Past

there will be no seminar in this week.

The first half of this seminar will discuss the hedging problem faced by a large sports betting agent who has to risk-manage an unwanted position in a bet on the simultaneous outcome of multiple football matches, by trading in moderately liquid simple bets on individual results. The resulting mathematical framework is that of a coupled system of multi-dimensional HJB equations.

This leads to the wider question of the numerical approximation of such problems. Dynamic programming with PDEs, while very accurate in low dimensions, becomes practically intractable as the dimensionality increases. Monte Carlo methods, while robust for computing linear expectations in high dimensions, are not per se well suited to dynamic programming. This leaves high-dimensional stochastic control problems to be considered computationally infeasible in general.

In the second half of the seminar, we will outline ongoing work in this area by sparse grid techniques and asymptotic expansions, the former exploiting smoothness of the value function, the latter a fast decay in the importance of principal components. We hope to instigate a discussion of other possible approaches including e.g. BSDEs.

15th Biennial OXFORD / CAMBRIDGE MEETING

PROGRAMME FOR THE

‘WOOLLY OWL TROPHY’

Invited Judges

John Harper

(Victoria University of Wellington, NZ)

Arash Yavari

(Georgia Tech, Atlanta, USA)

Sharon Stephen

(University of Birmingham, UK)

10:45 Morning Coffee The Maths Inst Common Room

... for Torelli groups of surfaces.

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.

Human T-lymphotropic virus type I (HTLV-I) is a persistent human retrovirus characterised by a high proviral load and risk of developing ATL, an aggressive blood cancer, or HAM/TSP, a progressive neurological and inflammatory disease. Infected individuals typically mount a large, chronically activated HTLV-I-specific CTL response, yet the virus has developed complex mechanisms to evade host immunity and avoid viral clearance. Moreover, identification of determinants to the development of disease has thus far been elusive.

 This model is based on a recent experimental hypothesis for the persistence of HTLV-I infection and is a direct extension of the model studied by Li and Lim (2011). A four-dimensional system of ordinary differential equations is constructed that describes the dynamic interactions among viral expression, infected target cell activation, and the human immune response. Focussing on the particular roles of viral expression and host immunity in chronic HTLV-I infection offers important insights to viral persistence and pathogenesis.

The talk will address two recent results concerning the Doi-Smoluchowski equation and the Onsager model for nematic liquid crystals. The first result concerns the existence of inertial manifolds for the Smloluchowski equation both in the presence and in the absence of external flows. While the Doi-Smoluchowski equation as a PDE is an infinite-dimensional dynamical system, it reduces to a system of ODEs on a set coined inertial manifold, to which all other solutions converge exponentially fast.  The proof uses a non-standard method, which consists in circumventing the restrictive spectral-gap condition, which the original equation fails to satisfy by transforming the equation into a form that does. 

The second result concerns the isotropic-nematic phase transition for the Onsager model on the circle using more complicated potentials than the Maier-Saupe potential. Exact multiplicity of steady-states on the circle is proven for the two-mode truncation of the Onsager potential.    

In 1986 Kardar, Parisi, and Zhang proposed a stochastic PDE for the motion of driven interfaces,
in particular for growth processes with local updating rules. The solution to the 1D KPZ equation
can be approximated through the weakly asymmetric simple exclusion process. Based on work of 
Tracy and Widom on the PASEP, we obtain an exact formula for the one-point generating function of the KPZ
equation in case of sharp wedge initial data. Our result is valid for all times, but of particular interest is
the long time behavior, related to random matrices, and the finite time corrections. This is joint work with 
Tomohiro Sasamoto.

: Backward error analysis is a technique that has been extremely successful in understanding the behaviour of numerical methods for ordinary differential equations.  It is possible to fit an ODE (the so called modified equation) to a numerical method to very high accuracy. Backward error analysis has been of particular importance in the numerical study of Hamiltonian problems, since it allows to approximate symplectic numerical methods by a perturbed Hamiltonian system, giving an approximate statistical mechanics for symplectic methods. 

Such a systematic theory in the case of numerical methods for stochastic differential equations (SDEs) is currently lacking. In this talk we will describe a general framework for deriving modified equations for SDEs with respect to weak convergence. We will start by quickly recapping of how to derive modified equations in the case of ODEs and describe how these ideas can be generalized in the case of SDEs. Results will be presented for first order methods such as the Euler-Maruyama and the Milstein method. In the case of linear SDEs, using the Gaussianity of the underlying solutions, we will derive a SDE that the numerical method solves exactly in the weak sense. Applications of modified equations in the numerical study of Langevin equations and in the calculation of effective diffusivities will also be discussed, as well as the use of modified equations  as a tool for constructing higher order methods for stiff stochastic differential equations.

