Joshua Greene
Boston College, MA·Mathematics & Statistics
Does every closed loop drawn in a plane pass through the four corners of some square? This old puzzle, the square peg problem, is one of several questions in low-dimensional topology, the study of shapes in one to four dimensions.
- Fields
Mathematics & Statistics
- Topics
low-dimensional topology
geometric topology
knot theory
- Grants
- 1 known grant, through Nov 2026
What they work on
- 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.
- Study exceptional Dehn surgery, cutting and regluing three-dimensional spaces along knots, using graphs that track how surfaces intersect.
- Investigate rational homology cobordism of three-dimensional spaces using Floer homology, an algebraic invariant, together with lattice-theoretic methods.
- Study ribbon concordance, a relationship between knots, using classical topological methods.
More about the research
These dimensions match how we perceive the physical world. They also behave differently from higher dimensions and need their own tools. This project tackles a set of major problems in the field by bringing in methods from nearby areas of mathematics.
Why it matters
This is pure mathematics. It aims to settle long-standing questions about curves, knots and three- and four-dimensional shapes, and to connect tools from combinatorics, symplectic geometry and Floer theory. The results deepen understanding of the dimensions that model physical space and could inform future work across geometry and topology.
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.
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.
The school
Boston College: 11 labs on PI Pages, $96M a year on research, 161 research doctorates a year.
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