NSF Org:	DMS Division Of Mathematical Sciences
Recipient:	TRUSTEES OF THE UNIVERSITY OF PENNSYLVANIA, THE
Initial Amendment Date:	July 2, 2003
Latest Amendment Date:	July 2, 2003
Award Number:	0239600
Award Instrument:	Standard Grant
Program Manager:	Christopher Stark DMS  Division Of Mathematical Sciences MPS  Directorate for Mathematical and Physical Sciences
Start Date:	July 1, 2003
End Date:	January 31, 2007 (Estimated)
Total Intended Award Amount:	$402,023.00
Total Awarded Amount to Date:	$402,023.00
Funds Obligated to Date:	FY 2003 = $192,151.00
History of Investigator:	John Etnyre (Principal Investigator) etnyre@math.gatech.edu
Recipient Sponsored Research Office:	University of Pennsylvania 3451 WALNUT ST STE 440A PHILADELPHIA PA  US  19104-6205 (215)898-7293
Sponsor Congressional District:	03
Primary Place of Performance:	University of Pennsylvania 3451 WALNUT ST STE 440A PHILADELPHIA PA  US  19104-6205
Primary Place of Performance Congressional District:	03
Unique Entity Identifier (UEI):	GM1XX56LEP58
Parent UEI:	GM1XX56LEP58
NSF Program(s):	GEOMETRIC ANALYSIS, TOPOLOGY
Primary Program Source:	app-0103
Program Reference Code(s):	0000, 1045, 1187, OTHR
Program Element Code(s):	126500, 126700
Award Agency Code:	4900
Fund Agency Code:	4900
Assistance Listing Number(s):	47.049

ABSTRACT

Proposal DMS-0239600
PI: John Etnyre, University of Pennsylvania
Title: CAREER: Knot Theory and Dynamics in Contact Geometry

ABSTRACT



The focus of this proposal is to study the relation between topology
and contact geometry in all (odd) dimensions and to apply contact
geometric techniques to questions in hydrodynamics. Many of the
connections between topology and contact geometry are mediated by
Legendrian knots (these are knots that are tangent to a contact
structure), thus the first main theme of the proposed research is
Legendrian knots. As part of this proposal the general structure
of Legendrian knots will be studied. The expected outcome will be
various classification results for certainLegendrian knots and contact
structures; and, moreover, a betterunderstanding of Legendrian surgery
(an important surgery construction of contact structures). Legendrian
knots in higher dimensions will also be studied using contact homology.
There is very little known about contactstructures, or Legendrian knots,
in dimensions above three. By investigating Legendrian knots in these
dimensions the nature of contactstructures should be illuminated, just
as the corresponding study revealed much about three dimensional contact
structures. The final part of the proposed research centers on the
connection between contact structures and hydrodynamics discovered a few
years ago by the Principal Investigator and R. Ghrist. Here work with
Ghrist will continue with the aim of understanding when, and what type of,
closed flow lines occur in fluid flows. We shall also study hydrodynamic
instability from the contact topological perspective. This naturally leads
into the study of energy minimization for fluid flows and relations between
contact and Riemannian geometry.

Contact structures are very natural objects, born over two centuries
ago,in the study of geometric optics and partial differential equations.
Through the centuries contactstructures have touched on many diverse areas
of mathematics and physics,including classical mechanics and thermodynamics.
In everyday life oneencounters contact geometry when ice skating, parallel
parking a car,using a refrigerator, or simply watching the beautiful play
of light ina glass of water. Many great mathematicians have devoted a lot of
their work to this subject but only in the last decade or two has it moved
into the foreground of mathematics. This renaissance is due to the recent
remarkable breakthroughs in contact topology, resulting in a rich and
beautiful theory with many applications. The most remarkable feature of
all this recent work is the intimate connections between contact
structures and topology in dimension three. Thus by studying this abstract
notion of a contact structure one can learn many subtle things about the
universe in which we live. For example, the study of contact geometry
has recently lead to some unexpected advances in our understanding of
the flow of idealized fluids. The Principal Investigator will explore
connections between contact structures and topology in all (odd)
dimensions, continue his study of idealized fluid flows (hydrodynamics)
via contact geometry and analyze intriguing new conjectures concerning
string theory and contact geometry. The Principal Investigator will
also engage in several educational endeavors, including the support and
encouragement of graduates students and the creation of introductory
and survey materials to bring the rapidly developing field of contact
geometry to a wider audience.

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