Past

The visceral endoderm (VE) is an epithelium of approximately 200 cells

encompassing the early post-implantation mouse embryo. At embryonic day

5.5, a subset of around 20 cells differentiate into morphologically

distinct tissue, known as the anterior visceral endoderm (AVE), and

migrate away from the distal tip, stopping abruptly at the future

anterior. This process is essential for ensuring the correct orientation

of the anterior-posterior axis, and patterning of the adjacent embryonic

tissue. However, the mechanisms driving this migration are not clearly

understood. Indeed it is unknown whether the position of the future

anterior is pre-determined, or defined by the movement of the migrating

cells. Recent experiments on the mouse embryo, carried out by Dr.

Shankar Srinivas (Department of Physiology, Anatomy and Genetics) have

revealed the presence of multicellular ‘rosettes’ during AVE migration.

We are developing a comprehensive vertex-based model of AVE migration.

In this formulation cells are treated as polygons, with forces applied

to their vertices. Starting with a simple 2D model, we are able to mimic

rosette formation by allowing close vertices to join together. We then

transfer to a more realistic geometry, and incorporate more features,

including cell growth, proliferation, and T1 transitions. The model is

currently being used to test various hypotheses in relation to AVE

migration, such as how the direction of migration is determined, what

causes migration to stop, and what role rosettes play in the process.

A celebrated result in mathematical general relativity is the uniqueness of the Kerr(-Newman) black-holes as regular solutions to the stationary and axially-symmetric Einstein(-Maxwell) equations. The axial symmetry can be removed if one invokes Hawking's rigidity theorem. Hawking's theorem requires, however, real analyticity of the solution. A recent program of A. Ionescu and S. Klainerman seeks to remove the analyticity requirement in the vacuum case. They were able to show that any smooth extension of "Kerr data" prescribed on the horizon, satisfying the Einstein vacuum equations, must be Kerr, using a characterization of Kerr metric due to M. Mars. In this talk I will give a characterization for the Kerr-Newman metric, and extend the rigidity result to cover the electrovacuum case.

We consider the motion of a viscous incompressible fluid described by

the Navier-Stokes equations in a bounded cylinder with slip boundary

conditions. Assuming that $L_2$ norms of the derivative of the initial

velocity and the external force with respect to the variable along the

axis of the cylinder are sufficiently small we are able to prove long

time existence of regular solutions. By the regular solutions we mean

that velocity belongs to $W^{2,1}_2 (Dx(0,T))$ and gradient of pressure

to $L_2(Dx(0,T))$. To show global existence we prolong the local solution

with sufficiently large T step by step in time up to infinity. For this purpose

we need that $L_2(D)$ norms of the external force and derivative

of the external force in the direction along the axis of the cylinder

vanish with time exponentially.

Next we consider the inflow-outflow problem. We assume that the normal

component of velocity is nonvanishing on the parts of the boundary which

are perpendicular to the axis of the cylinder. We obtain the energy type

estimate by using the Hopf function. Next the existence of weak solutions is

proved.

The subject of this talk is the structure of the space of homomorphisms from a free abelian group to a Lie group G as well as quotients spaces given by the associated space of representations.

These spaces of representations admit the structure of a simplicial space at the heart of the work here.

Features of geometric realizations will be developed.

What is the fundamental group or the first homology group of the associated space in case G is a finite, discrete group ?

This deceptively elementary question as well as more global information given in this talk is based on joint work with A. Adem, E. Torres, and J. Gomez.

A representation of Hermite polynomials of degree 2n + 1, as sum of an element in the polynomial ideal generated by the roots of the Hermite polynomial of degree n and of a reminder, suggests a folding of multivariate polynomials over a finite set of points. From this, the expectation of some polynomial combinations of random variables normally distributed is computed. This is related to quadrature formulas and has strong links with designs of experiments.

This is joint work with G. Pistone

We demonstrate that stochastic differential equations (SDEs) driven by fractional Brownian motion with Hurst parameter H > 1/2 have similar ergodic properties as SDEs driven by standard Brownian motion. The focus in this article is on hypoelliptic systems satisfying H\"ormander's condition. We show that such systems satisfy a suitable version of the strong Feller property and we conclude that they admit a unique stationary solution that is physical in the sense that it does not "look into the future".

