Past

Self-gravitating elastic bodies provide models for extended

objects in general relativity. I will discuss constructions of static

and rotating self-gravitating bodies, as well as recent results on the

initial value problem for self-gravitating elastic bodies.

I will give a new proof for the famous criteria by Novikov and Kazamaki, which provide sufficient conditions for the martingale property of a nonnegative local martingale. The proof is based on an extension theorem for probability measures that can be considered as a generalization of a Girsanov-type change of measure.

In the second part of my talk I will illustrate how a generalized Girsanov formula can be used to compute the distribution of the explosion time of a weak solution to a stochastic differential equation

Let (V, ≥) be a finite, partially ordered set. Say a directed forest on V is a set of directed edges [x,y> with x ≤ y such that no vertex has indegree greater than one.

Thus for a finite measure μ on some partially ordered measurable space D we may define a Poisson random forest by choosing a set of vertices V according to a Poisson point process weighted by the number of directed forests on V, and selecting a directed forest uniformly.

We give a necessary and sufficient condition for such a process to exist and show that the process may be expressed as a multi-type branching process with type space D.

We build on this observation, together with a construction of the simple birth death process due to Kurtz and Rodrigues [2011] to develop a coalescent theory for rapidly expanding populations.

There are many recent points of contact of model theory and other 
parts of mathematics: o-minimality and Diophantine geometry, geometric group 
theory, additive combinatorics, rigid geometry,...  I will probably 
emphasize  long-standing themes around stability, Diophantine geometry, and 
analogies between ODE's and bimeromorphic geometry.

Thermomechanical processes observed in deformable solids under intensive dynamic or quasi-static loadings consist of coupled mechanical, thermal and fracturing stages. The fracturing processes involve formation, motion and interaction of defects in crystals, phase transitions, breaking of bonds between atoms, accumulation of micro-structural damages (pores, cracks), etc. Irreversible deformations, zones of adiabatic shear micro-fractures are caused by these processes. Dynamic fracturing is a complicated multistage process, which includes appearance, evolution and confluence of micro-defects and formation of embryonic micro-cracks and pores that can grow and lead to the breaking-up of bodies with formation of free surfaces. This results in a need to use more advanced mathematical and numerical techniques.

This talk presents modelling of irreversible deformation near the tip of a crack in a porous domain containing oil and gas during the hydraulic fracturing process. The governing equations for a porous domain containing oil and gas are based on constructing a mathematical model of thermo-visco-elasto-plastic media with micro-defects (micro-pores) filled with another phase (e.g., oil or/and gas). The micro-pores can change their size during the process of dynamical irreversible deformation. The existing pores can expand or collapse. The model was created by using fundamental thermodynamic principles and, therefore, it is a thermodynamically consistent model. All the processes (i.e., irreversible deformation, fracturing, micro-damaging, heat transfer) within a porous domain are strongly coupled. An explicit normalized-corrected meshless method is used to solve the resulting governing PDEs. The flexibility of the proposed technique allows efficient running using a great number of micro- and macro- fractures. The results are presented, discussed and future studies are outlined.

We wish to discuss the role of Modelling in Health Care. While risk factor prevalences vary and change with time it is difficult to anticipate the change in disease incidence that will result without accurately modelling the epidemiology. When detailed study of the prevalence of obesity, tobacco and salt intake, for example, are studied clear patterns emerge that can be extrapolated into the future. These can give rise to estimated probability distributions of these risk factors across age, sex, ethnicity, social class groups etc into the future. Micro simulation of individuals from defined populations (eg England 2012) can then estimate disease incidence, prevalence, death, costs and quality of life. Thus future health and other needs can be estimated, and interventions on these risk factors can be simulated for their population effect. Health policy can be better determined by a realistic characterisation of public health. The Foresight microsimulation modelling of the National Heart Forum (UK Health Forum) will be described. We will emphasise some of the mathematical and statistical issues associated with so doing.

The study of difference algebraic geometry stems from the efforts of Macintyre and Hrushovski to

count the number of solutions to difference polynomial equations over fields with powers of Frobenius.

We propose a notion of multiplicity in the context of difference algebraic schemes and prove a first principle

of preservation of multiplicity. We shall also discuss how to formulate a suitable intersection theory of difference schemes.

Self-renewal and pluripotency of mouse embryonic stem (ES) cells are controlled by a complex transcriptional regulatory network (TRN) which is rich in positive feedback loops. A number of key components of this TRN, including Nanog, show strong temporal expression fluctuations at the single cell level, although the precise molecular basis for this variability remains unknown. In this talk I will discuss recent work which uses a genetic complementation strategy to investigate genome-wide mRNA expression changes during transient periods of Nanog down-regulation. Nanog removal triggers widespread changes in gene expression in ES cells. However, we found that significant early changes in gene expression were reversible upon re-induction of Nanog, indicating that ES cells initially adopt a flexible “primed” state. Nevertheless, these changes rapidly become consolidated irreversible fate decisions in the continued absence of Nanog. Using high-throughput single cell transcriptional profiling we observed that the early molecular changes are both stochastic and reversible at the single cell level. Since positive feedback commonly gives rise to phenotypic variability, we also sought to determine the role of feedback in regulating ES cell heterogeneity and commitment. Analysis of the structure of the ES cell TRN revealed that Nanog acts as a feedback “linchpin”: in its presence positive feedback loops are active and the extended TRN is self-sustaining; while in its absence feedback loops are weakened, the extended TRN is no longer self-sustaining and pluripotency is gradually lost until a critical “point-of-no-return” is reached. Consequently, fluctuations in Nanog expression levels transiently activate different sub-networks in the ES cell TRN, driving transitions between a (Nanog expressing) feedback-rich, robust, self-perpetuating pluripotent state and a (Nanog-diminished), feedback-depleted, differentiation-sensitive state. Taken together, our results indicate that Nanog- dependent feedback loops play a central role in controlling both early fate decisions at the single cell level and cell-cell variability in ES cell populations.

