Past

A dimer model on a surface with punctures is an embedded quiver such that its vertices correspond to the punctures and the arrows circle round the faces they cut out. To any dimer model Q we can associate two categories: A wrapped Fukaya category F(Q), and a category of matrix factorizations M(Q). In both categories the objects are arrows, which are interpreted as Lagrangian subvarieties in F(Q) and will give us certain matrix factorizations of a potential on the Jacobi algebra of the dimer in M(Q).

We show that there is a duality D on the set of all dimers such that for consistent dimers the category of matrix factorizations M(Q) is isomorphic to the Fukaya category of its dual,  F((DQ)). We also discuss the connection with classical mirror symmetry.

We say that a hypergraph is \emph{stable} if each sufficiently large subset of its vertices either spans many hyperedges or is very structured. Hypergraphs that arise naturally in many classical settings posses the above property. For example, the famous stability theorem of Erdos and Simonovits and the triangle removal lemma of Ruzsa and Szemeredi imply that the hypergraph on the vertex set $E(K_n)$ whose hyperedges are the edge sets of all triangles in $K_n$ is stable. In the talk, we will present the following general theorem: If $(H_n)_n$ is a sequence of stable hypergraphs satisfying certain technical conditions, then a typical (i.e., uniform random) $m$-element independent set of $H_n$ is very structured, provided that $m$ is sufficiently large. The above abstract theorem has many interesting corollaries, some of which we will discuss. Among other things, it implies sharp bounds on the number of sum-free sets in a large class of finite Abelian groups and gives an alternate proof of Szemeredi’s theorem on arithmetic progressions in random subsets of integers.

Joint work with Noga Alon, Jozsef Balogh, and Robert Morris.

The Market Selection Hypothesis is a principle which (informally) proposes that `less knowledgeable' agents are eventually eliminated from the market. This elimination may take the form of starvation (the proportion of output consumed drops to zero), or may take the form of going broke (the proportion of asset held drops to zero), and these are not the same thing. Starvation may result from several causes, diverse beliefs being only one.We firstly identify and exclude these other possible causes, and then

prove that starvation is equivalent to inferior belief, under suitable technical conditions. On the other hand, going broke cannot be characterized solely in terms of beliefs, as we show. We next present a remarkable example with two agents with different beliefs, in which one agent starves yet amasses all the capital, and the other goes broke yet consumes all the output -- the hungry miser and the happy bankrupt.

This example also serves to show that although an agent may starve, he may have long-term impact on the prices. This relates to the notion of price impact introduced by Kogan et al (2009), which we correct in the final section, and then use to characterize situations where asymptotically equivalent

pricing holds.

Attractive features of Lifshitz type theories are described with different

examples,

as the improvement of graphs convergence, the introduction of new

renormalizable

interactions, dynamical mass generation, asymptotic freedom, and other

features

related to more specific models. On the other hand, problems with the

expected

emergence of Lorentz symmetry in the IR are discussed, related to the

different

effective light cones seen by different particles when they interact.

We describe several bifurcation properties corresponding to various classes of nonlinear elliptic equations The purpose of this talk is two-fold. First, it points out different competition effects between the terms involved in the equations. Second, it provides several non standard phenomena that occur according to the structure of the differential operator.

A longstanding question in geometric group theory is the following. Suppose G is a hyperbolic group where all finitely generated subgroups of infinite index are free. Is G the fundamental group of a surface? This question is still open for some otherwise well understood classes of groups. In this talk, I will explain why the answer is affirmative for graphs of free groups with cyclic edge groups. I will also discuss the extent to which these techniques help with the harder problem of finding surface subgroups.

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.

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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.