Ralf Schiffler
University of Connecticut, CT·Mathematics & Statistics
Cluster algebras are algebraic structures, introduced in 2002, that capture combinatorial patterns first noticed in representation theory, the study of the symmetries of an algebraic structure rather than the structure itself. The same patterns have since turned up in branches of mathematics and physics that once seemed unrelated.
- Fields
Mathematics & Statistics
- Topics
cluster algebras
knot theory
Alexander polynomial
- Grants
- 1 known grant, through Jul 2027
What they work on
- 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.
- Classify the 2-Calabi-Yau tilted algebras, non-commutative algebras attached to clusters via categorification, that admit only finitely many Cohen-Macaulay modules.
- Prove that triangulations of a surface with marked points and a dissection correspond bijectively to maximal almost rigid modules over the algebra associated with that dissection.
More about the research
This project develops cluster algebras further and builds bridges to knot theory, the mathematics of tangled loops in space, and to the representation theory of algebras. Graduate students take part in the research.
Why it matters
This is pure mathematics, and its contribution is structural: new connections between cluster algebras, knot invariants and the representation theory of algebras. Because representation theory has applications in physics and chemistry, such connections could eventually inform work in those fields.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Arik Wilbert
University of South Alabama, AL·Mathematics & Statistics
Study the geometric, topological and combinatorial properties of Springer fibers, shapes from Lie theory tied to representations of Weyl groups, and of their generalizations.
Steven Sam
University of California-San Diego, CA·Mathematics & Statistics
Compute the equations and syzygies (relations among equations) of multiple root loci in spaces of binary forms, treating all degrees together using monad-type constructions.
Felix Janda
University of Illinois Urbana-Champaign, IL·Mathematics & Statistics·Early-career
Prove the localization formula for the logarithmic gauged linear sigma model (log GLSM), a curve-counting technique, and study the effective invariants that feed into it.
The school
University of Connecticut: 31 labs on PI Pages, $434M a year on research, 360 research doctorates a year.
Reach out
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