Past

I'll explain (following Kontsevich and Soibelman) how cluster transformations intertwine non-commutative DT invariants for CY3 algebras related by a tilt.

We have recently found a way to describe large-scale neural

activity in terms of non-equilibrium statistical mechanics.

This allows us to calculate perturbatively the effects of

fluctuations and correlations on neural activity. Major results

of this formulation include a role for critical branching, and

the demonstration that there exist non-equilibrium phase

transitions in neocortical activity which are in the same

universality class as directed percolation. This result leads

to explanations for the origin of many of the scaling laws

found in LFP, EEG, fMRI, and in ISI distributions, and

provides a possible explanation for the origin of various brain

waves. It also leads to ways of calculating how correlations

can affect neocortical activity, and therefore provides a new

tool for investigating the connections between neural

dynamics, cognition and behavior

In this talk, Professor Lions will first present several examples of numerical simulations of complex industrial systems. All these simulations rely upon some mathematical models involving partial differential equations and he will briefly explain the nature, history and role of such equations. Examples showing the importance of the mathematical analysis (i.e. ‘understanding’) of those models will be presented, concluding with a few trends and perspectives.

Pierre-Louis Lions is the son of the famous mathematician Jacques-Louis Lions and has himself become a renowned mathematician, making numerous important contributions to the theory of non-linear partial differential equations. He was awarded a Fields Medal in 1994, in particular for his work with Ron DiPerna giving the first general proof that the Boltzmann equation of the kinetic theory of gases has solutions. Other awards Lions has received include the IBM Prize in 1987 and the Philip Morris Prize in 1991. Currently he holds the position of Chair of Partial Differential Equations and their Applications at the prestigious Collège de France in Paris.

This lecture is given as part of the 7th ISAAC Congress • www.isaac2009.org

Clore Lecture Theatre, Huxley Building, Imperial College London,
South Kensington Campus, London SW7 2AZ

RSVP: Attendance is free, but with registration in advance
Michael Ruzhansky • @email

I will describe some recent joint work with Davide Gaiotto and Greg Moore, in which we explain the origin of the wall-crossing formula of Kontsevich and Soibelman, in the context of N=2 supersymmetric field theories in four dimensions. The wall-crossing formula gives a recipe for constructing the smooth hyperkahler metric on the moduli space of the field theory reduced on a circle to 3 dimensions. In certain examples this moduli space is actually a moduli space of ramified Higgs bundles, so we obtain a new description of the hyperkahler structure on that space.

When two inclusions (in a composite) get closer and their conductivities degenerate

to zero or infinity, the gradient of the solution to the

conductivity equation blows up in general. We show

that the solution to the conductivity equation can be decomposed

into two parts in an explicit form: one of them has a bounded

gradient and the gradient of the other part blows up. Using the

decomposition, we derive the best possible estimates for the blow-up

of the gradient. The decomposition theorem and estimates have an

important implication in computation of electrical field in

the presence of closely located inclusions.

This short course runs from Monday 29th June to Friday 3rd July. For details of the course and how to register, please visit http://www2.maths.ox.ac.uk/oxmos/meetings/moms/.

Preview of problems to be solved at the study Group in Limerick taking place in the following week.

I will explain what a perfect obstruction theory is, and how it gives rise to a "virtual" fundamental class of the right expected dimension, even when the dimension of the moduli space is wrong. These virtual fundamental classes are one of the main preoccupations of "modern" moduli theory, being the central object of study in Gromov-Witten and Donaldson-Thomas theory. The purpose of the talk is to remove the black-box status of these objects. If there is time I will do some cheer-leading for dg-schemes, and try to convince the audience that virtual fundamental classes are most happily defined to live in the dg-world.

Martin Bridson will give a "repeat" performance of his Abel Lecture which he delivered a few weeks ago in Oslo as part of the scientific programme in honour of Abel Prize laureate Mikhail Gromov.

Abstract:

Gromov has illuminated great swathes of mathematics with the bright light of geometry. By means of example, I hope to convey the sense of wonder that his work engenders and something of the profound influence he has had on the way my generation thinks about mathematics.

I shall focus particularly on Geometric Group Theory. Gromov's ideas turned the study of discrete groups on its head, infusing it with an array of revolutionary ideas and unveiling deep connections to many other branches of mathematics.