The Hodge group (aka special Mumford-Tate group) of a complex abelian variety $X$ is a certain linear reductive algebraic group over the rationals that is closely related to the endomorphism ring of $X$. (For example, the Hodge group is commutative if and only if $X$ is an abelian variety of CM-type.) In this talk I discuss" lower bounds" for the center of Hodge groups of superelliptic jacobians. (This is a joint work with Jiangwei Xue.)

We define a notion of conformal equivalence for discrete surfaces (surfaces composed of euclidean triangles). For example, multiplying the lengths of all edges incident with a single vertex by the same factor is considered to be a conformal change of metric. It turns out that finding a conformally equivalent flat metric on a given discrete surface amounts to minimizing a globally convex functional on the space of all metrics. This functional involves the Lobachevski function (known in the context of computing the volume of hyperbolic tetrahedra). This is not an accident, since surprisingly the whole theory is stongly related to hyperbolic geometry. There are important practical applications of our method to Computer Graphics in the context of texture mapping.

We shall revisit the Boltzmann equation for rarefied non-linear particle dynamics, of conservative or dissipative nature, and on the stochastic N-particle model, introduced by M. Kac.
Related to this equation, we consider a a probabilistic dynamics from generalizations to N-particle model which includes multi-particle interactions. From basic symmetries and invariances for a general class of stochastic interactions, we show existence and uniqueness of states and recover the longtime dynamics and decay rates approaching stable laws characterized by self-similar rescaling, with finite or infinity energy initial data. We classify the moments integrability and see that broad tails (Pareto type) attractors are possible.

There is a large class of applications to these models including classical elastic or inelastic Maxwell type interactions with or without a thermostat, and social dynamics such as information percolation models, or wealth distributions models with Pareto tail formation.

This is work in collaboration with A. Bobylev, C. Cercignani and H. Tharkabhushanam.