Physical & Biological Sciences Division
Distinguished Professor
Faculty
McHenry Library
McHenry Building Room #4124
Mathematics Department
M.S., Moscow Institute of Steel and Alloys
Ph.D., University of California, Berkeley
Viktor Ginzburg’s work from the last two decades has mainly been in symplectic dynamics, an area at the interface of symplectic topology and dynamical systems. He has worked on various aspects of the existence problem for periodic orbits: the Hamiltonian Seifert conjecture, the Conley conjecture, multiplicity of periodic orbits of Reeb flows in higher dimensions, periodic orbits of a charge in a magnetic field and several others. His more recent work focuses on connections between Floer theory and dynamics beyond periodic orbits, including dynamics of Hamiltonian pseudo-rotations in all dimensions, invariant sets, entropy, and Floer homology persistence modules.
In the 90s he worked in Poisson geometry and Hamiltonian actions of compact groups.
Viktor Ginzburg has worked in various areas of symplectic geometry including Poisson geometry, geometry of Hamiltonian group actions, geometric quantization, and symplectic topology. His current research lies at the interface of symplectic topology and Hamiltonian dynamical systems and focuses on the existence problem for periodic orbits of Hamiltonian systems. Among his recent results are:
- Counterexamples to the Hamiltonian Seifert conjecture.
- Existence results for periodic orbits of a charge in a magnetic field.
- A work on symplectic topology of coisotropic submanifolds (coisotropic intersections and rigidity), providing a common framework for the Arnold conjecture and the Weinstein conjecture for hypersurfaces.
- The proof of Conley’s conjecture on the existence of periodic points of Hamiltonian diffeomorphisms for a wide class of symplectic manifolds.
- V. L. Ginzburg: The Conley conjecture. Ann. of Math. 172 (2010) 1127-1180
- V. L. Ginzburg and B. Z. Gurel: Action and index spectra and periodic orbits in Hamiltonian dynamics. Geom. Topol. 13 (2009) 2745-2805
- V. L. Ginzburg and B. Z. Gurel: Periodic orbits of twisted geodesic flows and the Weinstein-Moser theorem. Comment. Math. Helv. 84 (2009) 865-907
- V. L. Ginzburg: Coisotropic intersections. Duke Math. J. 140 (2007) 111-163
- V. L. Ginzburg and B. Z. Gurel: A C2-smooth counterexample to the Hamiltonian Seifert conjecture in R4. Ann. of Math. 158 (2003) 953-976
- V. L. Ginzburg, V. Guillemin and Y. Karshon: Cobordisms and Hamiltonian groups actions. Mathematical Surveys and Monographs, vol. 98, American Mathematical Society, 2002
- V. L. Ginzburg: The Hamiltonian Seifert conjecture, examples and open problems. Proceedings of the Third European Congress of Mathematics, Barcelona, 2000; Birkhauser, Progress in Mathematics, 202 (2001), vol. II, pp. 547-555
