Past

We treat the first n terms of general orthogonal series evolving with n as the partial sum process, and proved that under Menshov-Rademacher condition, the partial sum process can be enhanced into a geometric 2-rough process. For Fourier series, the condition can be improved, with an equivalent condition on limit function identified.

Translation actions appear all over geometry, so it is not surprising that there are many cases of moduli problems which involve non-reductive group actions, where Mumford’s geometric invariant theory does not apply. One example is that of jets of holomorphic map germs from the complex line to a projective variety, which is a central object in global singularity theory. I will explain how to construct this moduli space using the test curve model of Morin singularities and how this can be generalized to study the quotient of projective varieties by a wide class of non-reductive groups. In particular, this gives information about the invariant ring. This is joint work with Frances Kirwan.

We consider the forest fire process on Z: on each site, seeds and matches fall at random, according to some independent Poisson processes. When a seed falls on a vacant site, a tree immediately grows. When a match falls on an occupied site, a fire destroys immediately the corresponding occupied connected component. We are interested in the asymptotics of rare fires. We prove that, under space/time re-scaling, the process converges (as matches become rarer and rarer) to a limit forest fire process.
Next, we consider the more general case where seeds and matches fall according to some independent stationary renewal processes (not necessarily Poisson). According to the tail distribution of the law of the delay between two seeds (on a given site), there are 4 possible scaling limits.
We finally introduce some related coagulation-fragmentation equations, of which the stationary distribution can be more or less explicitely computed and of which we study the scaling limit.

I will study the sequestering of blow-up fields through a CFT in a toroidal orbifold setting. In particular, I will examine the disk correlator between orbifold blow-up moduli and matter Yukawa couplings. Blow-up moduli appear as twist fields on the worldsheet which introduce a monodromy

condition for the coordinate field X. Thus I will focus on how the presence of twist field affects

the CFT calculation of disk correlators. Further, I will explain how the results are relevant to

suppressing soft terms to scales parametrically below the gravitino mass. Last, I want to explore the

relevance of our calculation for the case of smooth Calabi-Yaus.

In the SABR model of Hagan et al. [2002] a perturbative expansion approach yields a tractable approximation to the implied volatility smile. This approximation formula has been adopted across the financial markets as a means of interpolating market volatility surfaces. All too frequently - in stressed markets, in the long-dated FX regime - the limitations of this approximation are pronounced. In this talk a highly efficient conditional integration approach, motivated by the work of Stein and Stein [1991] and Willard [1997], will be presented that when applied to the SABR model not only produces a volatility smile consistent with the underlying SABR process but gives access to the joint distribution of the asset and its volatility. The latter is particularly important in understanding the dynamics of the volatility smile as it evolves through time and the subsequent effect on the pricing of exotic options.

William McGhee is Head of Hybrid Quantitative Analytics at The Royal Bank of Scotland and will also discuss within the context of this presentation the interplay of mathematical modelling and the technology infrastructure required to run a complex hybrids trading business and the benefits of highly efficient numerical algorithms."

• Derek Moulton - "Growth and morphology of seashells"

• Simon Cotter - "A Hybrid stochastic finite element method for solving Fokker-Planck equations"

• Apala Majumdar -"The theory of liquid crystals - analysis, computation and applications"

The following two topics are likely to be discussed.

A) Modelling the collective behaviour of chicken flocks. Marian Dawkins has a joint project with Steve Roberts in Engineering studying the patterns of optical flow in large flocks of commercial broiler chickens. They have found that various measurements of flow (such as skew and kurtosis) are predictive of future mortality. Marian would be interested in seeing whether we can model these effects.
B) Asymmetrical prisoners’ dilemma games. Despite massive theoretical interest, there are very few (if any) actual examples of animals showing the predicted behaviour of reciprocity with delayed reward. Marian Dawkins suspects that the reason for this is that the assumptions made are unrealistic and she would like to explore some ideas about this.

Please note the slightly early start to accommodate the OCCAM group meeting that follows.

In these two talks we will look at a recent paper by David Anderson and Des Higham: http://arxiv.org/pdf/1107.2181 This paper takes the Multilevel Monte Carlo method which I developed in 2006 for Brownian SDEs, and comes up with an elegant way of applying it to stochastic biochemical reaction networks.

In this meeting

This is a joint work with Emmanuel Ullmo.

This work is motivated by J.Pila's strategy to prove the Andre-Oort conjecture. One ingredient in the strategy is the following

conjecture:

Let S be a Shimura variety uniformised by a symmetric space X.

Let V be an algebraic subvariety of S. Maximal algebraic subvarieties of the preimage of V in X are precisely the

components of the preimages of weakly special subvarieties contained in V.

We will explain the proof of this conjecture in the case where S is compact.

Computational Fluid Dynamics (CFD) has become an

indispensable tool in designing turbomachinery components in all sectors of

Rolls-Royce business units namely, Aerospace, Industrial, Marine and Nuclear.

Increasingly sophisticated search and optimisation techniques are used based on

both traditional optimisation methods as well as, design of computer experiment

techniques, advanced surrogate methods, and evolutionary optimisation

techniques. Geometry and data representation as well as access, queuing and

loading control of large high performance computing clusters are areas of

research to establish the most efficient techniques for improving the

performance of an already highly efficient modern jet engine.

\\

\\

This presentation focuses on a high fidelity design

optimisation framework called SOPHY that is used in Rolls-Royce to provide

parametric geometry, automatic meshing, advanced design-space search

algorithms, accurate and robust CFD methodology and post-processing. The

significance of including the so-called real geometry features and interaction

of turbomachinery components in the optimisation cycle are discussed. Examples are drawn from real world

applications of the SOPHY design systems in an engine project.