The main technical result required for the analysis is a bound on the moments of the inverse of the Malliavin covariance matrix, conditional on the past of the driving noise.

I will outline two areas currently under study by myself and my co-workers, particularly Jonathan Holland: one concerns the relation between the exceptional Lie group G_2 and Einstein's gravity; the second will introduce and apply the concept of a causal geometry.

Abstract: The valuation of a finite resource, be it acopper mine, timber forest or gas field, has received surprisingly littleattention from the academic literature. The fact that a robust, defensible andaccurate valuation methodology has not been derived is due to a mixture ofdifficulty in modelling the numerous stochastic uncertainties involved and thecomplications with capturing real day-to-day mining operations. The goal ofproducing such valuations is not just for accounting reasons, but also so thatoptimal extraction regimes and procedures can be devised in advance for use atthe coal-face. This paper shows how one can begin to bring all these aspectstogether using contingent claims financial analysis, geology, engineering,computer science and applied mathematics.

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Synthetic Aperture Radars (SARs) produce high resolution images over large areas at high data rates. An aircraft flying at 100m/s can easily image an area at a rate of 1square kilometre per second at a resolution of 0.3x0.3m, i.e. 10Mpixels/sec with a dynamic range of 60-80dB (10-13bits). Unlike optical images, the SAR image is also coherent and this coherence can be used to detect changes in the terrain from one image to another, for example to detect the distortions in the ground surface which precede volcanic eruptions.

It is clearly very desirable to be able to compress these images before they are relayed from one place to another, most particularly down to the ground from the aircraft in which they are gathered.

Conventional image compression techniques superficially work well with SAR images, for example JPEG 2000 was created for the compression of traditional photographic images and optimised on that basis. However there is conventional wisdom that SAR data is generally much less correlated in nature and therefore unlikely to achieve the same compression ratios using the same coding schemes unless significant information is lost.

Features which typically need to be preserved in SAR images are:

o texture to identify different types of terrain

o boundaries between different types of terrain

o anomalies, such as military vehicles in the middle of a field, which may be of tactical importance and

o the fine details of the pixels on a military target so that it might be recognised.

The talk will describe how Synthetic Aperture Radar images are formed and the features of them which make the requirements for compression algorithms different from electro-optical images and the properties of wavelets which may make them appropriate for addressing this problem. It will also discuss what is currently known about the compression of radar images in general.

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We shall explain how to give substance to an old dream of Tits, to invent exotic new zeta functions, and discover the skeleton of algebraic varieties (toric manifolds and tropial geometry).

This talk will introduce L1-regularized optimization problems that arise in image processing, and numerical methods for their solution. In particular, we will focus on methods of the split-Bregman type, which very efficiently solve large scale problems without regularization or time stepping. Applications include image

denoising, segmentation, non-local filters, and compressed sensing.

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Descent theory is the art of gluing local data together to global data. Beside of being an invaluable tool for the working geometer, the descent philosophy has changed our perception of space and topology. In this talk I will introduce the audience to the basic results of scheme and descent theory and explain how those can be applied to concrete examples.

Standard quantum logic, as intitiated by Birkhoff and von Neumann, suffers from severe problems which relate quite directly to interpretational issues in the foundations of quantum theory. In this talk, I will present some aspects of the so-called topos approach to quantum theory, as initiated by Isham and Butterfield, which aims at a mathematical reformulation of quantum theory and provides a new, well-behaved form of quantum logic that is based upon the internal logic of a certain (pre)sheaf topos.

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We shall explain how to give substance to an old dream of Tits, to invent exotic new zeta functions, and discover the skeleton of algebraic varieties (toric manifolds and tropical geometry).

We shall explain how to give substance to an old dream of Tits, to invent exotic new zeta functions, and discover the skeleton of algebraic varieties (toric manifolds and tropical geometry).

The Green-Griffiths conjecture from 1979 says that every projective algebraic variety $X$ of general type contains a certain proper algebraic subvariety $Y$ such that all nonconstant entire holomorphic curves in $X$ must lie inside $Y$. In this talk we explain that for projective hypersurfaces of degree $d>dim(X)^6$ this is the consequence of a positivity conjecture in global singularity theory.

