Past

I will discuss two of my papers that develop PDE methods for weak KAM theory, in the context of a singular variational problem that can be interpreted as a regularization of Mather's variational principle by an entropy term. This is, sort of, a statistical mechanics approach to the problem. I will show how the Euler-Lagrange PDE yield approximate changes to action-angle variables for the corresponding Hamiltonian dynamics.

Random matrices were introduced by E. Wigner to model the excitation spectrum of large nuclei. The central idea is based on the hypothesis that the local statistics of the excitation spectrum for a large complicated system is universal. Dyson Brownian motion is the flow of eigenvalues of random matrices when each matrix element performs independent Brownian motions. In this lecture, we will explain the connection between the universality of random matrices and the approach to local equilibrium of Dyson Brownian motion. The main tools in our approach are the logarithmic Sobolev inequality and entropy flow. The method will be applied to the adjacency matrices of Erdos-Renyi graphs.

Direct products of finitely generated free groups have a surprisingly rich subgroup structure. We will talk about how the finiteness properties of a subgroup of a direct product relate to the way it is embedded in the ambient product. Central to this connection is a conjecture on finiteness properties of fibre products, which we will present along with different approaches towards solving it.

It is known that generic and universal structures and Ramsey classes are related. We explain this connection and complement it by some new examples. Particularly we disscuss universal and Ramsey classes defined by existence and non-existence of homomorphisms.

The problem of isometric embedding of a Riemannian Manifold into

Euclidean space is a classical issue in differential geometry and

nonlinear PDE. In this talk, I will outline recent work my

co-workers and I have done, using ideas from continuum mechanics as a guide,

formulating the problem, and giving (we hope) some new insight

into the role of " entropy".

The main mechanism for crystal plasticity is the formation and motion of a special class of defects, the dislocations. These are topological defects in the crystalline structure that can be identify with lines on which energy concentrates. In recent years there has been a considerable effort for the mathematical derivation of models that describe these objects at different scales (from an energetic and a dynamical point of view). The results obtained mainly concern special geometries, as one dimensional models, reduction to straight dislocations, the activation of only one slip system, etc.

The description of the problem is indeed extremely complex in its generality.

In the presentation will be given an overview of the variational models for dislocations that can be obtained through an asymptotic analysis of systems of discrete dislocations.

Under suitable scales we study the ``variational limit'' (by means of Gamma-convergence) of a three dimensional (static) discrete model and deduce a line tension anisotropic energy. The characterization of the line tension energy density requires a relaxation result for energies defined on curves.

The talk will explain how a geometric principle gave rise to a new variational description of information-theoretic entropy and how this led to the solution of a problem dating back to the 50's: whether the the central limit theorem is driven by an analogue of the second law of thermodynamics.

This problem is classically addressed by the so-called Skorohod Embedding problem. We instead develop a stochastic control approach. Unlike the previous literature, our formulation seeks the optimal no arbitrage bounds given the knowledge of the distribution at some (or various) point in time. This problem is converted into a classical stochastic control problem by means of convex duality. We obtain a general characterization, and provide explicit optimal bounds in some examples beyond the known classical ones. In particular, we solve completely the case of finitely many given marginals.

In spirit with John's talk we will discuss how topological invariants can be defined within a purely algebraic framework. After having introduced étale fundamental groups, we will discuss conjectures of Gieseker, relating those to certain "flat bundles" in finite characteristic. If time remains we will comment on the recent proof of Esnault-Sun.

• James Kirkpatrick - "Drift Diffusion modelling of organic solar cells: including electronic disorder".
• Timothy Reis - "Moment-based boundary conditions for the Lattice Boltzmann method".
• Matthew Moore - "Introducing air cushioning to Wagner theory".
• Matthew Hennessy - “Organic Solar Cells and the Marangoni Instability”.

In the last twelve years there has been much study of what happens when an algebraic curve in $n$-space is intersected with two multiplicative relations $x_1^{a_1} \cdots x_n^{a_n}~=~x_1^{b_1} \cdots x_n^{b_n}~=~1 \eqno(\times)$ for $(a_1, \ldots ,a_n),(b_1,\ldots, b_n)$ linearly independent in ${\bf Z}^n$. Usually the intersection with the union of all $(\times)$ is at most finite, at least in zero characteristic. In Oxford nearly three years ago I could treat a special curve in positive characteristic. Since then there have been a number of advances, even for additive relations $\alpha_1x_1+\cdots+\alpha_nx_n~=~\beta_1x_1+\cdots+\beta_nx_n~=~0 \eqno(+)$ provided some extra structure of Drinfeld type is supplied. After reviewing the zero characteristic situation, I will describe recent work, some with Dale Brownawell, for $(\times)$ and for $(+)$ with Frobenius Modules and Carlitz Modules.

