Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mccme.ru/dfc/2012/Program1/Buryak_summary.pdf Дата изменения: Sun Oct 14 20:47:08 2012 Дата индексирования: Mon Feb 4 16:38:52 2013 Кодировка: Поисковые слова: mercury surface

Summary
Alexandr Buryak The moduli space of a certain class of geometric ob jects parametrizes the isomorphism classes of these ob jects. These kind of spaces are studied very intensively during the last several decades both because of their rich internal geometric and topological structure and because of their close connections with physics. My research focuses on two types of moduli spaces: the moduli space of algebraic curves and the moduli space of framed torsion free sheaves on the pro jective plane. The moduli space of curves is the space of all possible complex structures on a given 2-dimensional surface of a fixed topological type. The interest in the moduli space of curves is inspired by applications in topological string theory and, from mathematical side, Gromov-Witten theory ­ the modern way to count algebraic curves in a fixed target variety. In the last 20 years it was observed that Gromov-Witten theory is strongly related to the theory of integrable hierarchies of PDEs of hydrodynamic type. On one side this connection allows to get a deep understanding of a structure of Gromov-Witten invariants and on the other side it gives a clue for a proof of important results about integrable hierarchies. The topology of the moduli space of sheaves is endowed with different types of algebraic structures which are important from the point of view of representation theory. The moduli space of sheaves is also a source of a large family of very interesting spaces ­ Naka jima's quiver varieties. They are examples of symplectic resolutions and are very important for algebraic geometers. Different combinatorial structures are hidden in the moduli space of sheaves. The topology of the moduli space of sheaves provides an understanding of different results in partition theory and in the theory of symmetric functions. One of the main problems in the topology of the moduli space of curves is a description of a certain subalgebra in the cohomology algebra ­ the so-called tautological ring (see [Fab]). This ring is one of the main ob jects in my research. In the joint work with S. Shadrin ([BS2]) we gave a new proof of Faber's conjecture about top intersections in this ring. I have two works in progress now: joint work with S. Shadrin, L. Spietz and D. Zvonkine about certain cycles in the moduli space of curves ­ double ramification cycles; a proof of Zvonkine's conjecture ([Zvon]) about the top tautological group of a certain version of the moduli space of curves. There are a lot of conjectures about the tautological ring. This is the direction of my future research. Relations in the tautological ring automatically give relations between Gromov-Witten invariants. My goal is to apply this techniques for a proof of the Virasoro conjecture in Gromov-Witten theory (see [Pan]). Dubrovin and Zhang ([DZ]) constructed a correspondence between Gromov-Witten theory and integrable hierarchies. There is an unproved part in their construction, that they left as a conjecture. In my joint works with H. Posthuma and S. Shadrin ([BPS1, BPS2]) we proved a half of this conjecture. Proving the remaining part of this conjecture is a goal of my future research. My approach is partially based on certain deformation formulas that we found in the joint work with S. Shadrin ([BS1]). The group GL2 (C) acts on the pro jective plane, this action lifts to the action on the moduli space of sheaves. H. Naka jima (see [Nak]) developed a rich theory about varieties that are fixed points of actions of finite subgroups of S L2 (C) on the moduli space of sheaves. My goal is to generalize this theory for subgroups of GL2 (C) that doesn't belong to S L2 (C) and study applications in combinatorics. In my work [B1] and joint works [BF1, BF2] I obtained several results in the case of a one-dimensional subtorus in GL2 (C). The other goal is study applications in combinatorics, in particular in the theory of plane partitions. I made a progress in this direction in my work [B2]. There are several papers that I wrote during my undergraduate studies and that are not related to my research interests now (see [B3, B4, B5, B6]).
1


2

Literature Publications of A. Buryak related to the topics of the project: [BF2] A. Buryak, B. L. Feigin. Generating series of the Poincare polynomials of quasihomogeneous Hilbert schemes. arXiv:1206.5640. To be published in Springer Proceedings in Mathematics and Statistics. [BPS2] A. Buryak, H. Posthuma, S. Shadrin. On deformations of quasi-Miura transformations and the Dubrovin-Zhang bracket. Journal of Geometry and Physics 62 (2012), no. 7, 1639-1651. [BPS1] A. Buryak, H. Posthuma, S. Shadrin. A polynomial bracket for the Dubrovin-Zhang hierarchies. Journal of Differential Geometry 91 (2012), 1-33. [B1] A. Buryak. The classes of the quasihomogeneous Hilbert schemes of points on the plane. Moscow Mathematical Journal 12 (2012) no. 1, 1-17. [BF1] A. Buryak, B. L. Feigin. Homogeneous components in the moduli space of sheaves and Virasoro characters. Journal of Geometry and Physics 62 (2012), no. 7, 1652-1664. [B2] A. Buryak. The moduli space of sheaves and the generalization of MacMahon's formula. arXiv:1101.0433. To be published in Functional Analysis and Its Applications. [BS2] A. Buryak, S. Shadrin. A new proof of Faber's intersection number conjecture. Advances in Mathematics 228 (2011), 22-42. [BS1] A. Buryak, S. Shadrin. A remark on deformations of Hurwitz Frobenius manifolds. Letters in Mathematical Physics 93 (2010), no. 3, 243-252. Other papers of A. Buryak: [B6] A. Buryak. Bott's residue formula for singular varieties. TWMS Journal of Pure and Applied Mathematics 2 (2011), no. 1, 17-21. [B5] A. Buryak. First nonzero terms for the Taylor expansion at 1 of the Conway potential function (Russian). Moscow University Mathematics Bul letin 66 (2011), no. 1, 41-43. [B4] A. Buryak. Existence of a singular pro jective variety with an arbitrary set of characteristic numbers. Mathematical Research Letters 17 (2010), no. 3, 395-400. [B3] A. Buryak. The Poincare series of divisorial filtration associated with a curve with one branch at infinity. Mathematical Notes 87 (2010), no. 1-2, 52-58. Furher references: [DZ] B. Dubrovin, Y. Zhang. Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants. arXiv:math/0108160. [Fab] C. Faber. A conjectural description of the tautological ring of the moduli space of curves. Moduli of curves and abelian varieties, 109-129, Aspects Math., E33, Vieweg, Braunschweig, 1999. [Nak] H. Naka jima. Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras. Duke Mathematical Journal 76 (1994), 365-416. [Pan] R. Pandharipande. Open problems (for AGNES). arXiv:1004.3259. [Zvon] D. Zvonkine. A conjectural structure of the tautological ring of Mg,n . Unpublished note, 2009.