A graph is $\chi$-bounded with a function $f$ is for all induced subgraph H of G, we have $\chi(H) \le f(\omega(H))$.  Here, $\chi(H)$ denotes the chromatic number of $H$, and $\omega(H)$ the size of a largest clique in $H$. We will survey several results saying that excluding various kinds of induced subgraphs implies $\chi$-boundedness. More precisely, let $L$ be a set of graphs. If a $C$ is the class of all graphs that do not any induced subgraph isomorphic to a member of $L$, is it true that there is a function $f$ that $\chi$-bounds all graphs from $C$? For some lists $L$, the answer is yes, for others, it is no.  

A decomposition theorems is a theorem saying that all graphs from a given class are either "basic" (very simple), or can be partitioned into parts with interesting relationship. We will discuss whether proving decomposition theorems is an efficient method to prove $\chi$-boundedness. 

The main aim is to incorporate the nonlinear term into non-Markovian Master equations for a continuous time random walk (CTRW) with non-exponential waiting time distributions. We derive new nonlinear evolution equations for the mesoscopic density of reacting particles corresponding to CTRW with arbitrary jump and waiting time distributions. We apply these equations to the problem of front propagation in the reaction-transport systems of KPP-type.

We find an explicit expression for the speed of a propagating front in the case of subdiffusive transport.

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We shall report on the use of algebraic geometry for the calculation of Feynman amplitudes (work of Bloch, Brown, Esnault and Kreimer). Or how to combine Grothendieck's motives with high energy physics in an unexpected way, radically distinct from string theory.

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Heike Gramberg - Flagellar beating in trypanosomes

Robert Whittaker - High-Frequency Self-Excited Oscillations in 3D Collapsible Tube Flows

Micron-sized bacteria or algae operate at very small Reynolds numbers.

In this regime, inertial effects are negligible and, hence, efficient

swimming strategies have to be different from those employed by fish

or bigger animals. Mathematically, this means that, in order to

achieve locomotion, the swimming stroke of a microorganism must break

the time-reversal symmetry of the Stokes equations. Large ensembles of

bacteria or algae can exhibit rich collective dynamics (e.g., complex

turbulent patterns, such as vortices or spirals), resulting from a

combination of physical and chemical interactions. The spatial extent

of these structures typically exceeds the size of a single organism by

several orders of magnitude. One of our current projects in the Soft

and Biological Matter Group aims at understanding how the collective

macroscopic behavior of swimming microorganisms is related to their

microscopic properties. I am going to outline theoretical and

computational approaches, and would like to discuss technical

challenges that arise when one tries to derive continuum equations for

these systems from microscopic or mesoscopic models.

There is a widespread use of mathematical tools in finance and its

importance has grown over the last two decades. In this talk we

concentrate on optimization problems in finance, in particular on

numerical aspects. In this talk, we put emphasis on the mathematical problems and aspects, whereas all the applications are connected to the pricing of derivatives and are the

outcome of a cooperation with an international finance institution.

As one example, we take an in-depth look at the problem of hedging

barrier options. We review approaches from the literature and illustrate

advantages and shortcomings. Then we rephrase the problem as an

optimization problem and point out that it leads to a semi-infinite

programming problem. We give numerical results and put them in relation

to known results from other approaches. As an extension, we consider the

robustness of this approach, since it is known that the optimality is

lost, if the market data change too much. To avoid this effect, one can

formulate a robust version of the hedging problem, again by the use of

semi-infinite programming. The numerical results presented illustrate

the robustness of this approach and its advantages.

As a further aspect, we address the calibration of models being used in

finance through optimization. This may lead to PDE-constrained

optimization problems and their solution through SQP-type or

interior-point methods. An important issue in this context are

preconditioning techniques, like preconditioning of KKT systems, a very

active research area. Another aspect is the preconditioning aspect

through the use of implicit volatilities. We also take a look at the

numerical effects of non-smooth data for certain models in derivative

pricing. Finally, we discuss how to speed up the optimization for

calibration problems by using reduced order models.

In 2008, Juhasz published the following result, which was proved using sutured Floer homology.

Let $K$ be a prime, alternating knot. Let $a$ be the leading coefficient of the Alexander polynomial of $K$. If $|a|

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