Past

Let C be a (smooth projective algebraic) curve. It is well known that the Jacobian J of C is a principally polarized abelian variety. In otherwords, J is self-dual in the sense that J is identified with the space of topologically trivial line bundles on itself.

Suppose now that C is singular. The Jacobian J of C parametrizes topologically trivial line bundles on C; it is an algebraic group which is no longer compact. By considering torsion-free sheaves instead of line bundles, one obtains a natural singular compactification J' of J.

In this talk, I consider (projective) curves C with planar singularities. The main result is that J' is self-dual: J' is identified with a space of torsion-free sheaves on itself. This autoduality naturally fits into the framework of the geometric Langlands conjecture; I hope to sketch this relation in my talk.

We describe a construction of the Brownian measure on Jordan curves with respect to the Weil-Petersson metric. The step from Brownian motion on the diffeomorphism group of the circle to Brownian motion on Jordan curves in the complex plane requires probabilistic arguments well beyond the classical theory of conformal welding, due to the lacking quasi-symmetry of canonical Brownian motion on Diff(S1). A new key step in our construction is the systematic use of a Kählerian diffusion on the space of Jordan curves for which the welding functional gives rise to conformal martingales.

Consistent truncations have proved to be powerful tools in the construction of new string theory solutions. Recently, they have been employed in the holographic description of condensed matter systems. In the talk, I will present a rich class of supersymmetric consistent truncations of higher-dimensional supergravity which are based on geometric structures, focusing on the tri-Sasakian case. Then I will discuss some applications, including a general result relating AdS backgrounds to solutions with non-relativistic Lifshitz symmetry.

Yves Couder and co-workers have recently reported the results of a startling series of experiments in which droplets bouncing on a fluid surface exhibit several dynamical features previously thought to be peculiar to the microscopic realm. In an attempt to 

develop a connection between the fluid and quantum systems, we explore the Madelung transformation, whereby Schrodinger's equation is recast in a hydrodynamic form. New experiments are presented, and indicate the potential value of this hydrodynamic approach to both visualizing and understanding quantum mechanics.

A new approach for data-based stochastic parametrisation of unresolved scales and processes in numerical weather and climate prediction models is introduced. The subgrid-scale model is conditional on the state of the resolved scales, consisting of a collection of local models. A clustering algorithm in the space of the resolved variables is combined with statistical modelling of the impact of the unresolved variables. The clusters and the parameters of the associated subgrid models are estimated simultaneously from data. The method is tested and explored in the framework of the Lorenz '96 model using discrete Markov processes as local statistical models. Performance of the scheme is investigated for long-term simulations as well as ensemble prediction. The present method clearly outperforms simple parametrisation schemes and compares favourably with another recently proposed subgrid scheme also based on conditional Markov chains.

In this paper we deal with the utility maximization problem with a

preference functional of expected utility type. We derive a new approach

in which we reduce the utility maximization problem with general utility

to the study of a fully-coupled Forward-Backward Stochastic Differential

Equation (FBSDE).

The talk is based on joint work with Ying Hu, Peter Imkeller, Anthony

Reveillac and Jianing Zhang.

10am Radius Health - Mark Evans

10:30am NAG - Mick Pont and Lawrence Mulholland

Please note, that Thales are also proposing several projects but the academic supervisors have already been allocated.

The mechanisms for the selection of the propagation speed of waves

connecting unstable to stable states will be discussed in the

spatially non-homogeneous case, the differences from the very

well-studied homogeneous version being emphasised.

Let $x$ be a CM point in the moduli space $\mathcal{A}_g(\mathbb{C})$ of principally

polarized complex abelian varieties of genus $g$, corresponding to an

Abelian variety $A$ with complex multiplication by a ring $R$. Edixhoven

conjectured that the size of the Galois orbit of x should grow at least

like a power of the discriminant ${\rm Disc}(R)$ of $R$. For $g=1$, this reduces to the

classical Brauer-Siegel theorem. A positive answer to this conjecture

would be very useful in proving the Andr\'e-Oort conjecture unconditionally.

We will present a proof of the conjectured lower bounds in some special

cases, including $g\le 6$. Along the way we derive transfer principles for

torsion in class groups of different fields which may be interesting in

their own right.

