Workshop on Recent Advances in Mathematics at IPMU, Nov 16 - 18, 2009

Date:   16-18 November 2009
Place:  IPMU, Seminar Room, Prefab B
Organizing Committee:  Alexey Bondal and Kyoji Saito

16 November (Mo)
9:45 - 10:45      Yuichi Nohara
11:00 - 12:00     Sergei Galkin
14:00 - 15:00     Alex Bene
15:30 - 16:30     Alexander Getmanenko

17 November (Tu)
9:45 - 10:45      Tathagata Basak
11:00 - 12:00     Paul Bressler
14:00 - 15:00     Mikael Pichot
15:30 - 16:30     Ken Shakleton

18 November (We)
13:30 - 15:00      Colin Ingalls
15:30 - 17:00      Timothy Logvinenko

***** Title and Abstract of Talks *****

Yuichi Nohara:
Title: Toric degenerations of Gelfand-Cetlin systems and potential functions
Abstract: It is well known that a polarized toric variety is related to a moment polytope in two different ways, monomial basis and the moment map. In the case of flag manifolds, certain polytopes, called Gelfand-Cetlin polytopes, also appear in similar ways: the Gelfand-Cetlin basis, a basis of an irreducible representation; and the Gelfand-Cetlin system, a completely integrable system. Furthermore the flag manifold admits a degeneration into a toric variety corresponding to the Gelfand-Cetlin polytope. Kogan and Miller proved that the Gelfand-Cetlin basis can be deformed into monomial basis on the toric variety under the degeneration. We show that the Gelfand-Cetlin system can be deformed into a moment map on the toric variety. We also apply the result to disk counting and calculate the potential function for a Lagrangian torus fiber of the Gelfand-Cetlin system. This is a joint work with T. Nishinou and K. Ueda.

Sergei Galkin:
Ttitle: Landau-Ginzburg models of Fano varieties
Abstract: TBA

Alex Bene:
Ttitle: Feynman diagrams and mapping class representations.
Abstract: In this talk, I will review how elementary moves on fatgraphs, a type of Feynman diagram with cyclically oriented vertices arising in 2D quantum gravity, defines the so-called Ptolemy groupoid, which can be viewed as an enlargement of the mapping class group of a bordered surface. This viewpoint allows for the possibility of certain mapping class representations to be "extended to the groupoid level."  I will discuss examples of such representations which have target a certain vector space generated by similar Feynman diagrams called Jacobi diagrams which have arisen in the field of finite type invariants and Chern-Simons theory.

Alexander Getmanenko:
Title: Towards proving existence of resurgent solutions of a linear ODE.
Abstract: The talk will be devoted to discussion of foundational issues in the mathematically rigorous hyperasymptotic, or "resurgent",theory of linear differential equations. We will look at Shatalov-Sternin's proof of existence of resurgent  solutions of a linear ODE and discuss the construction of analytic continuation to a common "Riemann surface'' of all terms of the von Neumann series appearing in their proof. A more modest statement will be presented that we could write up in a detailed and rigorous fashion. We will also mention possible applications of the theory.

Reference: arXiv:0907.2934

Tathagata Bassak:
Title: A complex hyperbolic reflection group and the monster.
Abstract: Let R be the reflection group of the complex Leech lattice plus a hyperbolic cell. Let D be the incidence graph of the projective plane with three elements. Let A(D) be the Artin group of D : generators of A(D) correspond to the vertices of D. Two generators braid if there is an edge between them, otherwise they commute.
It is surprising that both the bimonster and the reflection group R are quotients of A(D), when the generators are mapped to elements of order 2 and 3 respectively.
A conjecture by Daniel Allcock seeks to explain this connection between R and the bimonster via complex hyperbolic geometry. We shall present some of the evidence for this conjecture so far. An imprecise analogy with Weyl groups, will be guiding our investigation. 
Any previous encounter with the monster or the Leech lattice will not be assumed. 

Paul Bressler:
Title: Deformations of gerbes

Mikael Pichot:
Title: Groups of intermediate rank.
Abstract: I will introduce countable discrete groups which interpolate the classical (integer) values of the rank, especially between rank 1 and rank 2. This is joint work with S. Barre.

Ken Shakleton
Title: On the coarse geometry of Teichmueller space
Abstract: We discuss the synthetic geometry of the pants graph in comparison with the Weil-Petersson metric, whose geometry the pants graph coarsely models following work of Brock’s. We also restrict our attention to the pants graph of the 5-holed sphere, studying the Gromov bordification and the dynamics of pseudo-Anosov mapping classes.

Colin Ingalls
Title:  Rationality of the Brauer-Severi Varieties of Skylanin algebras
Abstract:  Iskovskih's conjecture states that a conic bundle over a surface is rational if and only if the surface has a pencil of rational curves which meet the discriminant in 3 or fewer points, (with one exceptional case).  We generalize Iskovskih's proof that such conic bundles are rational, to the case of projective space bundles of higher dimension.  The proof involves maximal orders and toric geometry.  As a corollary we show that the Brauer-Severi variety of a Sklyanin algebra is rational.
Timothy Logvinenko
Title: Derived functors between cotangent bundles of flag varieties
Abstract: This is a joint work with Rina Anno (UChicago). We construct a network of functors, which correspond to `generalized braid diagrams', between derived categories of coherent sheaves on cotangent bundles of full and partial flag varieties. We then prove that isotopic braid diagrams correspond to isomorphic functors. This generalises the classic braid group action of Khovanov-Thomas on a derived category of the cotangent bundle of a complete flag variety.