Past

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.

TBA

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.

TBA

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)

Speakers include:

* David Abrahams (Manchester, UK); * Stuart Antman (Maryland, USA); * Martine Ben Amar (Ecole Normale Supérieure, France); * Mary Boyce (MIT, USA); * John Hutchinson (Harvard, USA); * Nadia Lapusta (Caltech, USA); * John Maddocks (Lausanne, Switzerland); * Stefan Mueller (Bonn, Germany); * Christoph Ortner (Oxford, UK); * Ares Rosakis (Caltech, USA); * Hanus Seiner (Academy of Sciences, Czech Republic); * Eran Sharon (Hebrew University, Israel); * Lev Truskinovsky (Lab de Mécanique des Solids, France); * John Willis (Cambridge, UK).

• “Two Problems Relating to Sand Dune Formation” by Andrew Ellis

• “Interface Sharpening with a Lattice Boltzmann Equation” by Tim Reis

• “A Dual Porosity Model for the Uptake of Nutrients by Root Hairs” by Kostas Zygalakis

Atomistic computer simulation models are constructed to study a range of materials in which

the atoms appear in novel environments. Two key research areas are considered:

• The Growth and Structure Inorganic Nanotubes. A range of materials have been

observed to form nanotubular structures (inorganic nanotubes - INTs) analogous to those

well known for carbon. These INTs, which may have unique low-dimensional morphologies

not simply related to known bulk polymorphs, potentially offer unique mechanical and electronic properties. A useful synthetic pathway is to use carbon nanotubes as templates using

molten salts. Atomistic simulation models, in which the atom interactions are treated utilizing relatively simple potential energy functions, are developed and applied to understand

the INT formation and stability. INT morphologies are classified by reference to folding

two dimensional sheets. The respective roles of thermodynamics and kinetics in determining

INT morphology are outlined and the atomistic results used to develop an analytic model to

predict INT diameters.

• Ordering on Multiple Length-Scales in Network-forming Liquids. Intermediate-range order (IRO), in which systems exhibit structural ordering on length-scales beyond

the nearest-neighbour (short-range), has been identified in a wide range of materials and is

characterised by the appearance of the so-called first sharp diffraction peak (FSDP) at low

scattering angles. The precise structural origin of such ordering remains contentious and a full

understanding of the factors underlying this order is vital if such materials (many of which are

technologically significant) are to be produced in a controlled manner. Simulation models,

in which the ion-ion interactions are represented by relatively simple potential functions

which incorporate (many-body) polarisation and which are parameterised by reference to

well-directed electronic structure calculations, have been shown to reproduce such IRO and

allow the precise structural origin of the IRO to be identified. Furthermore, the use of

relatively simple (and hence computationally tractable) models allows for the study of the

relatively long length- and time-scales required. The underlying structures are analysed with

reference to both recent (neutron scattering) experimental results and high level electronic

structure calculations. The role of key structural units (corner and edge sharing polyhedra)

in determining the network topology is investigated in terms of the underlying dynamics and

the relationship to the glass transition considered.

We will begin by reviewing the construction of the symplectic quotient and the definition of the Kirwan map. Then we will give an overview of Kirwan's original proof of the surjectivity of this map and some generalizations of this result. Finally we will talk about the techniques that are being developed to construct right inverses for the Kirwan map.

Speakers include:

* Graeme Ackland (School of Physics and Astronomy, Edinburgh) * Andrea Braides (Rome II) * Thierry Bodineau (École Normale Supérieure, Paris) * Matthew Dobson (Minneapolis) * Laurent Dupuy (CEA, Saclay) * Ryan Elliott (Minneapolis) * Roman Kotecky (Warwick) * Carlos Mora-Corral (BCAM, Bilbao) * Stefano Olla (CEREMADE, Paris-Dauphine) * Bernd Schmidt (TU Munich) * Lev Truskinovsky (École Polytechnique, Palaiseau) * Min Zhou (Georgia Tech, Atlanta)

We study a stochastic control problem in the context of utility maximization under model uncertainty. The problem is formulated as /max min/ problem : /max /over strategies and consumption and /min/ over the set of models (measures).

For the minimization problem, we have showed in Bordigoni G., Matoussi,A., Schweizer, M. (2007) that there exists a unique optimal measure equivalent to the reference measure. Moreover, in the context of continuous filtration, we characterize the dynamic value process of our stochastic control problem as the unique solution of a generalized backward stochastic differential equation with a quadratic driver. We extend first this result in a discontinuous filtration. Moreover, we obtain a comparison theorem and a regularity properties for the associated generalized BSDE with jumps, which are the key points in our approach, in order to solve the utility maximization problem over terminal wealth and consumption. The talk is based on joint work with M. Jeanblanc and A. Ngoupeyou (2009).

HSBC Currency Trading has collaborated with the Oxford Maths Institute for over six years and is now working with its third DPhil student. In this workshop, we will look at the some of the academic research which has directly benefited the trading operation.