Steven Sam
University of California-San Diego, CA·Mathematics & Statistics
Many problems in algebra can be stated as computations, such as finding the equations that describe a geometric shape or space. This project looks at classical questions in combinatorics and algebra from new angles.
- Fields
Mathematics & Statistics
- Topics
combinatorial algebra
commutative algebra
representation theory
- Grants
- 1 known grant, through Jun 2027
What they work on
- 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.
- Calculate syzygies of determinantal-like varieties through the coherent cohomology of super homogeneous spaces and super analogues of the Grothendieck-Springer resolution, explaining their Lie superalgebra symmetries.
- Use the link between syzygies and super homogeneous spaces to seek super analogues of the Borel-Weil-Bott theorem, a classical result describing cohomology on homogeneous spaces.
- Develop curried algebras to import Lie theory techniques, such as Bernstein-Gelfand-Gelfand category O, into representation stability theory and prove new structural results there.
More about the research
A hard computation is recast as a different one that turns out to be far easier to solve. The work sits between representation theory, the study of how symmetries act on spaces, and commutative algebra, the study of polynomial equations. Ideas flow in both directions. It has three subprojects.
Why it matters
This is basic mathematics research. It aims to deepen understanding of how symmetry and polynomial equations interact, giving new, more tractable ways to compute equations and their relations. These tools could eventually inform further work across representation theory, commutative algebra and algebraic geometry.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Claudiu Raicu
University of Notre Dame, IN·Mathematics & Statistics
Compute the cohomology of line bundles on flag varieties in positive characteristic, where the Borel–Weil–Bott theorem no longer gives the answer, and describe how the result depends on the prime p.
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.
Cristian Lenart
University at Albany, NY·Mathematics & Statistics
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.
The school
University of California-San Diego: 84 labs on PI Pages, $1.9B a year on research, 637 research doctorates a year.
Reach out
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