VIKTOR L. GINZBURG Professor of Mathematics M.S., Moscow Institute of Steel and Alloys Ph.D., University of California, Berkeley My Page Selected Publications Curriculum Vitae

Phone: 831-459-2218 Office: Baskin 353C Office Hours: By Appt. Email: ginzburg_at_ucsc_dot_edu

RESEARCH INTERESTS

Viktor Ginzburg's research interest has been in various areas of global analysis. These areas are symplectic topology and Hamiltonian dynamics, Poisson Lie groups and Poisson manifolds, and Hamiltonian actions of Lie groups.

1. Ginzburg's research in symplectic topology and Hamiltonian dynamics focuses on the existence problem for periodic orbits of certain Hamiltonian systems. Hamiltonian systems without periodic orbits and the systems describing the motion of a charge in a magnetic field are his primary concern.

2. The geometry of Poisson Lie groups and Poisson manifolds is known to be strongly related to both the theory of quantum groups and deformation quantization. Ginzburg is interested in the explicit calculation of certain invariants of Poisson manifolds and the study of momentum mappings for Poisson Lie groups.

3. Applications of cobordism techniques to gain a new understanding of various "classic" and recent results in symplectic geometry is the center of his project undertaken jointly with V. Guillemin (MIT) and Y. Karshon (Hebrew University). The examples of such applications are new proofs of the Duistermaat-Heckman formula, the Jeffrey-Kirwan localization theorem, and the fact that geometric quantization commutes with reduction.

SELECTED PUBLICATIONS

V. L. Ginzburg, V. Guillemin and Y. Karshon: Assignments and abstract moment maps. Preprint. Math DG/9904117

V. L. Ginzburg: Hamiltonian dynamical systems without periodic orbits. Preprint, Math.DG/9811014 (1998)

V. L. Ginzburg: A smooth counterexample to the Hamiltonian Seifert conjecture in IR6. Intl. Math. Res. Notices (IMRN) (1997), 13:641-650.

V. L. Ginzburg: Momentum mappings and Poisson cohomology. Intl. J. Math (1996), 7:329-358.

V. L. Ginzburg: On closed trajectories of a charge in a magnetic field. An application of symplectic geometry.

In Contact and Symplectic Geometry, ed., C. B. Thomas. Publications of the Isaac Newton Institute for Mathematical Sciences 8. New York: Cambridge University Press (1996).

V. L Ginzburg: On the existence and non-existence of closed trajectories for some Hamiltonian flows. Math. Z. (1996) 223:397-409.

V. L. Ginzburg, V. Guillemin, and Y. Karshon: Cobordism theory and localization formulas for Hamiltonian group actions. Intl. Math. Res. Notices (IMRN) (1996) 5:221-234.

V. L. Ginzburg, A. Weinstein: Lie-Poisson structure on some Poisson Lie groups. J. Amer. Math. Soc. (1992), 5:445-453.

PUBLISHED BOOKS & ARTICLES

Published Books

Moment maps, Cobordisms, and Hamiltonian group actions
Co-authors: V. Guillemin and Y. Karshon
Mathematical Surveys and Monographs, vol. 98,
American Mathematical Society, 2002, 350pp.

Published Articles

(27) The Hamiltonian Seifert conjecture: examples and open problems, Proceedings of the Third European Congress of Mathematicians Barcelona, 2000, Vol. II, 547-555, Progr. Math., 202 Birkhäuser, Basel, 2001.

(28) Grothendieck groups of Poisson vector bundles, J. Symplectic Geometry, 1 (2001), 121-169.

(29) Periodic orbits of Hamiltonian flows near symplectic extrema, co-author: E. Kerman, Pacific J. Math., 206 (2002), 69-91.

(30) On the construction of a C²-counterexample to the Hamiltonian Seifert conjecture in R⁴, co-author B. Gürel, Electron. Res. Announc. Amer. Math. Soc., 8 (2002), 11-19 (electronic).

Articles Accepted for Publication

(31) Comments to some of Arnold's problems, prepared for the English translation of Zadachi Arnol'da (Arnold's problems) by V.I. Arnold, preface by M.B. Sevryuk and V.B. Filippov, Izdatel'stvo FAZIS, Moscow, 2000 (in Russian).

(32) A C²-smooth counterexample to the Hamiltonian Seifert conjecture in R⁴, co-author: B. Gürel, Preprint 2001; math.DG/0110047. To appear in the Ann. Math.

(33) Relative Hofer-Zehnder capacity and periodic orbits in twisted cotangent bundles, co-author: B. Gürel, Preprint 2003; math.DG/0301073. To appear in Duke Math. Journal.

(34) Symplectic homology and periodic orbits near symplectic submanifolds, co-authors; K. Cieliebak and E. Kerman, Preprint 2002; math.DG/0210468. To appear in the Comment. Math. Helv.

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