About the School
Abstracts of one-hour Lectures
"Stochastic differential equations and rough paths"
Abstract: Stochastic differential equation is an ordinary differential equation containing stochastic processes. One of the most typical stochastic process is a standard Brownian motion. The equations formally contain the derivative of the Brownian motion with respect to the time parameter. However, the Brownian path is not differentiable with respect to the time parameter. We can define integration with respect to a certain class of irregular paths which are not bounded variation by using the Young integration theory. Unfortunately, in the present case, Young integration is not applicable and the equation can be defined by using Ito's stochastic integral with respect to the Brownian motion.
Rough path analysis is invented by Terry Lyons and this gives another useful approach to "stochastic differential equations". The important note is that the rough path analysis is a natural extension of the Young integration theory and essentially has nothing to do with stochastic analysis. In this talk, we explain how rough path analysis can be used to analysis of "stochastic differential equations".
"Hodge-de Rham theorem of Landau-Ginzburg model"
Abstract: A simple Landau-Ginzburg (LG) model is the complex n-dimensional Euclidean space with a holomorphic function (superpotential). With this holomorphic function, we can get the twisted Cauchy-Riemann operator and a family of differential operators. In particular, we can study the spectrum problem of the corresponding twisted Laplacian operator. If the LG system satisfies a tame condition, then this twisted Laplacian operator has the pure discrete spectrum. Based on this fact, we can build the Hodge theory and then a L2-version De Rham theorem, which relates the space of n-dimensional harmonic forms to the Milnor ring of the superpotential. This theorem is analogous to the Hodge-de Rham theorem in Kähler geometry for compact Kähler manifold.
"Concavity properties for elliptic and parabolic equations"
Abstract: In the theory of elliptic and parabolic equations it is a natural question to ask whether some relevant geometric property of the boundary and/or initial data is preserved by solutions of a Cauchy-Dirichlet problem.@ In this lecture I talk about a short history and recent results on power concavity properties of solutions of elliptic and parabolic equations.
"Optimal Connections Problems and Geometry of Hausdorff and Gromov-Hausdorff Distances"
Abstract: This lecture is devoted to geometry of the Gromov-Hausdorff distance that measures gdifferenceh between two metric spaces. If metric spaces are isometric then the distance vanishes, that is why it is naturally to consider the distance on isometry classes of metric spaces. In the case when the metric spaces in consideration are compact, this distance is a metric, and the corresponding ghyperspaceh is called Gromov-Hausdorff space. Our talk mainly devoted to description of geometry of the Gromov- Hausdorff space.
The convergence w.r.t. Gromov-Hausdorff distance has many beautiful applications. In particular, it was used by Gromov to prove that any discrete group with polynomial growth contains a nilpotent subgroup of finite index. This distance has been applied in computer graphics and computational geometry to find correspondences between different shapes. In Cosmology, Gromov-Hausdorff distance was used to prove stability of the Friedmann model.
"On the n-sphere Sn"
"The topology of the generalized billiards bounded by arcs of the confocal quadrics"
Abstract: In the first part of the talk we consider the billiard system without friction in the flat domain, bounded by smooth curve. All reflections is standart and without loss of the speed. Then we define the class of integrable billiards bounded by arcs of the confocal quadrics (elementary billiards). And the we define generalized billiard the topological billiard and the billiard book which are also integrable. These billiards are the results of the gluing elementary domains by some of their common boundaries. We want to describe the topology of these system. The second part of the talk will be devoted to bases, main ideas and constructions of Fomenko-Zieschang theory of invariants of integrable Hamiltonian systems. In fact, such graph invariants help effectively describe the topology of phase space foliated on the closures of solutions of integrable systems. This invariant ia complete, so two such systems have fiber-wise diffeomorphic foliations if and only if their invariants coincide. Some their extension also classifies structures of trajectories of such systems in the sense of topological (orbital) equivalence. On the third part we present the Fomenko-Zieshang invariants of the topological billiards and some interesting examples of the billiard books. As a corollary of this result we received the interesting facts about the topology of the isoenergy surafaces of these systems. As the main result of the fourth part of the talk we see that the Fomenko-Zieschang invariants help to model complex integrable Hamiltonian systems by integrable billiards. More precisely the smooth integrable systems in mechanics and geodesic lows on orientable 2-surfaces often have the same structure of closures of trajectories. Even if billiard domain is a some complex, it is more simple to see interesting effects (bifurcation of Liouville tori, periodic critical trajectories, their stability) than in integrable systems of higher degree. So, it allows to say that "visual" billiard system in a suitable domains gives us a chance to describe the behavior of a complicated integrable system. This for many cases was done in the papers by Fomenko and Vedyushkina and also will be presented on the talk.