| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | July 19, 2019 |
| Latest Amendment Date: | July 19, 2019 |
| Award Number: | 1907654 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Swatee Naik
snaik@nsf.gov (703)292-4876 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | August 1, 2019 |
| End Date: | July 31, 2024 (Estimated) |
| Total Intended Award Amount: | $295,645.00 |
| Total Awarded Amount to Date: | $295,645.00 |
| 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, EPSCoR Co-Funding |
| 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
This project is jointly funded by the Topology program and the Established Program to Stimulate Competitive Research (EPSCoR). It centers on questions lying at the intersection of two fields of mathematics called contact geometry and low-dimensional topology. Contact structure are geometric objects on certain spaces known as manifolds that arise naturally in physics through differential equations, optics, and dynamics. Pioneering work of Eliashberg first showed that contact structures play an essential role in determining geometric and topological properties of three- and four-dimensional manifolds. Contact geometry has since entered a renaissance after featuring prominently in the resolution of several long-standing problems in low-dimensional topology. It is in this context that this project broadly seeks to better our understanding of how characteristics of contact structures determine either geometric properties of the spaces they live on, or influence powerful invariants used in their study. The project will have immediate impact in several fields, such as: low-dimensional topology, symplectic and contact topology, dynamics, and mathematical physics. The PI will also devote time to helping mentor graduate students and postdoctoral scholars as they transition to being independent researchers.
Recall that contact structures fall into one of two categories: tight or overtwisted. Understanding which three-manifolds support tight contact structures and the number of tight contact structures supported by a given three-manifold are the paramount goals of modern contact geometry. Accordingly, a primary goal of this project is to develop effective and computable invariants capable of determining tightness and distinguishing contact structures. Since the inception of these invariants, strong evidence has steadily built suggesting deep connections between contact structures on three-manifolds and their associated Floer-theoretic invariants. For instance, each of the various Floer homologies support an invariants which are capable of detecting tightness and distinguishing contact structures. An important goal of this project is to develop and explore refinements of these invariants which are simultaneously more effective in detecting tightness, and are more easily computable. In a parallel direction, according to work on the PI and others, much of the formal algebraic structure underpinning Floer theory appears to mirror natural geometric characteristics and constructions involving contact structures. In turn, another key goal of this project is to clarify connections and correspondences between the algebraic structure of Floer-theoretic invariants and natural contact geometric phenomena and constructions.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
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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.
Intellectual Merit: With Daemi, Lidman and Wong, the PI studied 4-dimensional cobordisms without 3-handles (which we call Ribbon Cobordisms). Such spaces arise naturally in contact and symplectic geometry and also knot theory. In knot theory, the Slice-Ribbon Conjecture can be rephrased in terms of concordances without 2-handles, the exteriors of which have handle decompositions without 3-handles according to Gordon. We provide the first systematic study Ribbon Cobordisms using tools from many corners of low-dimensional topology. We show that they interact nicely with Thurston geometries, character varieties, and instanton and Heegaard Floer homologies.
With Sabloff and Wong, the PI showed that any finite collection of null-homologous Legendrian links in a tight contact 3-manifold with a common rotation number has an upper bound with respect to the Lagrangian cobordism relation. This construction allows us to define a notion of minimal Lagrangian genus between any two Legendrian links with a common rotation number. With Sabloff, Wong and Wu, the PI studies the metric monoid formed by the set of Lagrangian zigzag concordance classes.
With Wong, the PI used bordered–sutured Floer theory to show that the complement of a tangle in an arbitrary 3-dimensional space satisfies an unoriented skein exact triangle, generalizing a theorem by Manolescu for links in the 3-sphere. In addition to a theoretical proof, we provide a combinatorial description of all maps involved and explicitly compute them.
With Baldridge and McCarty, the PI showed how to lift Lagrangian immersions in CP^{n-1} to produce Lagrangian cones in C^n, and use this process to produce several families of examples of Lagrangian cones and special Lagrangian cones. This is motivated by the SYZ Conjecture which focuses attention on Calabi-Yau 3-folds which can be viewed as a fibration by 3-tori with some singular fibers.
Broader impacts: During the years of support from this grant, the PI gave numerous invited in-persona and virtual talks on the above work. The PI has co-organized formal and informal seminars at LSU, several of which center on graduate research and training. In Summer 2019, 2020, 2021, 2022, 2023 and 2024, this award provided support for several of the PI's graduate students. In total seven different graduate students received summer support to advance their research. The PI has supervised the completed graduate studies of seven students. Five of these students received their PhDs during the years of this award. The PI currently oversees the graduate work of three graduate students. Each are pursuing research on problems consistent with the goals of this project. At the undergraduate level, the PI mentored roughly two dozen undergraduates as they pursued research consistent with the goals of this project.
Last Modified: 12/28/2024
Modified by: David S Vela-Vick
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