Past

In this talk we will give an introduction to the theory of p-adic (or rigid) cohomology. We will first define the theory for smooth affine varieties, then sketch the definition in general, next compute a simple example, and finally discuss some applications.

Free groups, free abelian groups and fundamental groups of

closed orientable surfaces are the most basic and well-understood

examples of infinite discrete groups. The automorphism groups of

these groups, in contrast, are some of the most complex and intriguing

groups in all of mathematics. In these lectures I will concentrate

on groups of automorphisms of free groups, while drawing analogies

with the general linear group over the integers and surface mapping

class groups. I will explain modern techniques for studying

automorphism groups of free groups, which include a mixture of

topological, algebraic and geometric methods.

Abstract : The question adressed in this talk is the following one : how are the extreme eigenvalues of a matrix X moved by a small rank perturbation P of X ?
We shall consider this question in its generic apporach, i.e. when the matrices X and P are chosen at random independently and in isotropic ways.
We shall give a general answer, uncovering a remarkable phase transition phenomenon: the limit of the extreme eigenvalues of the perturbed matrix differs from the original matrix if and only if the eigenvalues of the perturbing matrix are above a certain critical threshold. We also examine the consequences of this eigenvalue phase transition on the associated eigenvectors and generalize our results to examine the case of multiplicative perturbations or of additive perturbations for the singular values of rectangular matrices.

Abstract: We will discuss systems of diffusion processes on the real line, in which the dynamics of every single process is determined by its rank in the entire particle system. Such systems arise in mathematical finance and statistical physics, and are related to heavy-traffic approximations of queueing networks. Motivated by the applications, we address questions about invariant distributions, convergence to equilibrium and concentration of measure for certain statistics, as well as hydrodynamic limits and large deviations for these particle systems. Parts of the talk are joint works with Amir Dembo, Tomoyuki Ichiba, Ioannis Karatzas, Soumik Pal and Ofer Zeitouni

Abstract: We will discuss systems of diffusion processes on the real line, in which the dynamics of every single process is determined by its rank in the entire particle system. Such systems arise in mathematical finance and statistical physics, and are related to heavy-traffic approximations of queueing networks. Motivated by the applications, we address questions about invariant distributions, convergence to equilibrium and concentration of measure for certain statistics, as well as hydrodynamic limits and large deviations for these particle systems. Parts of the talk are joint works with Amir Dembo, Tomoyuki Ichiba, Ioannis Karatzas, Soumik Pal and Ofer Zeitouni

The moduli space of supersymmetric (eighth-BPS) giant gravitons in $AdS_5 \times S^5$ is a limit of projective spaces. Quantizing this moduli space produces a Fock space of oscillator states, with a cutoff $N$ related to the rank of the dual $U(N)$ gauge group. Fuzzy geometry provides the ideal set of techniques for associating points or regions of moduli space to specific oscillator states. It leads to predictions for the spectrum of BPS excitations of specific worldvolume geometries. It also leads to a group theoretic basis for these states, containing Young diagram labels for $U(N)$ as well as the global $U(3)$ symmetry group. The problem of constructing gauge theory operators corresponding to the oscillator states and  some recent progress in this direction are explained.

Free groups, free abelian groups and fundamental groups of

closed orientable surfaces are the most basic and well-understood examples

of infinite discrete groups. The automorphism groups of these groups, in

contrast, are some of the most complex and intriguing groups in all of

mathematics. I will give some general comments about geometric group

theory and then describe the basic geometric object, called Outer space,

associated to automorphism groups of free groups.

This Colloquium talk is the first of a series of three lectures given by

Professor Vogtmann, who is the European Mathematical Society Lecturer. In

this series of three lectures, she will discuss groups of automorphisms

of free groups, while drawing analogies with the general linear group over

the integers and surface mapping class groups. She will explain modern

techniques for studying automorphism groups of free groups, which include

a mixture of topological, algebraic and geometric methods.

