer solutions $\mathbf{x} \in \mathbb{Z}^n$. Roughly how common should be expect the solutions to be? I will tell you what your naive first guess ought to be, give a one-line reason why, and discuss the reasons why this first guess might be wrong.

I then will apply these ideas to explain the intriguing parallels between the handling of the Brauer-Manin obstruction by Heath-Brown/Skorobogotov [doi:10.1007/BF02392841] on the one hand and Wei/Xu [arXiv:1211.2286] on the other, despite the very different methods involved in each case.

I will present a formula that relates a Higgs bundle on an algebraic curve and Gromov-Witten invariants. I will start with the simplest example, which is a rank 2 bundle over the projective line with a meromorphic Higgs field. The corresponding quantum curve is the Airy differential equation, and the Gromov-Witten invariants are the intersection numbers on the moduli space of pointed stable curves. The formula connecting them is exactly the path that Airy took, i.e., from wave mechanics to geometric optics, or what we call the WKB method, after the birth of quantum mechanics. In general, we start with a Higgs bundle. Then we apply a generalization of the topological recursion, originally found by physicists Eynard and Orantin in matrix models, to this context. In this way we construct a quantization of the spectral curve of the Higgs bundle.