Solving partial differential equations (PDEs) on curved surfaces is

important in many areas of science. The Closest Point Method is a new

technique for computing numerical solutions to PDEs on curves,

surfaces, and more general domains. For example, it can be used to

solve a pattern-formation PDE on the surface of a rabbit.

A benefit of the Closest Point Method is its simplicity: it is easy to

understand and straightforward to implement on a wide variety of PDEs

and surfaces. In this presentation, I will introduce the Closest

Point Method and highlight some of the research in this area. Example

computations (including the in-surface heat equation,

reaction-diffusion on surfaces, level set equations, high-order

interface motion, and Laplace--Beltrami eigenmodes) on a variety of

surfaces will demonstrate the effectiveness of the method.

Chebyshev polynomials are arguably the most useful orthogonal polynomials for computational purposes. In one dimension they arise from the close relationship that exists between Fourier series and polynomials. We describe how this relationship generalizes to Fourier series on certain symmetric lattices, that exist in all dimensions. The associated polynomials can not be seen as tensor-product generalizations of the one-dimensional case. Yet, they still enjoy excellent properties for interpolation, integration, and spectral approximation in general, with fast FFT-based algorithms, on a variety of domains. The first interesting case is the equilateral triangle in two dimensions (almost). We further describe the generalization of Chebyshev polynomials of the second kind, and many new kinds are found when the theory is completed. Connections are made to Laplacian eigenfunctions, representation theory of finite groups, and the Gibbs phenomenon in higher dimensions.

The aim of this work is to show how to derive the electricity price from models for the

underlying construction of the bid-stack. We start with modelling the behaviour of power

generators and in particular the bids that they submit for power supply. By modelling

the distribution of the bids and the evolution of the underlying price drivers, that is

the fuels used for the generation of power, we can construct an spede which models the

evolution of the bids. By solving this SPDE and integrating it up we can construct a

bid-stack model which evolves in time. If we then specify an exogenous demand process

it is possible to recover a model for the electricity price itself.

In the case where there is just one fuel type being used there is an explicit formula for

the price. If the SDEs for the underlying bid prices are Ornstein-Uhlenbeck processes,

then the electricity price will be similar to this in that it will have a mean reverting

character. With this price we investigate the prices of spark spreads and swing options.

In the case of multiple fuel drivers we obtain a more complex expression for the price

as the inversion of the bid stack cannot be used to give an explicit formula. We derive a

general form for an SDE for the electricity price.

We also show that other variations lead to similar, though still not tractable expressions

for the price.

We will attempt to introduce fusion systems in a way comprehensible to a Geometric Group Theorist. We will show how Bass--Serre thoery allows us to realise fusion systems inside infinite groups. If time allows we will discuss a link between the above and $\mathrm{Out}(F_n)$.

Colonies of motile microorganisms, the cytoskeleton and its components, cells and tissues have much in common with soft condensed matter systems (i.e. liquid crystals, amphiphiles, colloids etc.), but also exhibit behaviors that do not appear in inanimate matter and that are crucial for biological functions.

These unique properties arise when the constituent particles are active: they consume energy from internal and external sources and dissipate it by moving through the medium they inhabit. In this talk I will give a brief introduction to the notion of "active matter" and present some recent results on the hydrodynamics of active nematics suspensions in two dimensions.

The effective cone of the Hilbert scheme of points in $P^2$ has

finitely many chambers corresponding to finitely many birational models.

In this talk, I will identify these models with moduli of

Bridgeland-stable two-term complexes in the derived category of

coherent sheaves on $P^2$ and describe a

map from (a slice of) the stability manifold of $P^2$

to the effective cone of the Hilbert scheme that would explain the

correspondence. This is joint work with Daniele Arcara and Izzet Coskun.

The 2-dimensional assignment problem (minimum cost matching) is solvable in polynomial time, and it is known that a random instance of size n, with entries chosen independently and uniformly at random from [0,1], has expected cost tending to π^2/6.  In dimensions 3 and higher, the "planar" assignment problem is NP-complete, but what is the expected cost for a random instance, and how well can a heuristic do?  In d dimensions, the expected cost is of order at least n^{2-d} and at most ln n times larger, but the upper bound is non-constructive.  For 3 dimensions, we show a heuristic capable of producing a solution within a factor n^ε of the lower bound, for any constant ε, in time of order roughly n^{1/ε}.  In dimensions 4 and higher, the question is wide open: we don't know any reasonable average-case assignment heuristic.