# NYC Dynamics Seminar at Yeshiva University

NYC Dynamics Seminar at Yeshiva University is a research level seminar with a broad agenda aimed at research mathematicians and graduate students whose interests include various aspects of the modern theory of dynamical systems and related topics in analysis, geometry, number theory and possibly other subjects. Its aim is to supplement more specialized seminars in the NYC area and provide a meeting place and a venue for discussions for mathematicians associated with various universities and colleges in the NYC metropolitan area working or interested in dynamical systems. The seminar will primarily feature speakers from outside the area specially invited for this purpose as well as mathematicians visiting various NYC universities. Core financial support for the seminar is provided by the Center for Mathematical Sciences at Yeshiva University. Support from other institutions who contribute to funding visits of seminar speakers will be acknowledged.

Organizers:

Location: Yeshiva University, 215 Lexington Ave, Room 506. The entrance to the building is on the southeast corner of 33rd St and Lexington Ave. Participants will be asked to present some form of identification to the security to be signed in.

Meeting Times: Various Wednesdays at 5pm in Spring 2016 and likely in future semesters. The typical lecture time will be 1 hour, followed by a 30 minute period for extra time for lecture and/or discussions.

Spring 2016 Seminar Poster

## Talks from Spring 2016:

February 24 Barak Weiss Tel Aviv University Dynamics on closed subsets of $${\mathbb R}^d$$ and questions of Danzer and Gowers

Let $$X$$ denote the collection of closed subsets of $${\mathbb R}^d$$. Equipped with the Chabauty-Fell topology, this is a compact metric space on which any group of homeomorphisms of $${\mathbb R}^d$$ acts. In joint work with Solan and Solomon, we show that the only minimal sets for the action of volume preserving affine maps, are the two fixed points.

As a consequence we solve a question of Gowers in discrete geometry, which is a variant of the following open question of Danzer: does there exist a discrete subset of the plane, with finite upper density, which intersects every convex subset of area one?

The talk will be elementary and self-contained.

March 2 Boris Hasselblatt Tufts University Godbillon-Vey classes for isotropic foliations and rigidity

For a contact manifold $$(M^{2m+1},A)$$ and an $$m+1$$-dimensional $$dA$$-isotropic $$C^2$$ foliation, we define Godbillon--Vey invariants $$\{\mathit{GV}_i\}_{i=0}^{m+1}$$ inspired by the Godbillon--Vey invariant of a codimension-one foliation, and we demonstrate the potential of this family as a tool in geometric rigidity theory. One ingredient for the latter is the Mitsumatsu formula for geodesic flows on (Finsler) surfaces.

March 9 Rafael de la Llave Georgia Tech A posteriori KAM theory. Some applications to regularity.

We present some recent proofs of KAM theory on the 'a posteriori' format. They show that given an approximate solution of an invariance equation, one can find a true solution nearby.

This proofs lead to efficient algorithms, but they also lead to some theoretical results such as the monogenic or Whitney differentiability with respect to the frequency.

(joint work with R. Calleja and A. Celletti)

April 6 Federico Rodriguez Hertz Penn State New developments in smooth rigidity of lattice actions

In joint work with A. Brown and Z. Wang we were able to show some new rigidity results for higher rank lattice actions that in one way or other display some hyperbolicity. The main technical tool is the notion of resonance that will be discussed in the talk. In particular we prove global rigidity results for Anosov actions on nilmanifolds and also for actions on low dimensions.

April 20 Bassam Fayad CNRS and University of Paris-6 Spectral properties of mixing flows on surfaces

How 'chaotic’ can smooth area preserving surface flows be? It is known for a long time (from the works of Kolmogorov, Katok and then Kochergin) that starting from very low regularity these flows, if they do not have fixed points, cannot be mixing. Via Poincaré sections, the latter phenomenon is due to a Denjoy type rigidity of discrete time one dimensional dynamics. However, Kochergin and then Khanin and Sinai showed that these flows can be mixing when they have singularities. Nothing however was known about their spectral type. We will explain why Kochergin flows with one (sufficiently strong) power like singularity typically have a maximal spectral type equivalent to Lebesgue measure on the circle. So, these quasi-minimal flows on the two torus, that have almost the same phase portrait as that of a minimal translation flow, share the same maximal spectral type as Anosov flows! In fact, the Lebesgue spectrum is rather reminiscent of the parabolic paradigm (of horocyclic flows for example) to which the Kochergin flows are related due to the shear along their orbits. We will discuss this relation and its consequences as well as several questions around mixing area preserving flows.