MAT 290-016 Gromov-Witten Theory and
Virasoro Constraints
Spring Quarter 2006, CRN = 80247
Classroom Meetings: Fridays 4:10 PM – 6:00, in 3106 MSB
Organizer: Motohico Mulase (3103 MSB, 752-6324)
Seminar Presentations
April 7
Speaker: Motohico Mulase
Title:
From Grassmannian to Gromov-Witten
Abstract:
The GW theory is an intersection theory of cohomology
classes on the moduli space of holomorphic maps from
a Riemann surface into a symplectic manifold. The Virasoro
conjecture states that the generating function of GW invariants
is a solution to the KP-type equations and satisfies the Virasoro
constraint conditions.
In this talk, which serves as an introduction to our journey, I will start
with reviewing the intersection theory of Schubert cycles on
Grassmannians and its relation to the KP theory. This forms a
prototype of the theory to be developed.
The seminar is aimed at learning the relations between
the Mirzakhani theory, string topology, and GW theory.
A brief description on each of these topics will be discussed.
April 14
Speaker: Brad Safnuk
Title: Mirzakhani recursion formula,
Virasoro constraints, and Atiyah-Bott localization theorem
Abstract: Brad will give us first a review of
Mirzakhani's new proof for the Witten-Kontsevich theorem and
the Virasoro structure in the theory discovered in his
recent paper with me.
He will then present a new result, identifying the whole
Mirzakhani recursion from the point of view of Atiyah-Bott localization
theory.
April 21 Attention: This week we meet 5:10 - 6:00 !
Speaker: Brad Safnuk
Title: Continuation
Abstract: Brad will continue his discussion of McShane identity, localization formula, and Virasoro.
April 28
Speaker: Nora Ganter, UIUP and MSRI
Title: Generalized Moonshine, Elliptic Cohomology, and Conformal
Field Theory
Abstract: Nora will give a very informal exposition of the connection between
generalized Moonshine, equivariant elliptic cohomology and
CFT's with finite gauge group.
Besides representation theory of the Virasoro algebra itself,
the "Virasoro constraint condition" appears in two places in
mathematics: one is in Gromov-Witten theory, and the other in
Borchards' theory of vertex (operator) algebras and Moonshine.
We plan to learn from Nora about the most recent development
on the subject listed in the title of the talk.
May 5
Speaker: Motohico Mulase
Title: Why do we expect a Virasoro
constraint condition for the GW invariants?
Abstract: We'll examine the moduli space of
holomorphic maps from a Riemann surface into a target
algebraic manifold. I'll then explain when the target is a Fano variety or a
Calabi-Yau, why an integrable system or a Virasoro constraint
should show up. This consideration makes the study of the GW invariants
for a projective line the most important building block.
This leads to the next talk by Andrew, because the GW of a projective
line is a combinatorial problem of counting so called Hurwitz coverings.
Speaker: Andrew Hodge
Title: Localization theory and
ELSV formula
Abstract: The ELSV formula relates the combinatorial
data of Hurwitz coverings of a projective line with the intersection
theory of the moduli space of algebraic curves. This is quite an
interesting and mysterious formula. Andrew will explain to us the
exact statement, the significance, and some ideas of the proof, of this
formula.
May 12
Speaker: Xiang Tang
Title: The ring structure on orbifold cohomology
Abstract: It's concrete: there will be a dinner
in honor of Xiang after his talk.
May 19
Speaker: Hiro-Fumi Yamada, Okayama University and UC Davis
Title: Virasoro and KP --- another point of view
Abstract: Certain Fock representation of the Virasoro algebra will be discussed.
This representation is completely reducible and decomposes
into the direct sum of irreducible highest weight representations
with central charge c=1. The highest weight vector of each
irreducible summand is given by a Schur function indexed
by a rectangular Young diagram.
The surprising fact is that each vector in the "lower part" of the irreducible
decomposition corresponds to Hirota's bilinear differential
equation of the KP hierarchy.
At present no intrinsic (geometric)
explanation of this phenomenon is known,
although one can prove this fact.
This is a joint work with Wakimoto.
May 27 Attention: This week we meet on Saturday (!!!) in 1147 MSB, and the time is 3:00 - 5:00.
Speaker: Kenji Fukaya, Kyoto University and MSRI
Title: Gromov-Witten theory based on bordered
Riemann surfaces
Abstract: We will learn from Professor Fukaya
the current status of Gromov-Witten theory in Symplectic side, and
his vision for future developments.
There will be a dinner in honor of Kenji after his talk.
June 2
Speaker: Anne Schilling
Title: k-Schur functions and
Gromov-Witten invariants -- a mosaic
Abstract: Anne will give us an overview of
recent developments from the combinatorial point
of view. A more official abstract is coming soon.
This talk will conclude this quarter's (surprisingly)
successful GW seminar.
There will be a dinner in honor of Anne after her talk.