Counter-examples to the Hasse principle are known for many families of geometrically rational varieties. We discuss how often such failures arise for Chatelet surfaces and certain higher-dimensional hypersurfaces. This is joint work with Regis de la Breteche.

The bounded coherent dg-category on (suitable versions of) the Steinberg stack of a reductive group G is a categorification of the affine Hecke algebra in representation theory.  We discuss how to describe the center and universal trace of this monoidal dg-category.  Many of the techniques involved are very general, and the description makes use of the notion of "odd micro-support" of coherent complexes.  This is joint work with Ben-Zvi and Nadler.

Morse theory gives an estimate of the dimensions of the cohomology groups of a manifold in terms of the critical points of a function.
One can do better and compute the cohomology in terms of this function using the so-called Witten complex.
Already implicit in work of Smale in the fifties, it was rediscovered by Witten in the eighties using techniques from (supersymmetric) quantum field theories.
I will explain Witten's (heuristic) arguments and describe the Witten complex.

The symmetric group S_{mn} acts naturally on the collection of set partitions of a set of size mn into n sets each of size m, and the resulting permutation character is the Foulkes character. These characters are the subject of the longstanding Foulkes Conjecture. In this talk, we define a deflation map which sends a character of the symmetric group S_{mn} to a character of S_n. The values of the images of the irreducible characters under this map are described combinatorially in a rule which generalises two well-known combinatorial rules in the representation theory of symmetric groups, the Murnaghan-Nakayama formula and Young's rule. We use this in a new algorithm for computing irreducible constituents of Foulkes characters and verify Foulkes’ Conjecture in some new cases. This is joint work with Anton Evseev (Birmingham) and Mark Wildon (Royal Holloway).

Many methods for solving high-dimensional statistical inverse problems are based on convex optimization problems formed by the weighted sum of a loss function with a norm-based regularizer.

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Particular examples include $\ell_1$-based methods for sparse vectors and matrices, nuclear norm for low-rank matrices, and various combinations thereof for matrix decomposition and robust PCA. In this talk, we describe an interesting connection between computational and statistical efficiency, in particular showing that the same conditions that guarantee that an estimator has good statistical error can also be used to certify fast convergence of first-order optimization methods up to statistical precision.

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Joint work with Alekh Agarwahl and Sahand Negahban Pre-print (to appear in Annals of Statistics)

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http://www.eecs.berkeley.edu/~wainwrig/Papers/AgaNegWai12b_SparseOptFul…

We consider the problem of optimizing a portfolio of medium to low frequency

quant strategies under heavy tailed distributions. Approaching this problem by modelling

returns through mixture distributions, we derive robust and relative robust methodologies

and discuss conic optimization approaches to solving these models.

The Kolmogorov 1941 theory (K41) is, in a way, the starting point for all

models of turbulence. In particular, K41 and corrections to it provide

estimates of small-scale quantities such as increments and energy spectrum

for a 3D turbulent flow. However, because of the well-known difficulties

involved in studying 3D turbulent flow, there are no rigorous results

confirming or infirming those predictions. Here, we consider a well-known

simplified model for 3D turbulence: Burgulence, or turbulence for the 1D

Burgers equation. In the space-periodic case with a stochastic white in

time and smooth in space forcing term, we give sharp estimates for

small-scale quantities such as increments and energy spectrum.

A topological space is called rigid if its only autohomeomorphism is the identity map. Using the Axiom of Choice it is easy to construct rigid subsets of the real line R, but sets constructed in this way always have size continuum. I will explore the question of whether it is possible to have rigid subsets of R that are small, meaning that their cardinality is smaller than that of the continuum. On the one hand, we will see that forcing can be used to produce models of ZFC in which such small rigid sets abound. On the other hand, I will introduce a combinatorial axiom that can be used to show the consistency with ZFC of the statement "CH fails but every rigid subset of R has size continuum". Only a working knowledge of basic set theory (roughly what one might remember from C1.2b) and topology will be assumed.

Self-similarity is a fundamental idea in many areas of mathematics. In this talk I will explain how it has entered group theory and the links between self-similar groups and other areas of research. There will also be pretty pictures.

I will give an introduction to The McKay Correspondence, relating the irreducible representations of a finite subgroup Γ ≤ SL2 (C), minimal resolutions of the orbit space C2 /Γ, and affine Dynkin diagrams.

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.