Several years later, Hopf presented three families of fibrations of round spheres by parallel great subspheres of dimensions 1, 3 and 7, which were later seen to encompass all possible instances of this phenomenon.

We will prove optimality of the above in the following sense:

(1) Given a Hopf fibration of a round sphere by parallel great subspheres, the projection map to the base space is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class.

(2) Given a Hopf fibration of a round sphere by parallel great circles, view a unit vector field tangent to the fibres as a cross-section of the unit tangent bundle of the sphere. Then this vector field is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class.

Previous attempts to find a mathematical sense in which Hopf fibrations and vector fields are optimal (typically in terms of volume or energy minimization) have met with limited success.

All of this represents joint work with Dennis DeTurck and Pete Storm.

The talk is intended to be accessible to graduate students.

Villani (in collaboration with Desvillettes) obtained the first results on the convergence rate for initial data not close to equilibrium. In joint work with Mouhot, he established nonlinear Landau damping for the kinetic equations of plasma physics, settling a long-standing debate. He is one of the pioneers in applications of optimal transport theory to geometric and functional inequalities.

examples of Poisson varieties such as Kleinian and other isolated surface singularities and their symmetric powers, and nilpotent cones of semisimple Lie algebras, using the Springer resolution.

As applications, we deduce bounds on numbers of irreducible representations of spherical symplectic reflection algebras (or on zero-dimensional symplectic leaves in the commutative case), and similarly for universal enveloping and W-algebras modulo a central character. No background on Poisson geometry or D-modules will be assumed. This is joint work with Pavel Etingof.

In this talk we shall present a survey of results of the asymptotic non-limiting theory of singular numbers of random matrices developed in approximately last six years.

slowly than usually assumed in perturbative analysis. For example in the case of Ginzburg-Landau vortices it is necessary to understand dispersive estimates for two dimensional magnetic Schrodinger operators in the presence of non-zero magnetic fluxes. This latter fact implies directly that the gauge potential necessarily has slow (critical) decay which means the problem cannot be treated as a perturbation of the free Schrodinger evolution. We consider model problems in which it is possible to develop an analytical understanding of this situation.

In Rozansky-Witten model, connection with monoidal deformations of the derived category of coherent sheaves will be discussed. In Chern-Simons theory, surface operators will be used to explain the algebraic construction of 2d rational conformal field theory.

This talk is based on joint works with Andrea Montanari and Nike Sun.

To study such a system we focus on the dual object: the convex hull, in the space of all real quadratic forms on $R^n$, of those quadratic forms involved in the system (n is the number of variables in the system).

It turns out that the homology of $X$ is determined by the arrangement of this convex hull with respect to the cone of degenerate forms. This approach allows to efficiently compute homology for a very big number of variables n as long as the number of linearly independent inequalities is limited. Moreover, it works also for systems of integral quadratic inequalities, i.e. in the infinite dimension, beyond the semi-algebraic context.

The calculations are organized in a spectral sequence whose member $E_2$ and the differential $d_2$ have a simple clear geometric interpretation.

This is a joint work with A. Agrachev.

was motivated by the celebrated Gunning's theorem and by another famous conjecture: the Jacobians of curves are exactly the indecomposable principally polarized abelian varieties whose theta-functions provide explicit solutions of the so-called KP equation. The latter was proposed earlier by Novikov and was unsettled at the time of the Welter's work. It was proved later by T.Shiota and until recently has remained the most effective solution of the classical Riemann-Schottky problem.

The characterization of the Jacobains proposed by the trisecant conjecture is much stronger. The proof of this conjecture based on an notion of integrable linear equations and new type cubic identities for the theta-functions valid for the case of Jacobians on the theta-divisor will be presented. We will also discuss applications of integrable equations of the soliton theory for the characterization problem of Prym varieties.

In the talk, I will try to explain the main ideas of the proof. The two highlights are an application of basic algebraic geometry such as ruled surfaces, and an application of the ham sandwich theorem.

problems. In both cases we uncover a large class of solutions, arising from smooth data, which break down in finite time via concentration of a corresponding harmonic map. The mechanisms for the problems however display substantial qualitative and quantitative differences. I will also discuss the connection between the singularity formation result for the Wave Map and the behavior of geodesics on moduli space, which had been the subject of the "geodesic hypothesis".