| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | April 12, 2011 |
| Latest Amendment Date: | April 12, 2011 |
| Award Number: | 1065718 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Christopher Stark
DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | September 1, 2011 |
| End Date: | August 31, 2014 (Estimated) |
| Total Intended Award Amount: | $23,025.00 |
| Total Awarded Amount to Date: | $23,025.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
110 INNER CAMPUS DR AUSTIN TX US 78712-1139 (512)471-6424 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
110 INNER CAMPUS DR AUSTIN TX US 78712-1139 |
| 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
This collaborative project will study the topology of smooth 4-dimensional manifolds, in connection with well-known problems in low-dimensional topology. We will focus on the construction of new smooth manifolds with symplectic structures, including Stein manifolds and symplectic fillings of certain contact 3-manifolds. Recent advances in techniques based on knot surgery and Luttinger surgery for creating exotic manifolds with small Euler characteristic will be coupled with computations of gauge-theoretic and symplectic invariants. We will make use of 4-dimensional handlebody techniques in these constructions, with an organizing principle being the search for 'corks' and 'plugs' as a technique for changing the smooth structure. Techniques of gauge theory and symplectic geometry will be used to investigate the classification of symplectic 4-manifolds and their symmetry groups.
The physical world of space and time is a 4-dimensional space whose local structure is well understood but whose large-scale (or topological) properties remain mysterious. This Focused Research Group will explore the global topology of 4-dimensional spaces, with a goal of understanding what kinds of spaces (called 4-dimensional manifolds) can exist as mathematical objects, and what the properties of such manifolds are. Of particular interest will be the problem of existence and uniqueness of symplectic structures, as well as that of determining the symmetries of a given manifold. The group will investigate how subtle changes in the smooth structure of a manifold can be achieved by gluing together pieces of different manifolds. Such changes will be detected by combining expertise from several disciplines, including powerful techniques derived from gauge theories of mathematical physics.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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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.
During the course of this project, the PI has worked on the topology of 4-dimensional manifolds which are some geometric objects modeling our universe (3-dimensions for the space and one extra dimension for the time). Specifically the PI worked on exotic smooth structures on 4-manifolds which can be thought of different ways of applying calculus techniques on 4-dimensional spaces. The main objects of the study were corks which can be thought of atomic pieces in the 4-dimensional spaces which are responsible from exoticness. It turns out that removing a cork from a 4-manifold and regluing it in another way sometimes result in an exotic smooth structure on a 4-manifold. This operation is called cork twisting. The PI and his CO-PI S. Akbulut gave sufficient conditions where cork twisting yield exotic smooth structures. These results were published in peer reviewed journals.
The PI was also actively involved in the organization of three workshops on topology of 4-manifolds which were aiming to bring together the experts in this area and the graduate students. Thanks to these workshops the graduate students were exposed to current topics in 4-manifold topology. Many advanced graduate students were given oppurtunuties to present their work in these workshops which allowed them to establish their own network in their field.
Last Modified: 04/02/2015
Modified by: Cagri Karakurt
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