NSF Org:	DMS Division Of Mathematical Sciences
Recipient:	UNIVERSITY OF CALIFORNIA, LOS ANGELES
Initial Amendment Date:	June 24, 2016
Latest Amendment Date:	June 23, 2018
Award Number:	1643401
Award Instrument:	Continuing 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, 2016
End Date:	July 31, 2020 (Estimated)
Total Intended Award Amount:	$308,667.00
Total Awarded Amount to Date:	$308,667.00
Funds Obligated to Date:	FY 2015 = $38,666.00 FY 2016 = $90,001.00 FY 2017 = $89,999.00 FY 2018 = $90,000.00
History of Investigator:	Sucharit Sarkar (Principal Investigator) sucharit@math.ucla.edu
Recipient Sponsored Research Office:	University of California-Los Angeles 10889 WILSHIRE BLVD STE 700 LOS ANGELES CA  US  90024-4200 (310)794-0102
Sponsor Congressional District:	36
Primary Place of Performance:	University of California-Los Angeles CA  US  90095-2000
Primary Place of Performance Congressional District:	36
Unique Entity Identifier (UEI):	RN64EPNH8JC6
Parent UEI:
NSF Program(s):	TOPOLOGY, Division Co-Funding: CAREER
Primary Program Source:	01001516DB NSF RESEARCH & RELATED ACTIVIT 01001617DB NSF RESEARCH & RELATED ACTIVIT 01001718DB NSF RESEARCH & RELATED ACTIVIT 01001819DB NSF RESEARCH & RELATED ACTIVIT
Program Reference Code(s):	1045
Program Element Code(s):	126700, 804800
Award Agency Code:	4900
Fund Agency Code:	4900
Assistance Listing Number(s):	47.049

ABSTRACT

The project will concentrate on two modern homological knot invariants: Khovanov homology and knot Floer homology. Both invariants associate chain complexes to knots whose chain homotopy types (and consequently, homology groups) are knot invariants. This project has two major research goals. The first goal is to extend various aspects of these homological invariants to stable homotopy types, i.e., construct new knot invariant topological spaces (well-defined up to stable homotopy equivalences) whose homology groups are the existing invariants. This will produce higher structures on the homological invariants which will be useful for studying certain geometric properties of knots, such as their four-ball genus. The second goal of this project is to study the relationship between the homological invariants. The project seeks to find new spectral sequences, and combinatorial reformulations of the existing spectral sequences, from the Khovanov homology invariants to the knot Floer homology invariants. This will lead to a better understanding of why these two invariants coming from very different origins share so many similarities.

Topology is the branch of mathematics that studies shapes of spaces; and low dimensional topology concentrates on spaces up to dimension four. Knot theory is an important sub-field of low dimensional topology where one studies one-dimensional objects inside three-dimensional spaces, for example, knotted pieces of strings inside the (three-dimensional Euclidean) space that we live in. In addition to being an extremely valuable tool in low dimensional topology, knot theory has also proven to be useful in real world applications: from analysing knotting in DNA to studying mixing in liquids, and from estimating energies of orbits inside a magnetic field to creating new data encryption schemes. A fundamental problem in knot theory is the knot isotopy problem: to determine if a given knot can transform into another one without tearing or crossing itself (such a transformation is called a knot isotopy). Knot invariants are mathematical objects, such as numbers or groups, that one associates to knots, and which remain unchanged during such a knot isotopy. Therefore, knot invariants are used extensively in the knot isotopy problem: if one finds some knot invariant that takes different values on the two given knots, then one concludes that the two knots are not isotopic. The current project is based on knot theory, and it seeks to study properties of certain previously known knot invariants, and to extend them to construct new knot invariants. The project will lead to dispersion of mathematical knowledge, particularly in the area of low-dimensional topology, via a variety of means. This project will fund undergraduate students for summer research, week-long workshops on low-dimensional topology, and a wiki-based website on knot theory.

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.

The project had two goals: finding connections between different theories low-dimensional topology and extending these theories to stronger theories. Topology is the study of shapes of spaces, and low-dimensional topology is the study of spaces up to dimension 4. A plethora of recent theories have been constructed in low-dimensional topology which associate chain complexes to low-dimensional objects; the homologies of these chain complexes are invariants of the underlying objects (that is, they only depend on the "shape" of the topological object, not on any extra data), and these homology groups provide a lot of valuable information about the topological objects being studied.

This project primarily studied two specific families of homology theories: Khovanov homology and Floer homology. This project constructed a new equivariant Floer theory in the presence of group actions, that is, when there is some underlying symmetry. Regarding Khovanov homology, the project produced spatial refinements (which carry more information) of the Khovanov homologies of tangles (open-ended strings inside 3-dimensional spaces), and of odd Khovanov homology (which is a variant of the usual even Khovanov homology). The project also constructed lower bounds of certain complexities of cobordisms (surfaces inside 4-dimensional objects) from Khovanov homology. The project also studied cobordisms inside the 4-dimensional complex projective plane and proved an adjunction formula in Khovanov homology, which is similar to existing formulas for Floer theories. 

In January 2018 and January 2020, two 3-day workshops were organized at UCLA from this grant, with around 40 participants and 15 speakers each. 

 

Last Modified: 11/20/2020
Modified by: Sucharit Sarkar

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