Abhishek Mallick
Dartmouth College, NH·Mathematics & Statistics
Topology studies the properties of shapes that survive stretching and bending. Its central objects include knots, three-dimensional spaces, and four-dimensional spaces.
- Fields
Mathematics & Statistics
- Topics
low-dimensional topology
Floer theory
Heegaard Floer homology
- Grants
- 1 known grant, through Aug 2029
What they work on
- Establish new structural results for instanton, monopole and Heegaard Floer theories, which are invariants built from physics-inspired equations, in settings where spaces carry symmetries such as rotations or reflections.
- Develop equivariant (symmetry-aware) versions of quantum knot invariants inspired by Khovanov homology, an algebraic tool that distinguishes knots.
- Use internal and external symmetries of manifolds to construct and detect small exotic four-manifolds and exotically knotted surfaces, and to obstruct smooth isotopies of four-manifold diffeomorphisms.
- Use the topological quantum field theory structure of Floer invariants to study the three-dimensional equivariant homology cobordism group and the knot concordance group.
More about the research
Four dimensions is especially puzzling. There, "exotic" spaces can exist: spaces that look the same to coarse topology but differ in their smooth structure. This project studies spaces that have symmetries, such as rotations or reflections, called equivariant spaces. It uses those symmetries to understand how one- and three-dimensional objects constrain four-dimensional ones.
Why it matters
This is pure mathematics. Its contribution is a deeper understanding of how symmetry interacts with the structure of three- and four-dimensional spaces and knots, especially the exotic smooth structures that make four dimensions unusual. The new symmetry-aware invariants could become tools for other questions in low-dimensional topology and its links to mathematical physics.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Adam Levine
Duke University, NC·Mathematics & Statistics
Investigate knot concordance questions, including homology slice knots, ribbon concordance and piecewise-linear concordance of knots in homology 3-spheres, using Heegaard Floer homology invariants.
Matthew Stoffregen
Michigan State University, MI·Mathematics & Statistics
Prove surgery exact triangles in Seiberg-Witten Floer homotopy theory, the algebraic relations connecting invariants of three-manifolds obtained from one another by cutting and regluing along a knot.
Joshua Greene
Boston College, MA·Mathematics & Statistics
Attack peg problems, such as whether every closed curve in the plane contains the corners of a square, using symplectic methods from the geometry of classical mechanics.
The school
Dartmouth College: 16 labs on PI Pages, $418M a year on research, 104 research doctorates a year.
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