We will discuss the use of hypergraph-based methods for orderings of sparse matrices in Cholesky, LU and QR factorizations. For the Cholesky factorization case, we will investigate a recent result on pattern-wise decomposition of sparse matrices, generalize the result and develop algorithmic tools to obtain effective ordering methods. We will also see that the generalized results help us formulate the ordering problem in LU much like we do for the Cholesky case, without ever symmetrizing the given matrix $A$ as $A+A^{T}$ or $A^{T}A$. For the QR factorization case, the use of hypergraph models is fairly standard. We will nonetheless highlight the fact that the method again does not form the possibly much denser matrix $A^{T}A$. We will see comparisons of the hypergraph-based methods with the most common alternatives in all three cases.

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This is joint work with Iain S. Duff.

Cubature on Wiener space" is a numerical method for the weak

approximation of SDEs. After an introduction to this method we present

some cases when the method is computationally expensive, and highlight

some techniques that improve the tractability. In particular, we adapt

the Multilevel Monte-Carlo framework and extend the Milstein-scheme

based version of Mike Giles to higher dimensional and higher degree cases.

We will give an introduction to the theory of d-manifolds, a new class of geometric objects recently/currently invented by Joyce (see http://people.maths.ox.ac.uk/joyce/dmanifolds.html). We will start from scratch, by recalling the definition of a 2-category and talking a bit about $C^\infty$-rings, $C^\infty$-schemes and d-spaces before giving the definition of what a d-manifold should be. We will then discuss some properties of d-manifolds, and say some words about d-manifold bordism and its applications.

We will start off with a crash course in General relativity, and then I'll describe a 'recipe' for a time machine. This will lead us to the question whether or not the topology of the universe can change. We will see that, in some sense, this is topologically allowed. However, the Einstein equation gives a certain condition on the Ricci tensor (which is violated by certain quantum effects) and meeting this condition is a more delicate problem.

In biological cells, molecules are transported actively or by diffusion and react with each other when they are close.

The reactions occur with certain probability and there are few molecules of some chemical species. Therefore, a stochastic model is more accurate compared to a deterministic, macroscopic model for the concentrations based on partial differential equations.

At the mesoscopic level, the domain is partitioned into voxels or compartments. The molecules may react with other molecules in the same voxel and move between voxels by diffusion or active transport. At a finer, microscopic level, each individual molecule is tracked, it moves by Brownian motion and reacts with other molecules according to the Smoluchowski equation. The accuracy and efficiency of the simulations are improved by coupling the two levels and only using the micro model when it is necessary for the accuracy or when a meso description is unknown.

Algorithms for simulations with the mesoscopic, microscopic and meso-micro models will be described and applied to systems in molecular biology in three space dimensions.

Cell motility is a crucial part of many biological processes including wound healing, immunity and embryonic development. The interplay between mechanical forces and biochemical control mechanisms make understanding cell motility a rich and exciting challenge for mathematical modelling. We consider the two-phase, poroviscous, reactive flow framework used in the literature to describe crawling cells and present a stripped down version. Linear stability analysis and numerical simulations provide insight into the onset of polarization of a stationary cell and reveal qualitatively distinct families of travelling wave solutions. The numerical solutions also capture the experimentally observed behaviour that cells crawl fastest when the surface they crawl over is neither too sticky nor too slippy.

We introduce the notion of a real cubing. Roughly speaking, real cubings are to CAT(0) cube complexes what real trees are to simplicial trees. We develop an analogue of the Rips’ machine and establish the structure of groups acting nicely on real cubings.

To identify the Martin boundary for a transient Markov chain with Green's function G(x,y), one has to identify all possible limits Lim G(x,y_n)/G(0,y_n) with y_n "tending to infinity". For homogeneous random walks, these limits are usually obtained from the exact asymptotics of Green's function G(x,y_n). For non-homogeneous random walks, the exact asymptotics af Green's function is an extremely difficult problem. We discuss several examples where Martin boundary can beidentified by using large deviation technique. The minimal Martin boundary is in general not homeomorphic to the "radial"  compactification obtained by Ney and Spitzer for homogeneous random walks in Z^d : convergence of a sequence of points y_n toa point on the Martin boundary does not imply convergence of the sequence y_n/|y_n| on the unit sphere. Such a phenomenon is a consequence of non-linear optimal large deviation trajectories.

