Cristian Lenart
University at Albany, NY·Mathematics & Statistics
Many deep questions in algebra and geometry become tractable once they are translated into counting and diagram problems. This project uses combinatorics, the mathematics of discrete structures such as permutations and graphs, to solve problems and carry out computations in representation theory and in the geometry and topology of flag manifolds, and to uncover new links between these areas.
- Fields
Mathematics & Statistics
- Topics
combinatorics
representation theory
Lie algebras
- Grants
- 1 known grant, through Aug 2027
What they work on
- Develop a refinement of the atomic decomposition of crystals, the colored directed graphs encoding quantum algebra representations, and relate it to the pre-canonical bases of Hecke algebras that interpolate between the standard and Kazhdan-Lusztig bases.
- Extend the quantum alcove model as a uniform combinatorial description of Kirillov-Reshetikhin crystals and their tensor products in all affine types, and find new applications for it.
- Apply the Chevalley multiplication formula in the quantum K-theory of flag manifolds, proved by the PI with Naito and Sagaki, to problems in modern Schubert calculus.
- Work out the combinatorics of the elliptic classes of Rimanyi and Weber in the equivariant elliptic cohomology of flag manifolds.
More about the research
Representation theory is the main tool for studying symmetry: it represents elements of abstract groups and algebras as linear transformations acting on vector spaces. Representations of Lie algebras and quantum groups have physical applications, such as computing the probability that a system of particles is in a particular state at a given time. On the geometric side, Schubert calculus grew out of enumerative geometry, which counts the lines or planes meeting a set of generic conditions, and now connects to quantum cohomology and K-theory and to elliptic cohomology.
Why it matters
Combinatorial models turn abstract algebraic and geometric objects into structures that can be computed with explicitly. The results would give more uniform and effective tools for representations of Lie algebras and quantum groups, which underlie calculations about particle systems in physics, and for counting problems in the geometry of flag manifolds, including their quantum K-theory and elliptic cohomology.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
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.
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.
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.
The school
University at Albany: 15 labs on PI Pages, $472M a year on research, 144 research doctorates a year.
Reach out
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