### Web publications:

**Math and Music**(talk at UCLA Fiat Lux seminar), 2005

**The Hofer-Zehnder capacity**(talk at the summer schoold on Hamiltonian Mechaics and Integrable Systems, 2004), pp. 39-56 in the Proceedings of the summer school [Download]

### Papers and preprints:

**Picard groups of topologically stable Poisson structures**(with D. Shlyakhtenko), Pacific J. Math. 224 (2006), no. 1, 151--183

**Abstract**: We compute the group of Morita self-equivalences (the Picard group) of a Poisson structure on an orientable surface, under the assumption that the degeneracies of the Poisson tensor are linear. The answer involves mapping class groups of surfaces, i.e., groups of isotopy classes of diffeomorphisms. We also show that the Picard group of these structures coincides with the group of outer Poisson automorphisms. [Arxiv]

**Gauge equivalence of Dirac structures and symplectic groupoids**(with H. Bursztyn), Ann. Inst. Fourier (Grenoble), 53, (2003), no. 1, 309-337.

**Abstract**: We study gauge transformations of Dirac structures and the relationship between gauge and Morita equivalences of Poisson manifolds. We describe how the symplectic structure of a symplectic groupoid is affected by a gauge transformation of the Poisson structure on its identity section, and prove that gauge-equivalent integrable Poisson structures are Morita equivalent. As an example, we study certain generic sets of Poisson structures on Riemann surfaces: we find complete gauge-equivalence invariants for such structures which, on the 2-sphere, yield a complete invariant of Morita equivalence. [Arxiv]

**A classification of topologically stable Poisson structures on a compact oriented surface,**J. Symplectic Geometry, 1 (2002), no. 3, 523-542

**Abstract**: Poisson structures vanishing linearly on a set of smooth closed disjoint curves are generic in the set of all Poisson structures on a compact connected oriented surface. We construct a complete set of invariants classifying these structures up to an orientation-preserving Poisson isomorphism. We show that there is a set of non-trivial infinitesimal deformations which generate the second Poisson cohomology and such that each of the deformations changes exactly one of the classifying invariants. As an example, we consider Poisson structures on the sphere which vanish linearly on a set of smooth closed disjoint curves. [Arxiv]

**Towards a classification of Poisson structures on surfaces: Quantization, Poisson brackets and beyond**(Manchester, 2001). Contemporary Mathematics, 315 (2002), 81-88

**Abstract**: We consider topologically stable Poisson structures on a Riemann surface. These structures are defined as having the simplest (i.e., linear) degeneracies and from a dense set in the space of all Poisson structures on a given surface. We give a classification of such topologically stable structures by constructing a complete set of explicit invariants. We also explain how to extend these results to structures with higher order degeneracies; in the case of isolated quadratic degeneracies we construct a complete set of invariants.

**A bicovariant differential algebra of a quantum group: Coherent states, differential and quantum geometry**(Bialowieza, 1997). Rep. Math. Phys. 43 (1999), no. 1-2, 313-322

**Abstract**: A bicovariant differential algebra of four basic objects (coordinate functions, differential 1-forms, Lie derivatives and inner derivations) within a differential calculus on a quantum group is shown to be produced by a direct application of the cross-product construction to the Woronowicz differential complex, whose Hopf algebra properties account for the bicovariance of the algebra. A correspondence with classical differential calculus, including Cartan identity, and some other useful relations are considered. An explicit construction of a bicovariant differential algebra on GLq(N) is given and its (co)module properties are discussed.

**On the algebraic structure of differential calculus on quantum groups**(with A. Vladimirov) J. Math. Phys. 38 (1997), no. 10, 5434-5446

**Abstract**: Intrinsic Hopf algebra structure of the Woronowicz differential complex is shown to generate quite naturally a bicovariant algebra of four basic objects within a differential calculus on quantum groups -- coordinate functions, differential 1-forms, Lie derivatives, and inner derivations -- as the cross-product algebra of two mutually dual graded Hopf algebras. This construction, properly taking into account Hopf-algebraic properties of Woronowicz's bicovariant calculus, provides a direct proof of the Cartan identity and of many other useful relations. A detailed comparison with other approaches is also given. [Arxiv]

**Deformational quantization of Poisson-Lie groups and Lie bialgebras,**review paper in PhysTech. Journal, 2 (1996), no. 6, 15-36.

**Some Invariants of Poisson Manifolds**, Ph.D. Thesis, UC Berkeley, May 2002 [Download]