Past

Complex systems emerging from many biochemical applications often exhibit multiscale and multiphysics (MSMP) features: The systems incorporate a variety of physical processes or subsystems across a broad range of scales. A typical MSMP system may come across scales with macroscopic, mesoscopic and microscopic kinetics,
deterministic and stochastic dynamics, continuous and discrete state space, fastscale and slow-scale reactions, and species of both large and small populations. These complex features present great challenges in the modeling and simulation practice. The goal of our research is to develop innovative computational methods and rigorous fundamental theories to answer these challenges. In this talk we will start with introduction of basic stochastic simulation algorithms for biochemical systems and multiscale
features in the stochastic cell cycle model of budding yeast. With detailed analysis of these multiscale features, we will introduce recent progress on simulation algorithms and computational theories for multiscale stochastic systems, including tau-leaping methods, slow-scale SSA, and the hybrid method. 

While usual percolation concerns the study of the connected components of

random subgraphs of an infinite graph, bootstrap percolation is a type of

cellular automaton, acting on the vertices of a graph which are in one of

two states: `healthy' or `infected'. For any positive integer $r$, the

$r$-neighbour bootstrap process is the following update rule for the

states of vertices: infected vertices remain infected forever and each

healthy vertex with at least $r$ infected neighbours becomes itself

infected. These updates occur simultaneously and are repeated at discrete

time intervals. Percolation is said to occur if all vertices are

eventually infected.

As it is often difficult to determine precisely which configurations of

initially infected vertices percolate, one often considers a random case,

with each vertex infected independently with a fixed probability $p$. For

an infinite graph, of interest are the values of $p$ for which the

probability of percolation is positive. I will give some of the history

of this problem for regular trees and present some new results for

bootstrap percolation on certain classes of randomly generated trees:

Galton--Watson trees.

***** PLEASE NOTE THIS SEMINAR WILL TAKE PLACE ON TUESDAY 19TH FEBRUARY *****

With "fully anisotropic" I describe diffusion models of the form u_t=\nabla \nabla (D(x) u), where the diffusion tensor appears inside both derivatives. This model arises naturally in the modeling of brain tumor spread and wolf movement and other applications. Since this model does not satisfy a maximum principle, it can lead to interesting spatial pattern formation, even in the linear case. I will present a detailed derivation of this model and discuss its application to brain tumors and wolf movement. Furthermore, I will present an example where, in the linear case, the solution blows-up in infinite time; a quite surprising result for a linear parabolic equation (joint work with K.J. Painter and M. Winkler).

In this talk we focus on the incompressible Navier–Stokes equations with variable

density. The aim is to prove existence and uniqueness results in the case of a discontinuous

initial density (typically we are interested in discontinuity along an interface).

In the first part of the talk, by making use of Fourier analysis techniques, we establish the existence of global-in-time unique solutions in a critical

functional framework, under some smallness condition over the initial data,

In the second part, we use another approach to avoid the smallness condition over the nonhomogeneity : as a matter of fact, one may consider any density bounded

and bounded away from zero and still get a unique solution. The velocity is required to have subcritical regularity, though.

In all the talk, the Lagrangian formulation for describing the flow plays a key role in the analysis.

Abstract: A central question in the theory of random walks on groups is how symmetries of the underlying space gives rise to structure and rigidity of the random walks. For example, for nilpotent groups, it is known that random walks have diffusive behavior, namely that the rate of escape, defined as the expected distance of the walk from the identity satisfies E|Xn|~=n^{1/2}. On nonamenable groups, on the other hand we have E|Xn| ~= n. (~= meaning upto (multiplicative) constants )

In this work, for every 3/4 <= \beta< 1 we construct a finitely generated group so that the expected distance of the simple random walk from its starting point after n steps is n^\beta (up to constants). This answers a question of Vershik, Naor and Peres. In other examples, the speed exponent can fluctuate between any two values in this interval.

Previous examples were only of exponents of the form 1-1/2^k or 1 , and were based on lamplighter (wreath product) constructions.(Other than the standard beta=1/2 and beta=1 known for a wide variety of groups) In this lecture we will describe how a variation of the lamplighter construction, namely the permutational wreath product, can be used to get precise bounds on the rate of escape in terms of return probabilities of the random walk on some Schreier graphs. We will then show how groups of automorphisms of rooted trees, related to automata groups , can be constructed and analyzed to get the desired rate of escape. This is joint work with Balint Virag of the University of Toronto. (Paper available at http://arxiv.org/abs/1203.6226)

No previous knowledge of random walks,automaton groups or wreath products is assumed.

