16-19 June 2003







The workshop will focus on mathematical aspects of quantum cohomology, especially those which are related to combinatorics, algebra, differential equations, and integrable systems. The workshop is financially supported by grants from the JSPS and the RIMS.

The program is given below, with abstracts of talks.

For further information please contact

Martin Guest,
Anatol Kirillov

To speakers: blackboard with chalk, and overhead slide projector, will be available.

RIMS home page (with local information and links)

TMUGS The conference is sponsored in part by a grant from the JSPS to Tokyo Metropolitan University. The Tokyo Metropolitan University Geometry Server contains further information about mathematics and mathematicians in Japan.

BRIEF SUMMARY (full details are below)

June 16       Nakajima Saito Yoshioka
June 17 Kim-1 Mare-1 lunch Buch-1 Jinzenji Maeno
June 18 Kim-2 Guest lunch Buch-2 Ikeda Iritani
June 19 Kim-3   lunch Buch-3 Mare-2 Kirillov


June 16

13:00 - 14:00
Hiraku Nakajima (Kyoto University)
Introduction to moduli spaces of instantons on R^4

ABSTRACT : In this introductory lecture, I will explain basic properties of moduli spaces of $SU(r)$-instantons on $R^4$. These spaces can be interpreted as framed moduli spaces of locally free sheaves on the projective plane, and hence have a natural smooth (partial) compactification (by adding non locally free sheaves). These give resolutions of singularities of Uhlenbeck (partial) compactifications. If I have time, I will also introduce quiver varieties as subvarieties of (partial) compactifications.

14:10 - 15: 10
Kyoji Saito (RIMS)
On the period primitive integrals (one origin of the B-model side).

Kota Yoshioka (Kobe University)
Nekrasov's partition function and the blow-up formula.

ABSTRACT: Nekrasov conjectured that integration of 1 over the framed moduli spaces of instantons on C^2 gives the prepotential of N=2 super Yang-Mills theory. We consider similar integration over the framed moduli
spaces of instantons on the blow-up of C^2. We discuss the relation of the two invariants. As an application, we show that Nekrasov's invariant satisfies the Whitham equation in the integrable system. This is joint work with Hiraku Nakajima (Kyoto University).

June 17

10:00 - 11:00
Bumsig Kim (Pohang University of Science and Technology)
Introduction to the Virasoro Conjecture (after Givental) - 1.

11:10 - 12:10
Augustin-Liviu Mare (University of Toronto)
Quantum cohomology of flag manifolds - 1: generators and relations.

ABSTRACT : A theorem of Bumsig Kim gives a presentation of the (small) quantum cohomology ring (H^*(G/B)\otimes R[q_i], \circ) of the generalized flag manifold G/B in terms of generators and relations. In the first talk I will present a set of sufficient conditions which, if they are satisfied by a product \star on H^*(G/B)\otimes R[q_i], then the ring (H^*(G/B)\otimes R[q_i], \star) is isomorphic to Kim's ring. I will also raise the question of whether it is possible to construct such products \star which are different from \circ. At the end of this talk I will discuss the possibility of extending these results to the (quantum cohomology of the) infinite dimensional flag manifolds.

13:30 -14:30
Anders Skovsted Buch (Aarhus University)
Quantum cohomology with elementary methods - 1.

ABSTRACT: The (small) quantum cohomology ring of a homogeneous space is a deformation of the usual cohomology ring. The structure constants are the three-point, genus zero Gromov-Witten invariants, which count the number of rational curves meeting general Schubert varieties. The structure of the quantum ring is so far only known for certain homogeneous spaces, and sophisticated intersection theory on moduli spaces has been used to prove structure theorems in the known cases. In my talks, I will explain a new elementary method for the computation of quantum cohomology, which relies on classical Schubert calculus applied to partial flags called the kernel and span of a curve. This method will be applied to prove the structure theorems for Grassmannians and partial flag varieties SL(n)/P. I will also present results with A. Kresch and H. Tamvakis, which use the kernel and span to express Gromov-Witten invariants on Grassmannians as classical intersection numbers on two-step partial flag varieties, and a related conjectural Littlewood-Richardson rule for Gromov-Witten invariants.

14:40 - 15:40
Masao Jinzenji (Hokkaido University)
Coordinate Change of Gauss-Manin systems and generalized mirror transformations.

ABSTRACT: Continuing the analysis of the relation between quantum cohomology rings and Gauss-Manin systems, we try to derive the generalized mirror transformation of the quantum cohomology of general type projective hypersurfaces that was introduced in our previous work.

