Past

A copolymer is a chain of repetitive units (monomers) that

are almost identical, but they differ in their degree of

affinity for certain solvents. This difference leads to striking

phenomena when the polymer fluctuates

in a non-homogeneous medium, for example made up by two solvents

separated by an interface.

One may observe, for exmple, the localization of the polymer at the

interface between the two solvents.

Much of the literature on the subject focuses on the most basic model

based on the simple symmetric random walk on the integers, but

E. Bolthausen and F. den Hollander (AP 1997) pointed out

the convergence of the (rescaled) free energy of such a discrete model

toward

the free energy of a continuum model, based on Brownian motion,

in the limit of weak polymer-solvent coupling. This result is

remarkable because it strongly suggests

a universal feature for copolymer models. In this work we prove that

this is indeed the case. More precisely,

we determine the weak coupling limit for a general class of discrete

copolymer models, obtaining as limits

a one-parameter (alpha in (0,1)) family of continuum models, based on

alpha-stable regenerative sets.

This is a review of hep-th/0912.4912

McBurnie: “Sound propagation through bubbly liquids”.

Hewett: "Switching on a time-harmonic acoustic source".

When the human eye looks at a distant object, the lens is held in a state of tension by a set of fibres (known as zonules) that connect the lens to the ciliary body. To view a nearby object, the ciliary muscle (which is part of the ciliary body) contracts. This reduces the tension in the zonules, the lens assumes a thicker and more rounded shape and the optical power of the eye increases.

This process is known as accommodation.

With increased age, however, the accommodation mechanism becomes increasingly ineffective so that, from an age of about 50 years onwards, it effectively ceases to function. This condition is known as presbyopia. There is considerable interest in the ophthalmic community on developing a better understanding of the ageing processes that cause presbyopia. As well as being an interesting scientific question in its own right, it is hoped that this improved understanding will lead to improved surgical procedures (e.g. to re-start the accommodation process in elderly cataract patients).

Let R be a number field (or a recursive subring of anumber field) and consider the polynomial ring R[T].

We show that the set of polynomials with integercoefficients is diophantine (existentially definable) over R[T].

Applying a result by Denef, this implies that everyrecursively enumerable subset of R[T]^k is diophantine over R[T].

Some years ago Hall and Smith in a number of papers developed a theory governing the interaction of vortices and waves in shear flows. In recent years immense interest has been focused on so-called self-sustained processes in turbulent shear flows where the importance of waves interacting with streamwise vortex flows has been elucidated in a number of; see for example the work of Waleffe and colleagues, Kerswell, Gibson, etc. These processes have a striking resemblance to coherent structures observed in turbulent shear flow and for that reason they are often referred to as exact coherent structures. It is shown that the structures associated with the so-called 'lower branch' state, which has been shown to play a crucial role in these self-sustained process, is nothing but a Rayleigh wave vortex interaction with a wave system generating streamwise vortices inside a critical layer. The theory enables the reduction of the 3D Navier Stokes equations to a coupled system for a steady streamwise vortex and an inviscid wave system. The reduced system for the streamwise vortices must be solved with jump conditions in the shear across the critical layer and the position of that layer constitutes a nonlinear pde eigenvalue problem. Remarkable agreement between the asymptotic theory and numerical simulations is found thereby demonstrating the importance of vortex-wave interaction theory in the mathematical description of coherent structures in turbulent shear flows. The theory offers the possibility of drag reduction in turbulent shear flows by controlling the flow to the neighborhood of the lower branch state. The relevance of the work to more general shear flows is also discussed.

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In the last few years, there has been renewed interest in stochastic

finite element methods (SFEMs), which facilitate the approximation

of statistics of solutions to PDEs with random data. SFEMs based on

sampling, such as stochastic collocation schemes, lead to decoupled

problems requiring only deterministic solvers. SFEMs based on

Galerkin approximation satisfy an optimality condition but require

the solution of a single linear system of equations that couples

deterministic and stochastic degrees of freedom. This is regarded as

a serious bottleneck in computations and the difficulty is even more

pronounced when we attempt to solve systems of PDEs with

random data via stochastic mixed FEMs.

In this talk, we give an overview of solution strategies for the

saddle-point systems that arise when the mixed form of the Darcy

flow problem, with correlated random coefficients, is discretised

via stochastic Galerkin and stochastic collocation techniques. For

the stochastic Galerkin approach, the systems are orders of

magnitude larger than those arising for deterministic problems. We

report on fast solvers and preconditioners based on multigrid, which

have proved successful for deterministic problems. In particular, we

examine their robustness with respect to the random diffusion

coefficients, which can be either a linear or non-linear function of

a finite set of random parameters with a prescribed probability

distribution.

After reviewing the salient details from last week's seminar, I will construct an explicit example of a spectral curve, using co-Higgs bundles of rank 2. The role of the spectral curve in understanding the moduli space will be made clear by appealing to the Hitchin fibration, and from there inferences (some of them very concrete) can be made about the structure of the moduli space. I will make some conjectures about the higher-dimensional picture, and also try to show how spectral varieties might live in that picture.

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I will describe some more of the deformation theory necessary for the first talk. This leads to a number of natural questions and counterexamples. This talk requires a strong stomach, or a fanatical devotion to symmetric obstruction theories.

