Jun Li
California State University-Northridge, CA·Mathematics & Statistics
Space-time is modeled as a four-dimensional manifold, a space that looks flat up close but can bend and twist globally, yet four-dimensional spaces remain surprisingly poorly understood. Topology studies features of a space that survive continuous bending and stretching.
- Fields
Mathematics & Statistics
- Topics
symplectic topology
symplectic geometry
four-manifolds
- Grants
- 1 known grant, through Aug 2027
What they work on
- 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.
- Compare the group of smooth symmetries (Diff) with the group of symplectic symmetries (Symp) of four-manifolds, using tools such as family Seiberg-Witten invariants.
- Compare the symplectic and Kähler cones of four-manifolds, and solve isotopy problems for certain Lagrangian submanifolds and symplectic surfaces using Gromov-Witten theory and almost complex inflation.
- Investigate the dynamical properties of symplectic maps, the structure-preserving self-maps of symplectic 4-manifolds.
More about the research
Dynamics studies maps of a space to itself that preserve its structure. Symplectic structures first arose from the equations of Hamiltonian mechanics, the mathematics of classical motion, and they are a powerful tool for studying four-dimensional spaces.
Why it matters
This is pure mathematics. It aims to deepen understanding of the geometry and symmetries of four-dimensional spaces, the dimension that models space-time and the one that is least understood. These results could eventually inform related areas of geometry, topology and mathematical physics, including the Hamiltonian mechanics from which symplectic structures arise.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Alexander Zupan
University of Nebraska-Lincoln, NE·Mathematics & Statistics
Prove new cases of the smooth four-dimensional Poincaré conjecture, which asks whether sphere-like four-dimensional spaces are truly standard spheres, by connecting four-dimensional handle calculus with Dehn surgery on knots and links.
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.
Daniel Freed
Harvard University, MA·Mathematics & Statistics
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.
The school
California State University-Northridge: 4 labs on PI Pages, $27M a year on research, 11 research doctorates a year.
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