Abstract: Hairer recently had the remarkable insight that Lyons' theory of rough paths can be used to make sense of nonlinear SPDEs that were previously ill-defined due to spatial irregularities. Since rough path theory deals with the integration of functions defined on the real line, the SPDEs studied by Hairer have a one-dimensional spatial index variable. I will show how to combine paraproducts, a notion from functional analysis, with ideas from the theory of controlled rough paths, in order to develop a formulation of rough path theory that works in any index dimension. As an application, I will present existence and uniqueness results for an SPDE with multidimensional spatial index set, for which previously it was not known how to describe solutions. No prior knowledge of rough paths or paraproducts is required for understanding the talk. This is joint work with Massimiliano Gubinelli and Peter Imkeller.

In this talk I will discuss recent developments in information theoretical approaches to fundamental

molecular processes that affect the cellular decision making processes. One of the challenges of applying

concepts from information theory to biological systems is that information is considered independently from

meaning. This means that a noisy signal carries quantifiably more information than a unperturbed signal.

This has, however, led us to consider and develop new approaches that allow us to quantify the level of noise

contributed by any molecular reactions in a reaction network. Surprisingly this analysis reveals an important and hitherto

often overlooked role of degradation reactions on the noisiness of biological systems. Following on from this I will outline

how such ideas can be used in order to understand some aspects of cell-fate decision making, which I will discuss with

reference to the haematopoietic system in health and disease.

InSAR (Interferometric Synthetic Aperture Radar) is an important space geodetic technique (i.e. a technique that uses satellite data to obtain measurements of the Earth) of great interest to geophysicists monitoring slip along fault lines and other changes to shape of the Earth. InSAR works by using the difference in radar phase returns acquired at two different times to measure displacements of the Earth’s surface. Unfortunately, atmospheric noise and other problems mean that it can be difficult to use the InSAR data to obtain clear measurements of displacement.

Persistent Scatterer (PS) InSAR is a later adaptation of InSAR that uses statistical techniques to identify pixels within an InSAR image that are dominated by a single back scatterer, producing high amplitude and stable phase returns (Feretti et al. 2001, Hooper et al. 2004). PS InSAR has the advantage that it (hopefully) chooses the ‘better’ datapoints, but it has the disadvantage that it throws away a lot of the data that might have been available in the original InSAR signal.

InSAR and PS InSAR have typically been used in isolation to obtain slip-rates across faults, to understand the roles that faults play in regional tectonics, and to test models of continental deformation. But could they perhaps be combined? Or could PS InSAR be refined so that it doesn’t throw away as much of the original data? Or, perhaps, could the criteria used to determine what data are signal and what are noise be improved?

The key aim of this workshop is to describe and discuss the techniques and challenges associated with InSAR and PS InSAR (particularly the problem of atmospheric noise), and to look at possible methods for improvement, by combining InSAR and PS InSAR or by methods for making the choice of thresholds.

Motivated by industrial and biological applications, the Waves

Group at Manchester has in recent years been interested in

developing methods for obtaining the effective properties of

complex composite materials. As time allows we shall discuss a

number of issues, such as differences between composites

with periodic and aperiodic distributions of inclusions, and

modelling of nonlinear composites.

I will explain the beautiful generalization recently discovered by Y. Tian of Heegner's original proof of the existence of infinitely many primes of the form 8n+5, which are congruent numbers. At the end, I hope to mention some possible generalizations of his work to other elliptic curves defined over the field of rational numbers.

Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$. In the third talk we will see that $\Lambda$ admits a canonical quantization if it is a "conification" of a compact exact Lagrangian submanifold of a cotangent bundle. We will see how to use this quantization to recover results of Fukaya-Seidel-Smith and Abouzaid on the topology of $\Lambda$.

Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$.

In the third talk we will see that $\Lambda$ admits a canonical quantization if it is a "conification" of a compact exact Lagrangian submanifold of a

cotangent bundle. We will see how to use this quantization to recover results of Fukaya-Seidel-Smith and Abouzaid on the topology of $\Lambda$.