Symplectic implosion is a construction in symplectic geometry due to Guillemin, Jeffrey and Sjamaar, which is related to geometric invariant theory for non-reductive group actions in algebraic geometry. This talk (based on joint work in progress with Andrew Dancer and Andrew Swann) is concerned with an analogous construction in hyperkahler geometry.

 In the recent series of papers Kleban, Simmons, and Ziff gave a non-rigorous computation  (base on Conformal Field Theory) of probabilities of several connectivity events for critical percolation. In particular they showed that the probability that there is a percolation cluster connecting two points on the boundary and a point inside the domain can be factorized in therms of pairwise connection probabilities. We are going to use SLE techniques to rigorously compute probabilities of several connectivity events and prove the factorization formula.

We compute three-point functions of single trace operators in planar N = 4 SYM. We consider the limit where one of the operators is much smaller than the other two. We find a precise match between weak and strong coupling in the Frolov-Tseytlin classical limit for a very general class of classical solutions. To achieve this match we clarify the issue of back-reaction and identify precisely which three-point functions are captured by a classical computation.

The standard approach to continuous-time finance starts from postulating a

statistical model for the prices of securities (such as the Black-Scholes

model). Since such models are often difficult to justify, it is

interesting to explore what can be done without any stochastic

assumptions. There are quite a few results of this kind (starting from

Cover 1991 and Hobson 1998), but in this talk I will discuss

probability-type properties emerging without a statistical model. I will

only consider the simplest case of one security, and instead of stochastic

assumptions will make some analytic assumptions. If the price path is

known to be cadlag without huge jumps, its quadratic variation exists

unless a predefined trading strategy earns infinite capital without

risking more than one monetary unit. This makes it possible to apply the

known results of Ito calculus without probability (Follmer 1981, Norvaisa)

in the context of idealized financial markets. If, moreover, the price

path is known to be continuous, it becomes Brownian motion when physical

time is replaced by quadratic variation; this is a probability-free

version of the Dubins-Schwarz theorem.

(Joint with Ronnie Nagloo.) I investigate algebraic relations between sets of solutions (and their derivatives) of the "generic" Painlev\'e equations I-VI, proving a somewhat weaker version of ``there are NO algebraic relations".

In this talk I shall discuss special polynomials associated with rational solutions of the Painlevé equations and of the soliton equations which are solvable by the inverse scattering method, including the Korteweg-de Vries, Boussinesq and nonlinear Schrodinger equations. Further I shall illustrate applications of these polynomials to vortex dynamics and rogue waves.

The Painlevé equations are six nonlinear ordinary differential equations that have been the subject of much interest in the past thirty years, and have arisen in a variety of physical applications. Further the Painlevé equations may be thought of as nonlinear special functions. Rational solutions of the Painlevé equations are expressible in terms of the logarithmic derivative of certain special polynomials. For the fourth Painlevé equation these polynomials are known as the generalized Hermite polynomials and generalized Okamoto polynomials. The locations of the roots of these polynomials have a highly symmetric (and intriguing) structure in the complex plane.

It is well known that soliton equations have symmetry reductions which reduce them to the Painlevé equations, e.g. scaling reductions of the Boussinesq and nonlinear Schrödinger equations are expressible in terms of the fourth Painlevé equation. Hence rational solutions of these equations can be expressed in terms of the generalized Hermite and generalized Okamoto polynomials.

I will also discuss the relationship between vortex dynamics and properties of polynomials with roots at the vortex positions. Classical polynomials such as the Hermite and Laguerre polynomials have roots which describe vortex equilibria. Stationary vortex configurations with vortices of the same strength and positive or negative configurations are located at the roots of the Adler-Moser polynomials, which are associated with rational solutions of the Kortweg-de Vries equation.

Further, I shall also describe some additional rational solutions of the Boussinesq equation and rational-oscillatory solutions of the focusing nonlinear Schrödinger equation which have applications to rogue waves.

The class of finite dimensional algebras over an algebraically closed field K

may be divided into two disjoint subclasses (tame and wild dichotomy).

One class

consists of the tame algebras for which the indecomposable modules

occur, in each dimension d, in a finite number of discrete and a

finite number of one-parameter families. The second class is formed by

the wild algebras whose representation theory comprises the

representation theories of all finite dimensional algebras over K.

Hence, the classification of the finite dimensional modules is

feasible only for the tame algebras. Frequently, applying deformations

and covering techniques, we may reduce the study of modules over tame

algebras to that for the corresponding simply connected tame algebras.

