Past

Living systems are subject to constant evolution through the two processes of mutations and selection, a principle discovered by C. Darwin. In a very simple, general and idealized description, their environment can be considered as a nutrient shared by all the population. This alllows certain individuals, characterized by a 'phenotypical trait', to expand faster because they are better adapted to use the environment. This leads to select the 'best fitted trait' in the population (singular point of the system). On the other hand, the new-born individuals undergo small variation of the trait under the effect of genetic mutations. In these circumstances, is it possible to describe the dynamical evolution of the current trait?

We will give a mathematical model of such dynamics, based on parabolic equations, and show that an asymptotic method allows us to formalize precisely the concepts of monomorphic or polymorphic population. Then, we can describe the evolution of the 'fittest trait' and eventually to compute various forms of branching points which represent the cohabitation of two different populations.

The concepts are based on the asymptotic analysis of the above mentioned parabolic equations once appropriately rescaled. This leads to concentrations of the solutions and the difficulty is to evaluate the weight and position of the moving Dirac masses that desribe the population. We will show that a new type of Hamilton-Jacobi equation, with constraints, naturally describes this asymptotic. Some additional theoretical questions as uniqueness for the limiting H.-J. equation will also be addressed.

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.