## Project Details

### Description

Overview

This project is devoted to the study of interplays between microlocal sheaf theory and symplectic geometry, and their various applications in representation theory. The PI will use microlocal sheaf theory to quantize Lagrangian submanifolds in an exact symplectic manifold, and will give the definition of a microlocal sheaf category of a symplectic manifold, which is expected to be equivalent to the Fukaya category but is of purely topological nature. The PI will develop a parallel story in the complex setting of holomorphic Lagrangians in an exact holomorphic symplectic manifold, which exhibits new and richer structures and which will have important applications in geometric representation theory. The PI will define a microlocal version of “holomorphic Fukaya category” and will study extensively for symplectic resolutions, a class of holomorphic symplectic manifolds in the center of modern representation theory. The PI will show that the microlocal version of “holomorphic Fukaya category” is equivalent to the category O defined in representation theory and will use techniques from microlocal sheaf theory to understand the intriguing phenomena of symplectic duality between category Os. Other applications to representation theory will involve calculations in the Hecke category using sheaf quantizations of the braid group action as symplectomorphisms and a proposal to realize the nonabelian Hodge theory using microlocal perverse sheaves.

Intellectual Merit

This proposal is a natural continuation of the PI’s current work on quantizations of real and complex Lagrangian submanifolds in cotangent bundles using microlocal sheaves and their applications in geometric representation theory. The proposed study will generalize the quantization procedure from cotangent bundles to a general exact symplectic manifold, which will lead to an important categorical invariant of a symplectic manifold expected to be equivalent to the Fukaya category. The pure topological nature of the invariant will allow combinatorial computations and will have remarkable applications in mirror symmetry. The generality of the quantizations over ring spectra will give rise to new interactions between symplectic geometry and stable homotopy theory. The quantizations in the holomorphic setting will open up new approaches to central topics in the crossroads of representation theory and mathematical physics, including symplectic duality, mixed Hodge modules and the nonabelian Hodge theory.

Broader Impacts

The proposed project lies in the intersection of several active branches of mathematics and physics. The PI will continue to disseminate her research results through conferences and other academic activities, with a goal of initiating collaborations with mathematicians specialized in different fields and proceeding on the discoveries of fascinating interdisciplinary connections. The PI has frequent communications with graduate students in her institute on research results and ideas. She has organized learning seminars on topics related to the proposed project. Some of the problems in the project are in its experimental stage and calculations for some specific cases can lead to research questions for undergraduate students. The PI will mentor some students to solve such questions.

This project is devoted to the study of interplays between microlocal sheaf theory and symplectic geometry, and their various applications in representation theory. The PI will use microlocal sheaf theory to quantize Lagrangian submanifolds in an exact symplectic manifold, and will give the definition of a microlocal sheaf category of a symplectic manifold, which is expected to be equivalent to the Fukaya category but is of purely topological nature. The PI will develop a parallel story in the complex setting of holomorphic Lagrangians in an exact holomorphic symplectic manifold, which exhibits new and richer structures and which will have important applications in geometric representation theory. The PI will define a microlocal version of “holomorphic Fukaya category” and will study extensively for symplectic resolutions, a class of holomorphic symplectic manifolds in the center of modern representation theory. The PI will show that the microlocal version of “holomorphic Fukaya category” is equivalent to the category O defined in representation theory and will use techniques from microlocal sheaf theory to understand the intriguing phenomena of symplectic duality between category Os. Other applications to representation theory will involve calculations in the Hecke category using sheaf quantizations of the braid group action as symplectomorphisms and a proposal to realize the nonabelian Hodge theory using microlocal perverse sheaves.

Intellectual Merit

This proposal is a natural continuation of the PI’s current work on quantizations of real and complex Lagrangian submanifolds in cotangent bundles using microlocal sheaves and their applications in geometric representation theory. The proposed study will generalize the quantization procedure from cotangent bundles to a general exact symplectic manifold, which will lead to an important categorical invariant of a symplectic manifold expected to be equivalent to the Fukaya category. The pure topological nature of the invariant will allow combinatorial computations and will have remarkable applications in mirror symmetry. The generality of the quantizations over ring spectra will give rise to new interactions between symplectic geometry and stable homotopy theory. The quantizations in the holomorphic setting will open up new approaches to central topics in the crossroads of representation theory and mathematical physics, including symplectic duality, mixed Hodge modules and the nonabelian Hodge theory.

Broader Impacts

The proposed project lies in the intersection of several active branches of mathematics and physics. The PI will continue to disseminate her research results through conferences and other academic activities, with a goal of initiating collaborations with mathematicians specialized in different fields and proceeding on the discoveries of fascinating interdisciplinary connections. The PI has frequent communications with graduate students in her institute on research results and ideas. She has organized learning seminars on topics related to the proposed project. Some of the problems in the project are in its experimental stage and calculations for some specific cases can lead to research questions for undergraduate students. The PI will mentor some students to solve such questions.

Status | Finished |
---|---|

Effective start/end date | 7/1/17 → 6/30/18 |

### Funding

- National Science Foundation (DMS-1710481)

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