Past

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 > 0}, \qquad u \geq 0

$$

to embrace a more general (two-phase) form

$$

\Delta u = \lambda_+ \chi_{u>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>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.

How many integer-points lie in a circle of radius $\sqrt{x}$?

A poor man's approximation might be $\pi x$, and indeed, the aim-of-the-game is to estimate

$$P(x) = \sharp\{(m, n) \in\mathbb{Z}: \;\; m^{2} + n^{2} \leq x\} -\pi x,$$

Once one gets the eye in to show that $P(x) = O(x^{1/2})$, the task is to graft an innings to reduce this bound as much as one can. Since the cricket-loving G. H. Hardy proved that $P(x) = O(x^{\alpha})$ can only possible hold when $\alpha \geq 1/4$ there is some room for improvement in the middle-order.

In this first match of the Junior Number Theory Seminar Series, I will present a summary of results on $P(x)$.

Physical space-time is a manifold with a Lorentzianmetric, but the more mathematical treatments of the theory usually prefer toreplace the metric with a positive - i.e. Riemannian - one. The passage fromLorentzian to Riemannian metrics is called 'Wick rotation'. In my talk I shallgive a precise description of what is involved, and shall explain some of itsimplications for physics.

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The subject of distributional symmetries and invarianceprinciples yields deep results on the structure of the underlying randomobjects. So it is of general interest to investigate if such an approach turnsout to be also fruitful in the quantum world. My talk will report recentprogress in the transfer of de Finetti's pioneering work to noncommutativeprobability. More precisely, an infinite sequence of random variables isexchangeable if its distribution is invariant under finite permutations. The deFinetti theorem characterizes such sequences as conditionally i.i.d. Recentlywe have proven a noncommutative analogue of this celebrated theorem. We willdiscuss the new symmetries `braidability'

and `quantum exchangeability' emerging from our approach.In particular, this brings our approach in close contact with Jones' subfactortheory and Voiculescu's free probability. Finally we will address that ourmethods give a new proof of Thoma's theorem on the general form of charactersof the infinite symmetric group. Quite surprisingly, Thoma's theorem turns outto be the spectral analysis of the tail algebra coming from a certainexchangeable sequence of transpositions. This is in part joint work with RolfGohm and Roland Speicher.

REFERENCES:

[1] C. Koestler. A noncommutative extended de Finettitheorem 258 (2010) 1073-1120.

[2] R. Gohm & C. Kostler. Noncommutativeindependence from the braid group $\mathbb{B}_\infty$. Commun. Math. Phys.289(2) (2009), 435-482.

[3] C. Koestler & R. Speicher. A noncommutative deFinetti theorem:

Invariance under quantum permutations is equivalent tofreeness with amalgamation. Commun. Math. Phys. 291(2) (2009), 473-490.

[4] R. Gohm & C. Koestler: An application ofexchangeability to the symmetric group $\mathbb{S}_\infty$. Preprint.

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In this talk I will be discussing some reformulations of string theory which promote T-duality to the level of a manifest symmetry namely Hull's Doubled Formalism and Klimcik and Severa's  Poisson-Lie T-duality.   Such formalisms double the number of fields but also incorporate some chirality-like constraint. Invoking this constraint leads one to consider sigma-models which, though duality invariant, do not possess manifest Lorentz Invariance.   Whilst such formalisms make sense at a classical level their quantum validity is less obvious.  I address this issue by examining the renormalization of these duality invariant sigma models.  This talk is based upon both forthcoming work and recent work in arXiv:0910.1345 [hep-th] and its antecedents arXiv:0708.2267, arXiv:0712.1121.

Regularization techniques based on the Golub-Kahan iterative bidiagonalization belong among popular approaches for solving large discrete ill-posed problems. First, the original problem is projected onto a lower dimensional subspace using the bidiagonalization algorithm, which by itself represents a form of regularization by projection. The projected problem, however, inherits a part of the ill-posedness of the original problem, and therefore some form of inner regularization must be applied. Stopping criteria for the whole process are then based on the regularization of the projected (small) problem.

We consider an ill-posed problem with a noisy right-hand side (observation vector), where the noise level is unknown. We show how the information from the Golub-Kahan iterative bidiagonalization can be used for estimating the noise level. Such information can be useful for constructing efficient stopping criteria in solving ill-posed problems.

This is joint work by Iveta Hn\v{e}tynkov\'{a}, Martin Ple\v{s}inger, and Zden\v{e}k Strako\v{s} (Faculty of Mathematics and Physics, Charles University, and Institute of Computer Science, Academy of Sciences, Prague)