In this work, we study equilibrium solutions for a LQ

control problem with state-dependent terms in the objective, which

destroy the time-consisitence of a pre-commited optimal solution.

We get a sufficient condition for equilibrium by a system of

stochastic differential equations. When the coefficients in the

problem are all deterministic, we find an explicit equilibrium

for general LQ control problem. For the mean-variance portfolio

selection in a complete financial market, we also get an explicit

equilibrium with random coefficient of the financial.

Chern-Simons theory is topological gauge theory in three dimensions that contains an interesting class of operators called Wilson lines/loops, which have connections with both physics and pure mathematics. In particular, it has been shown that computations with Wilson operators in Chern-Simons theory reproduce knot invariants, and are also related to Gauss linking invariants. We will discuss the complex generalizations of these ideas, which are known as holomorphic Chern-Simons theory, Wilson operators, and linking, in the setting of Calabi-Yau three-folds. This will (hopefully) include a definition of all three of these holomorphic analogues as well as an investigation into how these ideas can be translated into simple homological algebra, allowing us to propose the existence of "homological Feynman rules" for computing things like Wilson operators in a holomorphic Chern-Simons theory. If time permits I may say something about physics too.

We explore methods (deterministic and otherwise) of composing music using mathematical models. Musical examples will be provided throughout and the audience (with the speakers assistance) will compose a brand new piece.

Diffusive process with discontinuous coefficients provide significant computational challenges. We consider the solution of a diffusive process in a domain where the diffusion coefficient changes discontinuously across a curved interface. Rather than seeking to construct discretizations that match the interface, we consider the use of regularly-shaped meshes so that the interface "cuts'' through the cells (elements or volumes). Consequently, the discontinuity in the diffusion coefficients has a strong impact on the accuracy and convergence of the numerical method. We develop an adjoint based a posteriori error analysis technique to estimate the error in a given quantity of interest (functional of the solution). In order to employ this method, we first construct a systematic approach to discretizing a cut-cell problem that handles complex geometry in the interface in a natural fashion yet reduces to the well-known Ghost Fluid Method in simple cases. We test the accuracy of the estimates in a series of examples.

A polar space $\Pi$ is a geometry whose elements are the totally isotropic subspaces of a vector space $V$ with respect to either an alternating, Hermitian, or quadratic form. We may form a new geometry $\Gamma$ by removing all elements contained in either a hyperplane $F$ of $\Pi$, or a hyperplane $H$ of the dual $\Pi^*$. This is a \emph{biaffine polar space}.

We will discuss two specific examples, one with automorphism group $q^6:SU_3(q)$ and the other $G_2(q)$. By considering the stabilisers of a maximal flag, we get an amalgam, or "glueing", of groups for each example. However, the two examples have "similar" amalgams - this leads to a group recognition result for their automorphism groups.

Generalized Donaldson-Thomas invariants $\bar{DT}^\alpha(\tau)$ defined by Joyce and Song are rational numbers which 'count' both $\tau$-stable and $\tau$-semistable coherent sheaves with Chern character $\alpha$ on a Calabi-Yau 3-fold X, where $\tau$ denotes Gieseker stability for some ample line bundle on X. The theory of Joyce and Song is valid only over the field $\mathbb C$. We will extend it to algebraically closed fields $\mathbb K$ of characteristic zero.

We will describe the local structure of the moduli stack $\mathfrak M$ of coherent sheaves on X, showing that an atlas for $\mathfrak M$ may be written locally as the zero locus of an almost closed 1-form on an \'etale open subset of the tangent space of $\mathfrak M$ at a point, and use this to deduce identities on the Behrend

function $\nu_{\mathfrak M}$ of $\mathfrak M$. This also yields an extension of generalized Donaldson-Thomas theory to noncompact Calabi-Yau 3-folds.

Finally, we will investigate how our argument might yield generalizations of the theory to a even wider context, for example the derived framework using Toen's theory and to motivic Donaldson-Thomas theory in the style of Kontsevich and Soibelman.

An embedding of a graph H in a graph G is an injective mapping of the vertices of H to the vertices of G such that edges of H are mapped to edges of G. Embedding problems have been extensively studied. A very powerful tool in this area is Szemeredi's Regularity Temma. It approximates the host graph G by a quasirandom graph which inherits many of the properties of G. Unfortunately the direct use of Szemeredi's Regularity Lemma is useless if the host graph G is sparse.

During the talk I shall expose a technique to deal with embedding trees in sparse graphs. This technique has been developed by Ajtai, Komlos,Simonovits and Szemeredi to solve the Erdos-Sos conjecture. Presently the author together with Hladky, Komlos, Simonovits, Stein and Szemeredi apply this method to solve the related conjecture of Loebl, Komlos and Sos (approximate version).

Automorphisms of right-angled Artin groups interpolate between $Out(F_n)$ and $GL_n(\mathbb{Z})$. An active area of current research is to extend properties that hold for both the above groups to $Out(A_\Gamma)$ for a general RAAG. After a short survey on the state of the art, we will describe our recent contribution to this program: a study of how higher-rank lattices can act on RAAGs that builds on the work of Margulis in the free abelian case, and of Bridson and the author in the free group case.

We will consider a branching Brownian motion where particles have a drift $-\rho$, binary branch at rate $\beta$ and are killed if they hit the origin. This process is supercritical  if $\beta>\rho^2/2$ and we will discuss the survival probability in the regime as criticality is approached. (Joint work with Elie Aidekon)

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.

\\

\\

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.