Ocean climate models are unlikely routinely to have sufficient

resolution to resolve the turbulent ocean eddy field. The need for the

development of improved mesoscale eddy parameterisation schemes

therefore remains an important task. The current dominant mesoscale eddy

closure is the Gent and McWilliams scheme, which enforces the

down-gradient mixing of buoyancy. While motivated by the action of

baroclinic instability on the mean flow, this closure neglects the

horizontal fluxes of horizontal momentum. The down-gradient mixing of

potential vorticity is frequently discussed as an alternative

parameterisation paradigm. However, such a scheme, without careful

treatment, violates fundamental conservation principles, and in

particular violates conservation of momentum.

A new parameterisation framework is presented which preserves

conservation of momentum by construction, and further allows for

conservation of energy. The framework has one dimensional parameter, the

total eddy energy, and five dimensionless and bounded geometric

parameters. The popular Gent and McWilliams scheme exists as a limiting

case of this framework. Hence the new framework enables for the

extension of the Gent and McWilliams scheme, in a manner consistent with

key physical conservations.

We consider the pricing of a maturity guarantee, which is equivalent to the pricing of a European put option, in a regime-switching market model. Regime-switching market models have been empirically shown to fit long-term stockmarket data better than many other models. However, since a regime-switching market is incomplete, there is no unique price for the maturity guarantee. We extend the good-deal pricing bounds idea to the regime-switching market model. This allows us to obtain a reasonable range of prices for the maturity guarantee, by excluding those prices which imply a Sharpe Ratio which is too high. The range of prices can be used as a plausibility check on the chosen price of a maturity guarantee.

• Jean Charles Seguis - The fictitious domain method applied to hybrid simulations in biology
• Chris Farmer - Data assimilation and parameter estimation
• Mark Curtis - Stokes' flow, singularities and sperm

To each additive definable category there is attached its category of pp-imaginaries. This is abelian and every small abelian category arises in this way. The connection may be expressed as an equivalence of 2-categories. We describe two associated spectra (Ziegler and Zariski) which have arisen in the model theory of modules.

Brittle failure through multiple cracks occurs in a wide variety of contexts, from microscopic failures in dental enamel and cleaved silicon to geological faults and planetary ice crusts. In each of these situations, with complicated stress geometries and different microscopic mechanisms, pairwise interactions between approaching cracks nonetheless produce characteristically curved fracture paths. We investigate the origins of this widely observed "en passant" crack pattern by fracturing a rectangular slab which is notched on each long side and then subjected to quasistatic uniaxial strain from the short side. The two cracks propagate along approximately straight paths until they pass each other, after which they curve and release a lens-shaped fragment. We find that, for materials with diverse mechanical properties, each curve has an approximately square-root shape, and that the length of each fragment is twice its width. We are able to explain the origins of this universal shape with a simple geometrical model.

I will review the basic properties of the DFT and describe how these can be exploited to efficiently compute degree 1 L-functions.

Computing the eigenvalue decomposition of a symmetric matrix and the singular value decomposition of a general matrix are two of the central tasks in numerical linear algebra. There has been much recent work in the development of linear algebra algorithms that minimize communication cost. However, the reduction in communication cost sometimes comes at the expense of significantly more arithmetic and potential instability.

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In this talk I will describe algorithms for the two decompositions that have optimal communication cost and arithmetic cost within a small factor of those for the best known algorithms. The key idea is to use the best rational approximation of the sign function, which lets the algorithm converge in just two steps. The algorithms are backward stable and easily parallelizable. Preliminary numerical experiments demonstrate their efficiency.

I will give a brief introduction into how Elliptic curves can be used to define complex oriented

cohomology theories. I will start by introducing complex oriented cohomology theories, and then move onto

formal group laws and a theorem of Quillen. I will then end by showing how the formal group law associated

to an elliptic curve can, in many cases, allow one to define a complex oriented cohomology theory.

 Consider the solution of a scalar Helmholtz equation where the potential (or index) takes two positive values, one inside a disk or a ball (when d=2 or 3) of radius epsilon and another one outside. For this classical problem, it is possible to derive sharp explicit estimates of the size of the scattered field caused by this inhomogeneity, for any frequencies and any contrast. We will see that uniform estimates with respect to frequency and contrast do not tend to zero with epsilon, because of a quasi-resonance phenomenon. However, broadband estimates can be derived: uniform bounds for the scattered field for any contrast, and any frequencies outside of a set which tends to zero with epsilon.