My research interests.



My research interests lie in the area of mathematics called symplectic topology. Its origins lie in the study of Hamiltonian dynamics describing evolution of classical mechanical systems. Symplectic topology provides a proper geometric language for studying Hamiltonian dynamical systems and features deep analytical, geometric, topological and algebraic methods, some of them coming from modern mathematical physics, to prove the existence of certain kinds of trajectories in such systems. Interestingly enough, the geometric structures appearing in symplectic topology are of independent interest for other areas of geometry, topology and modern mathematical physics.

More specifically, my recent research concerns several themes:

1. Quasi-morphisms and quasi-states. These are certain ``almost homomorphisms" and ``almost linear" functionals that appeared in our joint work with L.Polterovich and that have a number of useful applications concerning Hamiltonian dynamics, functional-theoretic properties of Poisson brackets and algebraic and metric properties of symplectomorphism groups. A particularly useful application of quasi-states concerns rigidity of intersections in symplectic manifolds: certain sets in symplectic manifolds cannot be displaced from certain other sets, or from themselves, by symplectomorphisms, even though they can be displaced by smooth, or even volume-preserving isotopies. For more details and references see my survey

Quasi-morphisms and quasi-states in symplectic topology, preprint, 2014. To appear in the Proceedings of the International Congress of Mathematicians (Seoul, 2014).

In fact, the study of quasi-states is of interest already in the finite-dimensional setting where a quasi-state is function on a (real) finite-dimensional Lie algebra which is additive on any pair of commuting elements. For results on such quasi-states see:

Lie quasi-states (with L.Polterovich). Published in Journal of Lie theory 19:3 (2009), 613-637.



2. Connecting trajectories of Hamiltonian flows. Various deep methods of symplectic topology (including, but not restricted to the quasi-states mentioned above) can be applied to prove the existence of trajerctories of Hamiltonian flows connecting two sets (so-called Hamiltonian chords). The key feature of the results obtained in this way is that the assumptions on a Hamiltonian needed to prove the existence of Hamiltonian chords are stable in the uniform norm (that is, do not involve any derivatives or any convexity properties of the Hamiltonian). For more details see:

Poisson brackets and symplectic invariants (with L.Buhovsky and L.Polterovich). Published in Selecta Mathematica, 18:1 (2012), 89-157.



3. Rigidity of the Poisson brackets with respect to the uniform norm. The Poisson brackets of two functions on a symplectic manifold depend on the first derivatives of the functions. Nevertheless, as it was first discovered by F.Cardin and C.Viterbo, the uniform norm of the Poisson bracket cannot change arbitrarily when the functions are perturbed in the same uniform norm (that does not involve any derivatives). Deep methods of symplectic topology (including, but not restricted to the quasi-states mentioned above) can be used to prove various results on functional properties of the Poisson bracket with respect to the uniform norm. Interestingly enough, some of the results can be extended to certain expressions involving iterated Poisson brackets of two functions. For more details see:

Poisson brackets and symplectic invariants (with L.Buhovsky and L.Polterovich). Published in Selecta Mathematica, 18:1 (2012), 89-157.

Poisson brackets, quasi-states and symplectic integrators (with L.Polterovich and D.Rosen). Published in Discrete and Continuous Dynamical Systems, 28 (2010), 1455-1468.

C^0-rigidity of the double Poisson bracket (with L.Polterovich). Published in the International Mathematics Research Notices (2009), 1134-1158.

C^0-rigidity of Poisson brackets (with L.Polterovich). Published in Proceedings of the Joint Summer Research Conference on Symplectic Topology and Measure-Preserving Dynamical Systems (eds. A. Fathi, Y.-G. Oh and C. Viterbo), 25-32, Contemporary Mathematics 512, AMS, 2010.

Quasi-morphisms and the Poisson bracket (with L.Polterovich and F.Zapolsky). Published in Pure and Applied Mathematics Quarterly (the special issue dedicated to Gregory Margulis) 3:4 (2007).