| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | August 15, 2012 |
| Latest Amendment Date: | August 15, 2012 |
| Award Number: | 1249708 |
| 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, 2012 |
| End Date: | May 31, 2016 (Estimated) |
| Total Intended Award Amount: | $131,615.00 |
| Total Awarded Amount to Date: | $131,615.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
202 HIMES HALL BATON ROUGE LA US 70803-0001 (225)578-2760 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
LA US 70803-2701 |
| 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: |
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| 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
The principle investigator endeavors to deepen our understanding of geometric objects on 3-manifolds called contact structures. In recent years, contact structures have moved to the forefront of mathematical interest after featuring prominently in the resolution of several long-standing conjectures. The first goal of this project is to probe connections linking contact structures and Heegaard Floer invariants. Since it's introduction roughly a decade ago, Heegaard Floer theory has revolutionized the study of knots, 3-manifolds and smooth 4-manifolds. This project seeks to better understand how geometric properties of contact structures imprint themselves in the algebraic formalism of Heegaard Floer invariants. The project's second goal is to study connections between geometric characteristics of contact structures and topological properties of the open book decompositions that support them. Specifically, the principle investigator aims to develop obstructions to contact structures having support genus one and to find lower bounds for the binding number. Finally, the project seeks to broaden our understanding of Legendrian and transverse knot theory. To accomplish this, the principle investigator aims to develop new Legendrian and transverse invariants and to apply these and other known invariants to classify Legendrian and transverse representatives in a broad class of knot types.
The principle investigator seeks to broaden our understanding of 3 and 4-dimensional spaces by studying geometric objects called contact structures. Contact structures first appeared in physics through the work of Hamilton, Huygens and Jacobi on geometric optics. They provide a natural language for studying optics, classical mechanics and thermodynamics, and have applications in many subfields of physics and mathematics. They are a tool one can use to probe 3 and 4-dimensional spaces to better understand their shape and geometric structure. The development of techniques for studying these spaces ultimately helps to informs us about the topological and geometric characteristics of our own universe.
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PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
This project aimed to address three primary goals. First, it sought to better understand connections between geometric structures on 3-dimensional spaces known as "contact structures" and invariants of those same spaces known as "Heegaard Floer invariants". The second goal of the project, was to explore correspondences between geometric properties of contact structures and topological properties of so-called "open book decompositions". The final main goal of the project was to expand our understanding of Legendrian and transverse knot theory through the development and application of new and existing invariants.
The project made inroads toward each of the stated goals. In line with the first goal, the PI, in joint work with John Etnyre (Georgia Tech) and Rumen Zarev showed concretely how much of the formal algebraic structure present in Heegaard Floer theory can be obtained via natural contact-geometric constructions. These same constructions can be applied in more general contexts to obtain similar results for other Floer-type invariants. The PI made further inroads toward the first goal in joint work with John Baldwin (Boston College) by developing two refinements of an invariant of contact structures present in Heegaard Floer theory. In addition to applications toward detecting an important property of contact structures known as tightness, these refinements have significant topological applications.
Consistent with the second goal, the project made significant made progress toward bettering our understanding of connections and correspondences between contact structures and open book decompositions. The PI, in joint with Kenneth Baker (University of Miami), Jeremy Van HornMorris (University of Arkansas) and Vera Vertesi (Universite de Strasbourg) developed a general program for obtaining invariants of contact structures and open books coming from branched covers and generalized (transverse) braid theory. Over the course of the grant, substantial progress was made toward fleshing out the details of this program which, strikingly, is sufficient to give a complete and rigorous proof of the Giroux correspondence itself.
Finally, in line with the third stated goal of the project, the PI, in joint work with numerous other researchers developed new invariants of Legendrian and transverse knots and showed how they can be applied in various contexts. Notably, with John Etnyre and Rumen Zarev, the PI gave an alternative, geometric characterization of the Legendrian and transverse invariants previously defined in the context of Heegaard Floer theory. In this same body of work, invariants which track the "twisting" of contact structures in the complements of Legendrian and transverse knots were defined and some elementary applications explored. In joint work with Tye Lidman (North Carolina State University) and Sucharit Sarkar (University of California, LA) the PI showed how to obtain invariants of transverse knots by defining Heegaard Floer type invariants for infinite cyclic covers of transverse knots.
Last Modified: 09/14/2016
Modified by: David S Vela-Vick
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