Past

Associated to a pencil of algebraic curves with singular fibres is a bundle of Jacobians (which are abelian varieties off the discriminant locus of the family and semiabelian varieties over it). Normal functions, which are holomorphic sections of such a Jacobian bundle, were introduced by Poincare and used by Lefschetz to prove the Hodge Conjecture (HC) on algebraic surfaces. By a recent result of Griffiths and Green, an appropriate generalization of these normal functions remains at the center of efforts to establish the HC more generally and understand its implications. (Furthermore, the nature of the zero-loci of these normal functions is related to the Bloch-Beilinson conjectures on filtrations on Chow groups.)

Abel-Jacobi maps give the connection between algebraic cycles and normal functions. In this talk, we shall discuss the limits and singularities of Abel-Jacobi maps for cycles on degenerating families of algebraic varieties. These two features are strongly connected with the issue of graphing admissible normal functions in a Neron model, properly generalizing Poincare's notion of normal functions. Some of these issues will be passed over rather lightly; our main intention is to give some simple examples of limits of AJ maps and stress their connection with higher algebraic K-theory.

A very new theme in homological mirror symmetry concerns what the mirror of a normal function should be; in work of Morrison and Walcher, the mirror is related to counting holomorphic disks in a CY 3-fold bounding on a Lagrangian. Along slightly different lines, we shall briefly describe a surprising application of "higher" normal functions to growth of enumerative (Gromov-Witten) invariants in the context of local mirror symmetry.

Let $G=(V,E)$ be a graph with weights $\{w_e : e \in E\}$, and assume all weights are distinct. If $G$ is finite, then the well-known Prim's algorithm constructs its minimum spanning tree in the following manner. Starting from a single vertex $v$, add the smallest weight edge connecting $v$ to any other vertex. More generally, at each step add the smallest weight edge joining some vertex that has already been "explored" (connected by an edge) to some unexplored vertex.

If $G$ is infinite, however, Prim's algorithm does not necessarily construct a spanning tree (consider, for example, the case when the underlying graph is the two-dimensional lattice ${\mathbb Z}^2$, all weights on horizontal edges are strictly less than $1/2$ and all weights on vertical edges are strictly greater than $1/2$).

The behavior of Prim's algorithm for *random* edge weights is an interesting and challenging object of study, even
when the underlying graph is extremely simple. This line of research was initiated by McDiarmid, Johnson and Stone (1996), in the case when the underlying graph is the complete graph $K_n$. Recently Angel et. al. (2006) have studied Prim's algorithm on regular trees with uniform random edge weights. We study Prim's algorithm on $K_n$ and on its infinitary analogue Aldous' Poisson-weighted infinite tree. Along the way, we uncover two new descriptions of the Poisson IIC, the critical Poisson Galton-Watson tree conditioned to survive forever.

Joint work with Simon Griffiths and Ross Kang.

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