Matthew Stoffregen
Michigan State University, MI·Mathematics & Statistics
What shapes can exist in four dimensions? Four-dimensional geometry behaves unlike any other dimension, and deciding which shapes are possible, and which three-dimensional boundaries can bound them, remains one of the central open problems in topology.
- Fields
Mathematics & Statistics
- Topics
Floer homology
Seiberg-Witten theory
Floer homotopy type
- Grants
- 1 known grant, through Aug 2027
What they work on
- 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.
- Determine the Seiberg-Witten Floer homotopy type of Seifert spaces, a structured family of three-manifolds that provides computable test cases for the refined gauge-theoretic invariants.
- Develop practical techniques for calculating involutive Floer homology, the Hendricks-Manolescu refinement that adds a symmetry to standard Floer homology and has so far resisted computation.
- Apply the resulting computations to find new phenomena in the homology cobordism group of three-manifolds, which records which three-dimensional spaces bound well-behaved four-dimensional ones.
More about the research
The main tools for attacking such questions come from mathematical gauge theory, a set of techniques that originated in physics and translates equations for fields into algebraic invariants of a space. This project develops those tools further and uses them to constrain what four-dimensional shapes can look like.
Why it matters
This is pure mathematics: the work sharpens the tools used to decide which four-dimensional shapes and which three-dimensional boundaries are possible. Making involutive Floer homology and Seiberg-Witten Floer spectra computable would let topologists apply invariants that are currently understood only in principle, and new phenomena in the homology cobordism group would refine the structural picture of three- and four-dimensional spaces that gauge theory has been building since its origins in physics.
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.
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.
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
Michigan State University: 28 labs on PI Pages, $932M a year on research, 555 research doctorates a year.
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