We address, in the study of acoustic scattering by 2D and 3D planar screens, three inter-related and challenging questions. Each of these questions focuses particularly on the formulation of these problems as boundary integral equations. The first question is, roughly, does it make sense to consider scattering by screens which occupy arbitrary open sets in the plane, and do different screens cause the same scattering if the open sets they occupy have the same closure? This question is intimately related to rather deep properties of fractional Sobolev spaces on general open sets, and the capacity and Haussdorf dimension of their boundary. The second question is, roughly, that, in answering the first question, can we understand explicitly and quantitatively the influence of the frequency of the incident time harmonic wave? It turns out that we can, that the problems have variational formations with sesquilinear forms which are bounded and coercive on fractional Sobolev spaces, and that we can determine explicitly how continuity and coercivity constants depend on the frequency. The third question is: can we design computational methods, adapted to the highly oscillatory solution behaviour at high frequency, which have computational cost essentially independent of the frequency? The answer here is that in 2D we can provably achieve solutions to any desired accuracy using a number of degrees of freedom which provably grows only logarithmically with the frequency, and that it looks promising that some extension to 3D is possible.

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This is joint work with Dave Hewett, Steve Langdon, and Ashley Twigger, all at Reading.

The second order sensitivity of a trading position, the so

called gamma, has a very real and intuitive meaning to the traders.

People think that convex payoffs must generate convex prices. Being long

or short of gamma is a strategy used to balance risks in options books.

While the simples models, like Black Scholes, are consistent with this

intuition other popular models used in the industry are not. I will give

examples of simple and popular models which do not always convert a

convex payoff into a convex price. I will also give the necessary and

sufficient conditions under which the convexity is propagated.

The new schedule will follow shortly

First of all, I will give an overview of what the phenomenon of homological stability is and why it's useful, with plenty of examples. I will then introduce configuration spaces -- of various different kinds -- and give an overview of what is known about their homological stability properties. A "configuration" here can be more than just a finite collection of points in a background space: in particular, the points may be equipped with a certain non-local structure (an "orientation"), or one can consider unlinked embedded copies of a fixed manifold instead of just points. If by some miracle time permits, I may also say something about homological stability with local coefficients, in general and in particular for configuration spaces.

 We consider the emergence of classical correlations in macroscopic quantum systems, and its connection to monogamy relations for violation of Bell-type inequalities. We work within the framework of Abramsky and Brandenburger [1], which provides a unified treatment of non-locality and contextuality in the general setting of no-signalling empirical models. General measurement scenarios are represented by simplicial complexes that capture the notion of compatibility of measurements. Monogamy and locality/noncontextuality of macroscopic correlations are revealed by our analysis as two sides of the same coin: macroscopic correlations are obtained by averaging along a symmetry (group action) on the simplicial complex of measurements, while monogamy relations are exactly the inequalities that are invariant with respect to that symmetry. Our results exhibit a structural reason for monogamy relations and consequently for the classicality of macroscopic correlations in the case of multipartite scenarios, shedding light on and generalising the results in [2,3].More specifically, we show that, however entangled the microscopic state of the system, and provided the number of particles in each site is large enough (with respect to the number of allowed measurements), only classical (local realistic) correlations will be observed macroscopically. The result depends only on the compatibility structure of the measurements (the simplicial complex), hence it applies generally to any no-signalling empirical model. The macroscopic correlations can be defined on the quotient of the simplicial complex by the symmetry that lumps together like microscopic measurements into macroscopic measurements. Given enough microscopic particles, the resulting complex satisfies a structural condition due to Vorob'ev [4] that is necessary and sufficient for any probabilistic model to be classical.  The generality of our scheme suggests a number of promising directions. In particular, they can be applied in more general scenarios to yield monogamy relations for contextuality inequalities and to study macroscopic non-contextuality.

