| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
|
| Initial Amendment Date: | May 19, 2011 |
| Latest Amendment Date: | May 19, 2011 |
| Award Number: | 1103976 |
| 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: | July 1, 2011 |
| End Date: | December 31, 2015 (Estimated) |
| Total Intended Award Amount: | $214,701.00 |
| Total Awarded Amount to Date: | $214,701.00 |
| Funds Obligated to Date: |
|
| History of Investigator: |
|
| Recipient Sponsored Research Office: |
1200 E CALIFORNIA BLVD PASADENA CA US 91125-0001 (626)395-6219 |
| Sponsor Congressional District: |
|
| Primary Place of Performance: |
1200 E CALIFORNIA BLVD PASADENA CA US 91125-0001 |
| Primary Place of
Performance Congressional District: |
|
| Unique Entity Identifier (UEI): |
|
| Parent UEI: |
|
| NSF Program(s): | TOPOLOGY |
| Primary Program Source: |
|
| Program Reference Code(s): | |
| Program Element Code(s): |
|
| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
This project deals with invariants in low-dimensional topology which come from gauge theory and symplectic geometry, especially Heegaard Floer homology. The focus will be the applications of Heegaard Floer homology to low-dimensional topology, and the connection between Heegaard Floer homology and other aspects of low-dimensional topology. One problem we plan to study is unbalanced sutured manifold decompositions for Heegaard Floer homology. This is related to characterizing incompressible surfaces using Heegaard Floer homology. Another problem we will study is the botany problem in Heegaard Floer homology, namely, to what extent one may determine a manifold using its Heegaard Floer homology. These problems are related to questions about Dehn surgeries and Khovanov homology. We will also address the applications of Floer homology to 4-dimensional topology, for example, the topology of knot surgeries on the K3 surface.
In the microscopic world, macromolecules are often visualized as knots and links in the three-dimensional space. The knot invariants studied in this project thus provide important tools in analyzing the structures of macromolecules: Some questions our methods can study are, how firmly the macromolecules are interlocked, how to detect their chirality, and how to change their topological structure. These features are extremely significant in nanotechnology and pharmacology.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
Note:
When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external
site maintained by the publisher. Some full text articles may not yet be available without a
charge during the embargo (administrative interval).
Some links on this page may take you to non-federal websites. Their policies may differ from
this site.
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
Low-dimensional topology is a branch of mathematics that studies the topology of three- and four-dimensional manifolds. This project deals with invariants in low-dimensional topology
which come from gauge theory and symplectic geometry, especially Heegaard Floer homology. The focus
is the applications of Heegaard Floer homology to low-dimensional topology, and the connection between Heegaard Floer homology and other aspects of low-dimensional topology.
The PI has written 13 papers during the project period. All papers have been posted to the arXiv, and have been submitted to peer-reviewed journals.
Broader Impacts
Conferences and seminars
In 2013, the PI organized the N+7th Southern California Topology Colloquium/Conference. This was a one-day conference with four speakers: Ian Agol, Francis Bonahon, Ciprian Manolescu, and Lenhard Ng. About 50--60 people from
the southern California area attended this conference.
The PI co-organize the Geometry and Topology Seminar and the Physics and Geometry Seminar at Caltech, and also co-organize the Joint LA Topology Seminar with colleagues from UCLA and USC. Most speakers the PI invited are graduate students or postdoctoral scholars. Many speakers are women.
The PI has given talks in various conferences and seminars, including talks given in Australia, Japan, China (including Hong Kong and Taipei), and France.
Mentoring students and postdoctors
Two graduate students of the PI have graduated: Liling Gu (2014) and Nakul Dawra (2015). Gu wrote a thesis on the surgery realization problem of type-T,O,I spherical manifolds. Dawra wrote a paper on L-space links. Dawra was awarded a Troesh Family Fellowship during the last year of his graduate study. The PI is currently supervising three graduate students: Caili Shen, Siqi He, and Gahye Jeong. Siqi He has already written a paper on the Kapustin--Witten equation.
The PI has supervised three undergraduate students for summer research: Eileen Li in 2012, Tim Holland and Laura Shou in 2013. The PI has since written a paper with Li. Holland and Shou have written a program to compute the correction terms for some 3-manifolds.
The PI has supervised two postdoctoral scholars: Zhongtao Wu (2010--2013) and Faramarz Vafaee (2014--now). Wu is currently an assistant professor at the Chinese University of Hong Kong.
The PI organizes a reading seminar for the graduate students and postdoctors. The participants need to present papers of other people in the seminar. In this way, they can not only learn different topics related to Floer homology, but also
improve their presentation techniques.
The PI taught two graduate topics courses during the project period: one on symplectic geometry and one on Seiberg--Witten theory.
Educational outreach
During 2012--2014, the PI served on the graduate admission committee of the Department of Mathematics at Caltech.
In 2014, the PI served on the award committee (Los Angeles branch) of the Chinese Government Award for Outstanding Self-Financed Students Abroad.
In 2015, the PI gave a talk on knot theory at the Caltech Harvey Mudd Math Competition. More than 300 people were in the audience, including the high school students participating the competition and their teachers/parents.
Last Modified: 02/13/2016
Modified by: Yi Ni
Please report errors in award information by writing to: awardsearch@nsf.gov.
