Arik Wilbert
University of South Alabama, AL·Mathematics & Statistics
A knot is simply a tangled loop, yet telling two knots apart can be a deep mathematical problem. Knot theory studies these tangled loops in space and has uses in biology, chemistry and physics.
- Fields
Mathematics & Statistics
- Topics
categorification
knot theory
Khovanov homology
- Grants
- 1 known grant, through Sep 2027
What they work on
- Study the geometric, topological and combinatorial properties of Springer fibers, shapes from Lie theory tied to representations of Weyl groups, and of their generalizations.
- Use convolution algebras from Springer fibers of Lie types B, C and D to build new homological knot invariants, extending the type A geometric picture of Khovanov homology.
- Find out how these Springer fiber algebras relate to invariants of knots and links in spaces beyond ordinary 3D space, including manifolds and orbifolds.
More about the research
One powerful approach is categorification. This means replacing a simple mathematical object, such as a number or a polynomial, with a richer structure that holds more information. This richer structure can help settle hard open questions.
Why it matters
This is basic mathematics. It aims to create new tools for open problems in knot theory and to deepen the links between algebra, geometry and topology. Knot theory has uses in biology, medicine, chemistry and physics, so new knot invariants could eventually inform those fields. The MaPP Challenge outreach also aims to bring more students into STEM.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Ralf Schiffler
University of Connecticut, CT·Mathematics & Statistics
Construct a cluster algebra from an arbitrary knot or link so that one of its clusters has cluster variables specializing to the Alexander polynomial, a standard invariant distinguishing knots.
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.
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
University of South Alabama: 5 labs on PI Pages, $70M a year on research, 53 research doctorates a year.
Reach out
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