SPECIAL LECTURE SERIES

## Professor Vadim Kaloshin

University of Maryland

Title and abstract:

**LECTURE 1:Monday, June 12, 2017 at 15:30**

**TITLE**: Stochastic Arnold diffusion

**ABSTRACT**: In 1964, Arnold constructed an example of a nearly integrable deterministic system exhibiting instabilities. In the 1970s, Chirikov, a physicist, coined the term “Arnold diffusion” for this phenomenon, where diffusion refers to the stochastic nature of instability.One of the most famous examples of stochastic instabilities for nearly integrable systems,discovered numerically by Wisdom, an astronomer, is the dynamics of Asteroids in Kirkwood gaps in the Asteroid belt. In the talk we will describe a class of nearly integrable deterministic systems, where we prove stochastic diffusive behavior. Namely, we show that distributions given by a deterministic evolution of certain random initial conditions weakly converge to a diffusion process.This result is conceptually different from known mathematical results, where the existence of “diffusing orbits” is shown. This work is based on joint papers with Castejon, Guardia, J.Zhang, and K.Zhang.

**LECTURE 2: Tuesday, June 13, 2017 at 15:30**

**TITLE:** Birkhoff Conjecture for convex planar billiards

**ABSTRACT:** G.D.Birkhoff introduced a mathematical billiard inside of a convex domain as the motion of a massless particle with elastic reflection at the boundary. A theorem of Poncelet says that the billiard inside an ellipse is integrable, in the sense that the neighborhood of the boundary is foliated by smooth closed curves and each billiard orbit near the boundary is tangent to one and only one such curve (in this particular case, a confocal ellipse). A famous conjecture by Birkhoff claims that ellipses are the only domains with this property. We show a local version of this conjecture – namely, that a small perturbation of an ellipse has this property only if it is itself an ellipse. This is based on several papers with Avila, De Simoi, G.Huang, Sorrentino.

**LECTURE 3:Thursday, June 15, 2017 at 15:30**

**TITLE:** Collisions for the Newtonian 3-Body Problem are rare, aren’t they?

**ABSTRACT**: Siegel and Alekseev asked if there is an open set of initial conditions for a 3-Body Problem which has a dense subset of collision solutions, i.e. initial conditions in the phase space leading to a double collision. For the Restricted Circular Planar 3-Body Problem with the mass ratio of the primaries μ, we present an open set of initial conditions in which the collision solutions form a ~μ1/30 dense subset. As μ → 0 this subset is asymptotically dense. This is a joint work with Guardia and J.Zhang.

All lectures will take place at Auditorium 232, Amado Mathematics Building, Technion