16:00 -17:00
Toshiaki Maeno (Kyoto University)
Quantum Schubert calculus for Coxeter groups.

ABSTRACT: The quantum cohomology ring of a flag variety gives a natural deformation of the coinvariant ring of the corresponding Weyl group. We show the existence of natural deformations of coinvariant rings of noncrystallographic
finite Coxeter groups. In the noncrystallographic case, the number of deformation parameters is smaller than the rank of the root system. An analog of the Pieri formula holds also in our deformed ring.


10:00 - 11:00
Bumsig Kim
Introduction to the Virasoro Conjecture (after Givental) - 2.

ABSTRACT: See above

11:10 - 12:10
Martin Guest (Tokyo Metropolitan University)
Gromov-Witten invariants of flag manifolds, via D-modules.

ABSTRACT: The quantum cohomology algebra of the (full) flag manifold is a fundamental example in quantum cohomology theory, with connections to combinatorics, algebraic geometry, and integrable systems. Using the approach introduced in "Quantum cohomology via D-modules" (math.DG/0206212), we give an algorithm for computing the multiplicative structure constants of this algebra, the 3-point genus zero Gromov-Witten invariants. The algorithm involves a Grobner basis calculation and the solution by quadrature of a system of differential equations. In particular we obtain the quantum Schubert polynomials in a natural fashion. This is joint work with A. Amarzaya.

13:30 -14:30
Anders Skovsted Buch
Quantum cohomology with elementary methods - 2.

ABSTRACT: See above

14:40 - 15:40
Kaoru Ikeda (Keio University)
Geometry of the quantum Toda lattice.

ABSTRACT: We consider the iso-level set of the Toda lattice. We show that the compactification of the iso-level set is homeomorphic to a flag variety. To show this fact we consider the line bundle of eigenvectors of the Lax operator on the iso-level set. We consider the canonical quantization of the Lax operator. The line bundle on the iso-level set of the eigenvectors of the quantum Lax operator can also be defined. In the quantum case, the fiber of line bundle is the fundamental solution of a partial differential equation. We can define the "quantum" 1st Chern class from this line bundle. We also show that the algebra generated by these "quantum" 1st Chern classes is isomorphic to the quantum cohomology algebra of the flag variety.

16:00 - 17:00
Hiroshi Iritani (Kyoto University)
Quantum D-module and equivariant Floer theory for free loop spaces.

ABSTRACT: We try to give an explanation of "Homological Geometry" invented by Alexander Givental. He conjectured that quantum D-module of a symplectic manifold is isomorphic to the S^1-equivariant Floer cohomology for the free loop space.First, motivated by the work of Martin Guest, we formulate the abstract notion of quantum D-module which generalizes the D-module defined by the small quantum cohomology ring. Second, we construct the equivariant Floer cohomology modules for symplectic toric super-manifolds explicitly. This is shown to satisfy the axioms of an abstract quantum D-module and coincide with the quantum D-module defined by the small quantum cohomology.

June 19

10:00 - 11:00
Bumsig Kim
Introduction to the Virasoro Conjecture (after Givental) - 3.

ABSTRACT: See above

11:10 - 12:10

13:30 -14:30
Anders Skovsted Buch
Quantum cohomology with elementary methods - 3

ABSTRACT: See above

14:40 - 15:40
Augustin-Liviu Mare (University of Toronto)
Quantum cohomology of flag manifolds - 2: the quantum Giambelli problem.

ABSTRACT: In order to be able to determine completely the quantum cohomology ring of G/B (i.e. determine the Gromov-Witten invariants), one is looking for solutions of the "quantum Giambelli problem": find polynomial representatives of the Schubert classes in Kim's presentation. I will show how the "quantum Chevalley formula" (proved by D. Peterson and W. Fulton and C. Woodward) leads to general formulas for such polynomials. For G=SL(n,C) similar results were obtained by Fomin, Gelfand and Postnikov, and by A. Kirillov and T. Maeno. A purely combinatorial definition of the quantum cohomology ring of G/B will be outlined at the end: I will show how the methods of these two talks enable us to say `"everything" about that combinatorial quantum cohomology ring, i.e. describe it in terms of generators and relations and find solutions of the quantum Giambelli problem.

Anatol N. Kirillov (RIMS)
Bracket algebras and quantum cohomology of flag varieties.

ABSTRACT: We will describe an amazing and mysterious connection between certain quadratic algebras ( so-called "bracket algebras"), and classical and quantum Schubert calculus for flag varieties ( of type A ).