Adam Levine
Duke University, NC·Mathematics & Statistics
Three- and four-dimensional spaces hold surprises that higher dimensions do not. Many shape questions are harder in four dimensions than in five or more.
- Fields
Mathematics & Statistics
- Topics
low-dimensional topology
knot theory
4-manifolds
- Grants
- 1 known grant, through Dec 2026
What they work on
- 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.
- Construct new exotic 4-manifolds and develop invariants that distinguish knotted 2-spheres in four-dimensional space.
- Resolve technical issues in how Heegaard Floer homology behaves under surgery, the cut-and-reglue operation on 3- and 4-dimensional spaces.
- Construct a spectral sequence linking Khovanov homology to knot Floer homology in characteristic 2, clarifying how these two knot invariants relate.
More about the research
Low-dimensional topology studies the overall shapes of these spaces and of the knots and surfaces inside them. These questions connect to ideas ranging from the shape of the universe to the knotting of DNA. Answering them needs tools beyond classical algebraic topology. This project uses two families of invariants developed in the early 2000s, Heegaard Floer homology and Khovanov homology. Invariants are algebraic objects that tell shapes apart.
Why it matters
This is pure mathematics. It aims to deepen understanding of why four-dimensional space behaves so differently from other dimensions and how knot and surface invariants relate to each other. Better tools could eventually inform fields that rely on topology, such as mathematical physics and the study of knotted molecules like DNA.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Abhishek Mallick
Dartmouth College, NH·Mathematics & Statistics
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.
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.
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.
The school
Duke University: 61 labs on PI Pages, $1.6B a year on research, 417 research doctorates a year.
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