| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | May 20, 2008 |
| Latest Amendment Date: | November 19, 2008 |
| Award Number: | 0805942 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Joanna Kania-Bartoszynska
jkaniaba@nsf.gov (703)292-4881 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | June 1, 2008 |
| End Date: | May 31, 2012 (Estimated) |
| Total Intended Award Amount: | $139,327.00 |
| Total Awarded Amount to Date: | $153,242.00 |
| Funds Obligated to Date: |
FY 2009 = $13,915.00 |
| History of Investigator: |
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| Recipient Sponsored Research Office: |
426 AUDITORIUM RD RM 2 EAST LANSING MI US 48824-2600 (517)355-5040 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
426 AUDITORIUM RD RM 2 EAST LANSING MI US 48824-2600 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): | TOPOLOGY |
| Primary Program Source: |
01000910DB NSF RESEARCH & RELATED ACTIVIT |
| Program Reference Code(s): |
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| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
This project concerns applications of techniques from classical 3-dimensional topology and hyperbolic geometry to the study of polynomial link invariants and finite type invariants of links and 3-manifolds:
(a) The PI has proved results suggesting deep connections between the volume of hyperbolic links and the coefficients of their Jones polynomial. She would like to further investigate these connections and establish a bridge between quantum topology and hyperbolic geometry.
(b) Investigate the connections of link polynomial invariants to a graph theoretic polynomials (e.g. the Bollob\'as ?Riordan polynomial). She hopes that studying link invariants within this framework, can lead to new connections between link polynomials and Khovanov homology and geometric structures of link complements.
(c) Use the general machinery developed to understand how 3-manifolds change under surgery to explore the topological relations captured by the finite type knot invariants. The main tools here include sutured 3-manifold theory, combinatorial techniques from Dehn surgery and surface mapping group techniques.
The research of the project lies in the area of 3-dimensional topology the central objects of study of which are spaces called 3-manifolds. A 3-manifold is an object that locally looks like the ordinary 3-dimensional space but whose global structure can be complicated. An important part of 3-dimensional topology is also the study of knots (loops embedded in some tangled way in 3-manifolds) and their classification. One of the ways that topologists have been approaching these problems is through the use of invariants. In the recent years, ideas originated in physics, lead mathematicians to the discovery of a variety of invariants of knots and 3-manifolds. The central theme of the PI's project is to understand the properties of these invariants, using ideas from 3-dimensional topology and geometry and from physics, and investigate the extent to which they distinguish knots and 3-manifolds.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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