We shall discuss the problem concerning connection between the

tameness of simply connected algebras and the weak nonnegativity of

the associated Tits quadratic forms, raised in 1975 by Sheila Brenner.

We will study the question of whether the adjoint of a given matrix can be written as a rational function in the matrix. After showing necessary and sufficient conditions, rational interpolation theory will help to characterize the most important existing cases. Several topics related to our question will be explored. They range from short recurrence Krylov subspace methods to the roots of harmonic polynomials and harmonic rational functions. The latter have recently found interesting applications in astrophysics, which will briefly be discussed as well.

: Recently strict local martingales have been used to model

exchange rates. In such models, put-call parity does not hold if one

assumes minimal superreplicating costs as contingent claim prices. I

will illustrate how put-call parity can be restored by changing the

definition of a contingent claim price.

More precisely, I will discuss a change of numeraire technique when the

underlying is only a local martingale. Then, the new measure is not

necessarily equivalent to the old measure. If one now defines the price

of a contingent claim as the minimal superreplicating costs under both

measures, then put-call parity holds. I will discuss properties of this

new pricing operator.

To illustrate this techniques, I will discuss the class of "Quadratic

Normal Volatility" models, which have drawn much attention in the

financial industry due to their analytic tractability and flexibility.

This talk is based on joint work with Peter Carr and Travis Fisher.

We will explain Bridgelands results on the stabiltiy manifold of a K3 surface. As an application we will define the stringy Kaehler moduli space of a K3 surface and comment on the mirror symmetry picture.

I will begin by defining the notion of a characteristic class of surface bundles, and constructing the MMM (Miller-Morita-Mumford) classes as examples. I will then talk about a recent theorem of Church, Farb, and Thibault which shows that the characteristic numbers associated to certain MMM-classes do not depend on how the total space is fibred as a surface bundle - they depend only on the topology of the total space itself. In particular they don't even depend on the genus of the fibre. Hence there are many 'coincidences' between the characteristic numbers of very different-looking surface bundles.

A corollary of this is an obstruction to low-genus fiberings: given a smooth manifold E, the non-vanishing of a certain invariant of E implies that any surface bundle with E as its total space must have a fibre with genus greater than a certain lower bound.

Also, following the paper of Church-Farb-Thibault, I will sketch how to construct examples of 4-manifolds which fibre in two distinct ways as a surface bundle over another surface, thus giving concrete examples to which the theorem applies.

There exist two constructions of families of exotic monotone Lagrangian tori in complex projective spaces and products of spheres, namely the one by Chekanov and Schlenk, and the one via the Lagrangian circle bundle construction of Biran. It was conjectured that these constructions give Hamiltonian isotopic tori. I will explain why this conjecture is true in the complex projective plane and the product of two two-dimensional spheres.

Hex was discovered independently by Piet Hein in Copenhagen in 1942 and byJohn Nash in Princeton in 1948.  The game is interesting because its rules are very simple, yet it is not known how to play best possible.  For example, a winning first move for the first player (who does have  a winning strategy) is still unknown. The talk will tell the history of the game and give simple proofs for basic results about it. Also the reverse game of HEX,sometimes called REX, will be discussed. New results about REX are under publication in Discrete Mathematics in a paper:  How to play Reverse Hex (joint work with Ryan Hayward and Phillip Henderson).

Quaternionic quantum Hamiltonians describing nonrelativistic spin particles

require the ambient physical space to have five dimensions. The quantum

dynamics of a spin-1/2 particle system characterised by a generic such

Hamiltonian is described. There exists, within the structure of quaternionic

quantum mechanics, a canonical reduction to three spatial dimensions upon

which standard quantum theory is retrieved. In this dimensional reduction,

three of the five dynamical variables oscillate around a cylinder, thus

behaving in a quasi one-dimensional manner at large distances. An analogous

mechanism exists in the case of octavic Hamiltonians, where the ambient

physical space has nine dimensions. Possible experimental tests in search

for the signature of extra dimensions at low energies are briefly discussed.

(Talk based on joint work with Eva-Maria Graefe, Imperial.)

This is a joint work with Craig Evans. We study the partial regularity of minimizers for certain functionals in the calculus of variations, namely the modified Landau-de Gennes energy functional in nematic liquid crystal theory introduced by Ball and Majumdar.