[1] Samson Abramsky and Adam Brandenburger, The sheaf-theoretic structure of non-locality and contextuality, New Journal of Physics 13 (2011), no. 113036.
[2] MarcinPawłowski and Caslav Brukner, Monogamy of Bell’s inequality violations in nonsignaling theories, Phys. Rev. Lett. 102 (2009), no. 3, 030403.
[3] R. Ramanathan, T. Paterek, A. Kay, P. Kurzynski, and D. Kaszlikowski, Local realism of macroscopic correlations, Phys. Rev. Lett. 107 (2011), no. 6, 060405.
[4] N.N.Vorob’ev, Consistent families of measures and their extensions, Theory of Probability and its Applications VII (1962), no. 2, 147–163, (translated by N. Greenleaf, Russian original published in Teoriya Veroyatnostei i ee Primeneniya).

Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$.

In the second talk we will introduce a stack on $\Lambda$ by localization of the category of sheaves on $M$. We deduce topological obstructions on $\Lambda$ for the existence of a quantization.

A number is called transcendental if it is not algebraic, that is it does not satisfy a polynomial equation with rational coefficients. It is easy to see that the algebraic numbers are countable, hence the transcendental numbers are uncountable. Despite this fact, it turns out to be very difficult to determine whether a given number is transcendental. In this talk I will discuss some famous examples and the theorems which allow one to construct many different transcendental numbers. I will also give an outline of some of the many open problems in the field.

I propose to present modeling and experimental data about the organization of telomeres (ends of the chromosomes): the 32 telomeres in Yeast form few local aggregates. We built a model of diffusion-aggregation-dissociation for a finite number of particles to estimate the number of cluster and the number of telomere/cluster and other quantities. We will further present based on eingenvalue expansion for the Fokker-Planck operator, asymptotic estimation for the mean time a piece of a polymer (DNA) finds a small target in the nucleus.

We discuss the problem to what extend a group action determines geometry of the space. 
More precisely, we show that for a large class of actions measurable isomorphisms must preserve 
the geometric structure as well. This is a joint work with Bader, Furman, and Weiss.

Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$. In the first talk we will see that the graph of a Hamiltonian isotopy admits a canonical quantization and we deduce a new proof of Arnold's non-displaceability conjecture.

Several recent works by D. Tamarkin, D. Nadler, E. Zaslow make use of the microlocal theory of sheaves of M. Kashiwara and P. Schapira to obtain results in symplectic geometry. The link between sheaves on a manifold $M$ and the symplectic geometry of the cotangent bundle of $M$ is given by the microsupport of a sheaf, which is a conic co-isotropic subset of the cotangent bundle. In the above mentioned works properties of a given Lagrangian submanifold $\Lambda$ are deduced from the existence of a sheaf with microsupport $\Lambda$, which we call a quantization of $\Lambda$.

In the first talk we will see that the graph of a Hamiltonian isotopy admits a canonical quantization and we deduce a new proof of Arnold's non-displaceability conjecture.

I will describe how a search for the answer to an old question about the existence of monotone arithmetic progressions in permutations of positive integers led to the study of infinite words with bounded additive complexity. The additive complexity of a word on a finite subset of integers is defined as the function that, for a positive integer $n$, counts the maximum number of factors of length $n$, no two of which have the same sum.

Defect measures have successfully been used, in a variety of

contexts, as a tool to quantify the lack of compactness of bounded

sequences of square-integrable functions due to concentration and

oscillation effects. In this talk we shall present some results on the

structure of the set of possible defect measures arising from sequences

of solutions to the linear Schrödinger equation on a compact manifold.

This is motivated by questions related to understanding the effect of

geometry on dynamical aspects of the Schrödinger flow, such as

dispersive effects and unique continuation.

It turns out that the answer to these questions depends strongly on

global properties of the geodesic flow on the manifold under

consideration: this will be illustrated by discussing with a certain

detail the examples of the the sphere and the (flat) torus.

In 2005, Bonk and Kleiner showed that a hyperbolic group admits a

quasi-isometrically embedded copy of the hyperbolic plane if and only if the

group is not virtually free. This answered a question of Papasoglu. I will

discuss a generalisation of their result to certain relatively hyperbolic

groups (joint work with Alessandro Sisto). Key tools involved are new

existence results for quasi-circles, and a better understanding of the

geometry of boundaries of relatively hyperbolic groups.

(Joint work with P.E. Chaudru de Raynal and F. Delarue)

Several problems in financial mathematics, in particular that of contingent claim pricing, may be casted as decoupled Forward Backward Stochastic Differential Equations (FBSDEs). It is then of a practical interest to find efficient algorithms to solve these equations numerically with a reasonable complexity.

