### Description

OVERVIEW

The project will advance mathematics by elucidating a rich web of connections between sheaves, cluster varieties knots, and symplectic and algebraic geometry. It will also link these structures to new results in mirror symmetry and string physics.

INTELLECTUAL MERIT

Principal Investigator Zaslow, in a number of works, spearheaded the use of sheaf theory in symplectic geometry and mirror symmetry. His recent work on invariants of Legendrian knots and Lagrangian surfaces found new connections to cluster theory.

In the present work, Zaslow will extend these linkages to the dimension of great interest to the phyiscs of M5-branes in string theory: three-dimensional Lagrangians and two-dimensional Legendrian surfaces. New and interesting phenomena occur at this critical dimension. Zaslow will:

1. Find superpotentials encoding counts of holomorphic disks bounding Lagrangian fillings of Legendrian surfaces. Establish Ooguri-Vafa integrality in all framings.

2. Explain how such counts are determined by (Seiberg-like) mutations from cluster theory: at each genus, Donaldson-Thomas transformations relate them to simple building blocks.

3. Establish ``framing duality.'' That is, prove that the genus-g moduli space computes, via different framings, DT invariants of *all* symmetric quivers with g nodes.

4. Extend these results in C^3 to the non-exact setting of the resolved conifold by incorporating Q-deformations into the sheaf theory via twisted sheaves.

5. Apply this machinery to knot conormals to explain large-N duality using sheaf theory.

6. Prove the Lagrangians found by Zaslow-Treumann have special-Lagrangian embeddings.

7. Explain the Goncharov-Kenyon-Beauville integrable system via mirror symmetry.

BROADER IMPACTS

The broader impacts of this proposal lie in the professional development of young mathematicians and in a variety of outreach efforts undertaken by PI Zaslow.

Professional Development:

1. Zaslow will continue to supervise graduate students and postdocs.

2. Zaslow will travel to institutions and interact widely with colleagues from various fields.

Specifically, to Berkeley as a Miller Visiting Professor and to the Max Planck Institute for Mathematics-Bonn in 2017.

3. Zaslow will continue to disseminate his research widely to young scholars,

both in written form and at national and international conferences (String Math, HMS Conferences,...)

4. Zaslow will continue to support and supervise the Geometry/Physics seminar at NU.

Outreach:

1. Zaslow created and runs the Evanston Math Circle (EMC), which brings mathematics enrichment to interested middle- and high-school students in Evanston, Illinois. EMC is in its fifth year.

2. Zaslow, working with the Office of the Dean of the Weinberg College of Arts and Science (Northwestern), created the curriculum of the Northwestern Bridge Program. The program serves to prepare first-generation and low-income students interested in STEM fields and Economics, who are about to matriculate at Northwestern University. Zaslow will continue his deep involvement, and continue to teach the course he created expressly for this purpose, Quantitative Reasoning.

3. Zaslow will continue to be involved with and support the GROW conference at NU.

The project will advance mathematics by elucidating a rich web of connections between sheaves, cluster varieties knots, and symplectic and algebraic geometry. It will also link these structures to new results in mirror symmetry and string physics.

INTELLECTUAL MERIT

Principal Investigator Zaslow, in a number of works, spearheaded the use of sheaf theory in symplectic geometry and mirror symmetry. His recent work on invariants of Legendrian knots and Lagrangian surfaces found new connections to cluster theory.

In the present work, Zaslow will extend these linkages to the dimension of great interest to the phyiscs of M5-branes in string theory: three-dimensional Lagrangians and two-dimensional Legendrian surfaces. New and interesting phenomena occur at this critical dimension. Zaslow will:

1. Find superpotentials encoding counts of holomorphic disks bounding Lagrangian fillings of Legendrian surfaces. Establish Ooguri-Vafa integrality in all framings.

2. Explain how such counts are determined by (Seiberg-like) mutations from cluster theory: at each genus, Donaldson-Thomas transformations relate them to simple building blocks.

3. Establish ``framing duality.'' That is, prove that the genus-g moduli space computes, via different framings, DT invariants of *all* symmetric quivers with g nodes.

4. Extend these results in C^3 to the non-exact setting of the resolved conifold by incorporating Q-deformations into the sheaf theory via twisted sheaves.

5. Apply this machinery to knot conormals to explain large-N duality using sheaf theory.

6. Prove the Lagrangians found by Zaslow-Treumann have special-Lagrangian embeddings.

7. Explain the Goncharov-Kenyon-Beauville integrable system via mirror symmetry.

BROADER IMPACTS

The broader impacts of this proposal lie in the professional development of young mathematicians and in a variety of outreach efforts undertaken by PI Zaslow.

Professional Development:

1. Zaslow will continue to supervise graduate students and postdocs.

2. Zaslow will travel to institutions and interact widely with colleagues from various fields.

Specifically, to Berkeley as a Miller Visiting Professor and to the Max Planck Institute for Mathematics-Bonn in 2017.

3. Zaslow will continue to disseminate his research widely to young scholars,

both in written form and at national and international conferences (String Math, HMS Conferences,...)

4. Zaslow will continue to support and supervise the Geometry/Physics seminar at NU.

Outreach:

1. Zaslow created and runs the Evanston Math Circle (EMC), which brings mathematics enrichment to interested middle- and high-school students in Evanston, Illinois. EMC is in its fifth year.

2. Zaslow, working with the Office of the Dean of the Weinberg College of Arts and Science (Northwestern), created the curriculum of the Northwestern Bridge Program. The program serves to prepare first-generation and low-income students interested in STEM fields and Economics, who are about to matriculate at Northwestern University. Zaslow will continue his deep involvement, and continue to teach the course he created expressly for this purpose, Quantitative Reasoning.

3. Zaslow will continue to be involved with and support the GROW conference at NU.

Status | Active |
---|---|

Effective start/end date | 9/15/17 → 8/31/20 |

### Funding

- National Science Foundation (DMS-1708503)

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