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Abstract: In this talk we will review some recentadvances in order to construct geometric or weakly geometric rough paths abovea multidimensional fractional Brownian motion, with a special emphasis on thecase of a Hurst parameter H<1/4. In this context, the natural piecewiselinear approximation procedure of Coutin and Qian does not converge anymore,and a less physical method has to be adopted. We shall detail some steps ofthis construction for the simplest case of the Levy area.

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Many problems from combinatorics, number theory, quantum field theory and topology lead to power series of a special kind called q-hypergeometric series. Sometimes, like in the famous Rogers-Ramanujan identities, these q-series turn out to be modular functions or modular forms. A beautiful conjecture of W. Nahm, inspired by quantum theory, relates this phenomenon to algebraic K-theory.

In a different direction, quantum invariants of knots and 3-manifolds also sometimes seem to have modular or near-modular properties, leading to new objects called "quantum modular forms".

We study a class of Markovian optimal stochastic control problems in which the controlled process $Z^\nu$ is constrained to satisfy an a.s.~constraint $Z^\nu(T)\in G\subset \R^{d+1}$ $\Pas$ at some final time $T>0$.  When the set is of the form $G:=\{(x,y)\in \R^d\x \R~:~g(x,y)\ge 0\}$, with $g$ non-decreasing in $y$, we provide a Hamilton-Jacobi-Bellman  characterization of the associated value function. It gives rise to a state constraint problem where the constraint can be expressed in terms of an auxiliary value function $w$ which characterizes the set $D:=\{(t,Z^\nu(t))\in [0,T]\x\R^{d+1}~:~Z^\nu(T)\in G\;a.s.$ for some $ \nu\}$. Contrary to standard state constraint problems, the domain $D$ is not given a-priori and we do not need to impose conditions on its boundary. It is naturally incorporated in the auxiliary value function $w$ which is itself a viscosity solution of a non-linear parabolic PDE.  Applying ideas recently developed in Bouchard, Elie and Touzi (2008), our general result also allows to consider optimal control problems with moment constraints of the form $\Esp{g(Z^\nu(T))}\ge 0$ or $\Pro{g(Z^\nu(T))\ge 0}\ge p$.

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Each problem to be solved at the study group will be discussed.

I'll give a brief survey of what is known about the density of rational points on definable sets in o-minimal expansions of the real field, then discuss improving these results in certain cases.

Patterns of sources and sinks in the complex Ginzburg-Landau equation Jonathan Sherratt, Heriot-Watt University The complex Ginzburg-Landau equation is a prototype model for self-oscillatory systems such as binary fluid convection, chemical oscillators, and cyclic predator-prey systems. In one space dimension, many boundary conditions that arise naturally in applications generate wavetrain solutions. In some contexts, the wavetrain is unstable as a solution of the original equation, and it proves necessary to distinguish between two different types of instability, which I will

explain: convective and absolute. When the wavetrain is absolutely unstable, the selected wavetrain breaks up into spatiotemporal chaos. But when it is only convectively stable, there is a different behaviour, with bands of wavetrains separated by sharp interfaces known as "sources" and "sinks". These have been studied in great detail as isolated objects, but there has been very little work on patterns of alternating sources and sinks, which is what one typically sees in simulations. I will discuss new results on source-sink patterns, which show that the separation distances between sources and sinks are constrained to a discrete set of possible values, because of a phase-locking condition.

I will present results from numerical simulations that confirm the results, and I will briefly discuss applications and the future challenges. The work that I will describe has been done in collaboration with Matthew Smith (Microsoft Research) and Jens Rademacher (CWI, Amsterdam).

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A brief introduction to the field of complex networks is carried out by means of some examples. Then, we focus on the topics of defining and applying centrality measures to characterise the nodes of complex networks. We combine this approach with methods for detecting communities as well as to identify good expansion properties on graphs. All these concepts are formally defined in the presentation. We introduce the subgraph centrality from a combinatorial point of view and then connect it with the theory of graph spectra. Continuing with this line we introduce some modifications to this measure by considering some known matrix functions, e.g., psi matrix functions, as well as new ones introduced here. Finally, we illustrate some examples of applications in particular the identification of essential proteins in proteomic maps.

PLEASE NOTE THE CHANGE OF TIME FOR THIS WEEK: 13.30 instead of 12.

In the first of two talks, I will simultaneously introduce the notion of a co-Higgs vector bundle and the notion of the spectral curve associated to a compact Riemann surface equipped with a vector bundle and some extra data. I will try to put these ideas into both a historical context and a contemporary one. As we delve deeper, the emphasis will be on using spectral curves to better understand a particular moduli space.

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The paper for this first session is "Every discrete, finite image is uniquely determined by its dipole histogram" by Charles Chubb and John I. Yellott

We will look at CAT(0) spaces, their isometries and boundaries (defined through Busemann functions).

The Cox ring of a variety is an analogue of the homogeneous coordinate ring of projective space. Cox rings are not defined for every variety and even when they are defined, they need not be finitely generated. Varieties for which the Cox ring is finitely generated are called Mori dream spaces and, as the name suggests, they are particularly well-suited for the Minimal Model Program. Such varieties include toric varieties and del Pezzo surfaces.

I will report on joint work with T. Várilly and M. Velasco where we introduce a class of smooth projective surfaces having finitely generated Cox ring. This class of surfaces contains toric surfaces and (log) del Pezzo surfaces.

Future discovery and control in biology and medicine will come from

the mathematical modeling of large-scale molecular biological data,

such as DNA microarray data, just as Kepler discovered the laws of

planetary motion by using mathematics to describe trends in

astronomical data. In this talk, I will demonstrate that

mathematical modeling of DNA microarray data can be used to correctly

predict previously unknown mechanisms that govern the activities of

DNA and RNA.

First, I will describe the computational prediction of a mechanism of

regulation, by using the pseudoinverse projection and a higher-order

singular value decomposition to uncover a genome-wide pattern of

correlation between DNA replication initiation and RNA expression

during the cell cycle. Then, I will describe the recent

experimental verification of this computational prediction, by

analyzing global expression in synchronized cultures of yeast under

conditions that prevent DNA replication initiation without delaying

cell cycle progression. Finally, I will describe the use of the

singular value decomposition to uncover "asymmetric Hermite functions,"

a generalization of the eigenfunctions of the quantum harmonic

oscillator, in genome-wide mRNA lengths distribution data.

These patterns might be explained by a previously undiscovered asymmetry

in RNA gel electrophoresis band broadening and hint at two competing

evolutionary forces that determine the lengths of gene transcripts.

I will present existence and uniqueness results for theCauchy problem as in the title.

In this talk I will present recent developments of the obstacle type problems, with various applications ranging

from: Industry to Finance, local to nonlocal operators, and one to multi-phases.

The theory has evolved from a single equation

$$

\Delta u = \chi_{u &gt; 0}, \qquad u \geq 0

$$

to embrace a more general (two-phase) form

$$

\Delta u = \lambda_+ \chi_{u&gt;0} - \lambda_- \chi_{u0$.

The above problem changes drastically if one allows $\lambda_\pm$ to have the incorrect sign (that appears in composite membrane problem)!

In part of my talk I will focus on the simple {\it unstable} case

$$

\Delta u = - \chi_{u&gt;0}

$$

and present very recent results (Andersson, Sh., Weiss) that classifies the set of singular points ($\{u=\nabla u =0\}$) for the above problem.

The techniques developed recently by our team also shows an unorthodox approach to such problems, as the classical technique fails.

At the end of my talk I will explain the technique in a heuristic way.