Daniel Freed
Harvard University, MA·Mathematics & Statistics
Some geometric shapes carry an unusually rich mix of structure that links distant parts of mathematics and theoretical physics. Symplectic resolutions are one such class.
- Fields
Mathematics & Statistics
Physics
- Topics
symplectic geometry
Fukaya categories
symplectic resolutions
- Grants
- 1 known grant, through Jul 2027
What they work on
- Investigate expected relationships between Fukaya categories of symplectic resolutions, which are algebraic structures built by counting pseudoholomorphic curves, and structures from representation theory, using microlocal sheaf theory.
- Apply recent microlocal sheaf techniques developed by Ganatra, Pardon and Shende to compute and describe Fukaya categories of complex symplectic manifolds.
- Develop the foundations of microlocal sheaf theory, especially the theory of perverse microlocal sheaves, as tools for studying symplectic resolutions.
More about the research
Because so many fields meet in these objects, new results about them can have wide impact. This project studies algebraic structures attached to them, called Fukaya categories, and tests long-standing expectations that these categories connect to representation theory, the study of symmetry through linear algebra.
Why it matters
This is pure mathematics. It would deepen understanding of how symplectic geometry relates to representation theory, and strengthen the foundations of microlocal sheaf theory. Because symplectic resolutions interest both mathematicians and theoretical physicists, the results could eventually inform work across several areas of mathematics and physics.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Christine Berkesch
University of Minnesota, MN·Mathematics & Statistics
Develop algebraic versions of symplectic geometry techniques such as "stop manipulation" to uncover structure in multigraded polynomial rings, rings whose variables carry several degrees at once, that parallels classical results for standard graded rings.
Mohammed Abouzaid
Stanford University, CA·Mathematics & Statistics
Train problem-specific neural networks and reinforcement-learning agents to search for constructions in the symplectic ball-packing problem, which asks how efficiently balls fit into a region while preserving signed surface areas.
Jun Li
California State University-Northridge, CA·Mathematics & Statistics
Explore whether results on braids and Dehn twists (basic twisting symmetries) along Lagrangian spheres extend to general symplectic 4-manifolds, using surgery on Lefschetz fibrations.
The school
Harvard University: 42 labs on PI Pages, $1.5B a year on research, 712 research doctorates a year.
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