An efficient numerical approach using cubature on Wiener spaces was introduced by Crisan and Manoralakis [1]. The algorithm uses an approximation scheme requiring the calculation of conditional expectations, a task achieved through the cubature kernel approximation. This algorithm features several advantages, notably the fact that it is possible to solve with the same cubature tree several decoupled FBSDEs with different boundary conditions. There is, however, a drawback of this method: an exponential growth of the algorithm's complexity.

In this talk, we introduce a modification on the cubature method based on the recombination method of Litterer and Lyons [2] (as an alternative to Tree Branch Based Algorithm proposed in [1]). The main idea of the method is to modify the nodes and edges of the cubature trees in such a way as to preserve, up to a constant, the order of convergence of the expectation and conditional expectation approximations obtained via the cubature method, while at the same time controlling the complexity growth of the algorithm.

We have obtained estimations on the order of convergence and complexity growth of the algorithm under smoothness assumptions on the coefficients of the FBSDE and uniform ellipticity of the forward equation. We discuss that, just as in the case of the plain cubature method, the order of convergence of the algorithm may be degraded as an effect of solving FBSDEs with rougher boundary conditions. Finally, we illustrate the obtained estimations with some numerical tests.

References

[1] Crisan, D., and K. Manolarakis. “Solving Backward Stochastic Differential Equations Using the Cubature Method. Application to Nonlinear Pricing.” In Progress in Analysis and Its Applications, 389–397. World Sci. Publ., Hackensack, NJ, 2010.

[2] Litterer, C., and T. Lyons. “High Order Recombination and an Application to Cubature on Wiener Space.” The Annals of Applied Probability 22, no. 4 (August 2012): 1301–1327. doi:10.1214/11-AAP786.

In this talk I will consider the Burgers equation with a homogeneous Possion process as a forcing potential. In recent years, the randomly forced Burgers equation, with forcing that is ergodic in time, received a lot of attention, especially the almost sure existence of unique global solutions with given average velocity, that at each time only depend on the history up to that time. However, in all these results compactness in the space dimension of the forcing was essential. It was even conjectured that in the non-compact setting such unique global solutions would not exist. However, we have managed to use techniques developed for first and last passage percolation models to prove that in the case of Poisson forcing, these global solutions do exist almost surely, due to the existence of semi-infinite minimizers of the Lagrangian action. In this talk I will discuss this result and explain some of the techniques we have used.

This is joined work Yuri Bakhtin and Konstantin Khanin.

We discuss sparse portfolio optimization in continuous time.

Optimization objective is to maximize an expected utility as in the

classical Merton problem but with regularizing sparsity constraints.

Such constraints aim for asset allocations that contain only few assets or

that deviate only in few coordinates from a reference benchmark allocation.

With a focus on growth optimization, we show empirical results for various

portfolio selection strategies with and without sparsity constraints,

investigating different portfolios of stock indicies, several performance

measures and adaptive methods to select the regularization parameter.

Sparse optimal portfolios are less sensitive to estimation

errors and performance is superior to portfolios without sparsity

constraints in reality, where estimation risk and model uncertainty must

not be ignored.

We are interested in finding the Probability Density Function (PDF) of high dimensional chaotic systems such as a global atmospheric circulation model. The key difficulty stems from the so called “curse of dimensionality”. Representing anything numerically in a high dimensional space seems to be just too computationally expensive. Methods applied to dodge this problem include representing the PDF analytically or applying a (typically linear) transformation to a low dimensional space. For chaotic systems these approaches often seem extremely ad-hoc with the main motivation being that we don't know what else to do.

The Lorenz 95 system is one of the simplest systems we could come up with that is both chaotic and high dimensional. So it seems like a good candidate for initial investigation. We look at two attempts to approximate the PDF of this system to an arbitrary level of accuracy, firstly using a simple Monte-Carlo method and secondly using the Fokker-Planck equation. We also describe some of the (sometimes surprising) difficulties encountered along the way.

• Jean-Charles Seguis - Simulation in chemotaxis and comparison of cell models
• Laura Kimpton (née Gallimore) - A viscoelastic two-phase flow model of a crawling cell
• Benjamin Franz - Particles and PDEs and robots