This is joint work with A. Abdulle (EPFL). D. Cohen (Basel), G. Vilmart (EPFL).

Graeme Segal shall describe some of Dan Quillen’s work, focusing on his amazingly productive period around 1970, when he not only invented algebraic K-theory in the form we know it today, but also opened up several other lines of research which are still in the front line of mathematical activity. The aim of the talk will be to give an idea of some of the mathematical influences which shaped him, of his mathematical perspective, and also of his style and his way of approaching mathematical problems.

When managing risk, frequently only imperfect hedging instruments are at hand.

We show how to optimally cross-hedge risk when the spread between the hedging

instrument and the risk is stationary. At the short end, the optimal hedge ratio

is close to the cross-correlation of the log returns, whereas at the long end, it is

optimal to fully hedge the position. For linear risk positions we derive explicit

formulas for the hedge error, and for non-linear positions we show how to obtain

numerically effcient estimates. Finally, we demonstrate that even in cases with no

clear-cut decision concerning the stationarity of the spread it is better to allow for

mean reversion of the spread rather than to neglect it.

The talk is based on joint work with Georgi Dimitroff, Gregor Heyne and Christian Pigorsch.

I'll start by defining the zeta function and stating the Weil conjectures (which have actually been theorems for some time now). I'll then go on by saying things like "Weil cohomology", "standard conjectures" and "Betti numbers of the Grassmannian". Hopefully by the end we'll all have learned something, including me.

Recent papers by Butler-Campbell-Kovàcs, Rump, Prihoda-Puninski and others introduce over an order O over a Dedekind domain D a notion of "generalized lattice", meaning a D-projective O-module.

We define a similar notion over Dedekind-like rings -- a class of rings intensively studied by Klingler and Levy. We examine in which cases every generalized lattices is a direct sum of ordinary -- i.e., finitely generated -- lattices. We also consider other algebraic and model theoretic questions about generalized lattices.

Voltammetry is a powerful method for interrogating electrochemical systems. A voltage is applied to an electrode and the resulting current response analysed to determine features of the system under investigation, such as concentrations, diffusion coefficients, rate constants and thermodynamic potentials. Here we will focus on ac voltammetry, where the voltage signal consists of a high frequency sine-wave superimposed on a linear ramp. Using multiple scales analysis, we find analytical solutions for the harmonics of the current response and show how they can be used to determine the system parameters. We also include the effects of capacitance due to the double-layer at the electrode surface and show that even in the presence of large capacitance, the harmonics of the current response can still be isolated using the FFT and the Hanning window.

I show how invariants of curves of genus 2 can be used for explicitly constructing class fields of

certain number fields of degree 4.

Following the framework of the heterogeneous multiscale method, we present a numerical method for nonlinear elliptic homogenization problems. We briefly review the numerical, relying on an efficient coupling of macro and micro solvers, for linear problems. A fully discrete analysis is then given for nonlinear (nonmonotone) problems, optimal convergence rates in the H¹ and L² norms are derived and the uniqueness of the method is shown on sufficiently fine macro and micro meshes.

Numerical examples confirm the theoretical convergence rates and illustrate the performance and versatility of our approach.

We consider a multidimensional structural credit model, where each company follows a jump-diffusion process and is connected with other companies via global factors. We assume that a company can default both expectedly, due to the diffusion part, and unexpectedly, due to the jump part, by a sudden fall in a company's value as a result of a global shock. To price CDOs efficiently, we use ideas, developed by Bush et al.

for diffusion processes, where the joint density of the portfolio is approximated by a limit of the empirical measure of asset values in the basket. We extend the method to jump-diffusion settings. In order to check if our model is flexible enough, we calibrate it to CDO spreads from pre-crisis and crisis periods.

For both data sets, our model fits the observed spreads well, and what is important, the estimated parameters have economically convincing values.

We also study the convergence of our method to basic Monte Carlo and conclude that for a CDO, that typically consists of 125 companies, the method gives close results to basic Monte Carlo."

The class of fields with a given absolute Galois group is in general not an elementary class. Looking instead at abstract elementary classes we can show that this class, as well as the class of pairs (F,K), where F is a function field in one variable over a perfect base field K with a fixed absolute Galois group, is abstract elementary. The aim is to show categoricity for the latter class. In this talk, we will be discussing some consequences of basic properties of these two classes.