In this talk we describe a general framework for deriving

modified equations for stochastic differential equations with respect to

weak convergence. We will start by quickly recapping of how to derive

modified equations in the case of ODEs and describe how these ideas can

be generalized in the case of SDEs. Results will be presented for first

order methods such as the Euler-Maruyama and the Milstein method. In the

case of linear SDEs, using the Gaussianity of the underlying solutions,

we will derive a SDE that the numerical method solves exactly in the

weak sense. Applications of modified equations in the numerical study

of Langevin equations and in the calculation of effective diffusivities

will also be discussed.

Canonical quantization of gravitational field will beconsidered. Examples for which the procedure can be completed (without reducingthe degrees of freedom) will be presented and discussed. The frameworks appliedwill be: Loop Quantum Gravity, relational construction of the Dirac observablesand deparametrization.

In the talk we discuss some results of [1]. We apply our previous methods [2] to geometry and to the mappings with bounded distortion.

\textbf{Theorem 1}.  Let $v:\Omega\to\mathbb{R}$ be a $C^1$-smooth function on a domain (open connected set) $\Omega\subset\mathbb{R}^2$. Suppose

$$ (1)\qquad \operatorname{Int} \nabla v(\Omega)=\emptyset. $$

Then $\operatorname{meas}\nabla v(\Omega)=0$.

Here $\operatorname{Int}E$ is the interior of ${E}$, $\operatorname{meas} E$ is the Lebesgue measure of ${E}$. Theorem 1 is a straight consequence of the following two results.

\textbf{ Theorem 2 [2]}.  Let $v:\Omega\to\mathbb{R}$ be a $C^1$-smooth function on a domain $\Omega\subset\mathbb{R}^2$. Suppose (1) is fulfilled. Then the graph of $v$ is a normal developing surface. 

Recall that a $C^1$-smooth manifold $S\subset\mathbb{R}^3$ is called  a normal developing surface [3] if for any $x_0\in S$ there exists a straight segment $I\subset S$ (the point $x_0$ is an interior point of $I$) such that the tangent plane to $S$ is stationary along $I$.

\textbf{Theorem 3}.  The spherical image of any $C^1$-smooth normal developing surface $S\subset\mathbb{R}^3$ has the area (the Lebesgue measure) zero.

Recall that the spherical image of a surface $S$ is the set $\{\mathbf{n}(x)\mid x\in S\}$, where $\mathbf{n}(x)$ is the unit normal vector to $S$ at the point~$x$. From Theorems 1--3 and the classical results of A.V. Pogorelov (see [4, Chapter 9]) we obtain the following corollaries Corollary 4. Let the spherical image of a $C^1$-smooth surface $S\subset\mathbb{R}^3$ have no interior points. Then this surface is a surface of zero extrinsic curvature in the sense of Pogorelov.

\textbf{ Corollary 5}. Any $C^1$-smooth normal developing surface $S\subset\mathbb{R}^3$ is a surface of zero extrinsic curvature in the sense of Pogorelov.

\textbf{Theorem 6}. Let $K\subset\mathbb{R}^{2\times 2}$ be a compact set and the topological dimension of $K$ equals 1. Suppose there exists $\lambda> 0$ such that $\forall A,B\in K, \, \, |A-B|^2\le\lambda\det(A-B).$

Then for any Lipschitz mapping $f:\Omega\to\mathbb R^2$ on a domain $\Omega\subset\mathbb R^2$ such that $\nabla f(x)\in K$ a.e. the identity $\nabla f\equiv\operatorname{const}$ holds.

Many partial cases of Theorem 6 (for instance, when $K=SO(2)$ or $K$ is a segment) are well-known (see, for example, [5]).

The author was supported by the Russian Foundation for Basic Research (project no. 08-01-00531-a).

[1] {Korobkov M.\,V.,} {``Properties of the $C^1$-smooth functions whose gradient range has topological dimension~1,'' Dokl. Math., to appear.}

[2] {Korobkov M.\,V.} {``Properties of the $C^1$-smooth functions with nowhere dense gradient range,'' Siberian Math. J., \textbf{48,} No.~6, 1019--1028 (2007).}

[3] { Shefel${}'$ S.\,Z.,} {``$C^1$-Smooth isometric imbeddings,'' Siberian Math. J., \textbf{15,} No.~6, 972--987 (1974).}

[4] {Pogorelov A.\,V.,} {Extrinsic geometry of convex surfaces, Translations of Mathematical Monographs. Vol. 35. Providence, R.I.: American Mathematical Society (AMS). VI (1973).}

[5] {M\"uller ~S.,} {Variational Models for Microstructure and Phase Transitions. Max-Planck-Institute for Mathematics in the Sciences. Leipzig (1998) (Lecture Notes, No.~2. http://www.mis.mpg.de/jump/publications.html).}

In this talk, we will focus on the asymptotic behavior in time of the solution of a model equation for Bose-Einstein condensation, in the case where the trapping potential varies randomly in time.

The model is the so called Gross-Pitaevskii equation, with a quadratic potential with white noise fluctuations in time whose amplitude tends to zero.

The initial condition is a standing wave solution of the unperturbed equation We prove that up to times of the order of the inverse squared amplitude the solution decomposes into the sum of a randomly modulatedmodulation parameters.

In addition, we show that the first order of the remainder, as the noise amplitude goes to zero, converges to a Gaussian process, whose expected mode amplitudes concentrate on the third eigenmode generated by the Hermite functions, on a certain time scale, as the frequency of the standing wave of the deterministic equation tends to its minimal value.

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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).