[geometry-ml:00839] Seminar at IPMU on 2009/4/27

[Satoshi Kondo]

Dear colleagues:

We announce the following seminar at IPMU
(the Institute for the Physics and Mathematics of the Universe).

Regards,

Satoshi Kondo (IPMU)

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Speaker: Todor Eliseev Milanov (North Carolina State University)

Title: Gromov-Witten Theory and Integrable Hierarchies

Date: April 27, 2009, 15:30-17:00

Place: Seminar Room at IPMU Prefab. B

Abstract: (for pdf, please see http://db.ipmu.jp/seminar/?seminar_id=24)

By definition the Gromov--Witten (GW for brevity) invariants of a
symplectic manifold $X$, enumerate holomorphic curves in $X$,
satisfying various incidence constraints. Their precise definition is
quite involved and uses some of the latest achievements of
mathematics. It is natural to organize the GW invariants in a
generating function $\D_X$. One of the fundamental conjectures in GW
theory says that $\D_X$ is a highest weight vector for the Virasoro
algebra. Apparently there is some exciting and deep interaction
between representations of infinite-dimensional Lie algebras and the
topology of moduli spaces of holomorphic curves and their
generalizations.

In the first part of my talk, I would like to describe an approach to
GW invariants based on mirror symmetry and integrable hierarchies. It
is known that for a quite large class of manifolds, such as the
Grassmanians, toric manifolds, and flag manifolds, the corresponding
GW invariants can be described in terms of the asymptotical expansions
of certain families of oscillating integrals. Each manifold $X$ comes
with its own family, which is known as the mirror model of $X$. It
turns out that many representations of infinite-dimensional Lie
algebras and integrable hierarchies can be interpreted in terms of
oscillating integrals as well. Therefore, it looks very promising that
the mirror model of a symplectic manifold $X$ could provide a natural
language for describing the representations and the integrable
hierarchies which govern the GW theory of $X$.

In the second part of my talk I would like to explain the relation
between twisted representations of lattice vertex algebras and
singularity theory. In particular, in the case of simple singularities
such representations lead naturally to the so called
$\mathcal{W}$-constraints. My motivation to construct
$\mathcal{W}$-constraints in singularity theory is that the same ideas
should be applicable  to the mirror models of $X$ and hence to obtain
a characterization of the Gromov--Witten invariants of $X$ in terms of
$\mathcal{W}$-constraints.

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You can check the location from
http://www.ipmu.jp/access/img/KashiwaCampusMap.png

The schedule of the seminar can be checked from
http://db.ipmu.jp/seminar/?mode=seminar_recent