NSF Org:	DMS Division Of Mathematical Sciences
Recipient:	UNIVERSITY OF WISCONSIN SYSTEM
Initial Amendment Date:	May 14, 2003
Latest Amendment Date:	March 3, 2005
Award Number:	0306299
Award Instrument:	Continuing Grant
Program Manager:	Christopher Stark DMS  Division Of Mathematical Sciences MPS  Directorate for Mathematical and Physical Sciences
Start Date:	June 15, 2003
End Date:	December 31, 2006 (Estimated)
Total Intended Award Amount:	$242,079.00
Total Awarded Amount to Date:	$242,079.00
Funds Obligated to Date:	FY 2003 = $78,948.00 FY 2004 = $79,834.00 FY 2005 = $20,481.00
History of Investigator:	Eleny-Nicoleta Ionel (Principal Investigator) ionel@math.stanford.edu
Recipient Sponsored Research Office:	University of Wisconsin-Madison 21 N PARK ST STE 6301 MADISON WI  US  53715-1218 (608)262-3822
Sponsor Congressional District:	02
Primary Place of Performance:	University of Wisconsin-Madison 21 N PARK ST STE 6301 MADISON WI  US  53715-1218
Primary Place of Performance Congressional District:	02
Unique Entity Identifier (UEI):	LCLSJAGTNZQ7
Parent UEI:
NSF Program(s):	GEOMETRIC ANALYSIS, TOPOLOGY
Primary Program Source:	app-0103  app-0104  app-0105
Program Reference Code(s):	0000, OTHR
Program Element Code(s):	126500, 126700
Award Agency Code:	4900
Fund Agency Code:	4900
Assistance Listing Number(s):	47.049

ABSTRACT

Abstract

Award: DMS-0306299
Principal Investigator: Eleny-Nicoleta Ionel

This proposal aims to understand the structure of the
Gromov-Witten invariants of symplectic manifolds by combining
together ideas coming from different fields of research. The
first project is motivated by a conjecture made by two
physicists, Gopakumar and Vafa. The conjecture implies several
surprising restrictions on the Gromov invariants of a Calabi-Yau
3-fold. The goal is to prove the conjectured formula by adapting
some analytical techiques developed by Taubes to relate the
Seiberg-Witten and Gromov invariants in 4-dimensions. The second
project's goal is to obtain new relations in the cohomology ring
of the moduli space of complex structures on a marked Riemann
surface. Using the techniques introduced in a previous paper, the
PI found several families of interesting relations, one of which
proves a ten year old conjecture of Faber. The last project seeks
to extend the sum formula for Gromov-Witten invariants to
deformations more general then those appearing from a symplectic
sum. There already seems to be several interesting new phenomena
appearing in the general case.

The proposed work lies at the intersection of string theory and
symplectic topology. String theory developed as a potential
candidate for a unifying theory of the universe, which extends
Eistein relativity theory. It is based on the idea that
elementary particles (like electrons, photons) should be thought
not as points, but rather small vibrating loops. Working out the
details of this theory turned out to be quite delicate, and has
in turn inspired many remarkable results in mathematics. But also
fundamental results in mathematics have inspired many new
discoveries in physics. It is hoped that this project will
contribute to the increased interaction between mathematics and
high energy physics. In the same time, one of the projects will
involve graduate students.

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