Jeremy Usatine
Florida State University, FL·Mathematics & Statistics
Algebraic geometry studies varieties, which are shapes defined by polynomial equations, like the surface of a sphere or of a donut. Both of those examples are smooth: zoom in far enough and they look flat.
- Fields
Mathematics & Statistics
- Topics
algebraic geometry
singularities
motivic integration
- Grants
- 1 known grant, through Jul 2027
What they work on
- Use motivic integration for smooth Artin stacks, a class of generalized geometric spaces, to study stringy invariants of singular varieties.
- Describe stringy Hodge numbers in terms of crepant resolutions via Artin stacks, working toward a cohomological interpretation of these invariants.
- Develop a McKay correspondence for reductive groups, connecting group symmetries to geometry, using techniques related to motivic integration.
- Define quantum cohomology rings for arbitrary matroids by studying the Gromov-Witten theory of wonderful models of hyperplane arrangements, using tropical curves and polyhedral geometry.
More about the research
Smooth shapes appear throughout science. For example, the expectation that small changes in a physical system's starting point cause only small changes in its short-term behavior amounts to assuming that an underlying geometric object is smooth. Many of the best tools for studying smooth varieties, however, depend on varieties that have singularities, which are sharp or pointy spots. Understanding singularities is therefore a central goal of the field.
Why it matters
This is pure mathematics. It aims to deepen understanding of singular geometric shapes and of invariants that arise in theoretical physics, especially string theory. Results on stringy and Gromov-Witten invariants and on matroids could strengthen the tools mathematicians use to study smooth varieties, which underlie models across the sciences, and could tie algebraic geometry more closely to combinatorics.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
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.
Christopher Rogers
University of Nevada-Reno, NV·Mathematics & Statistics
Gather evidence for the Alekseev-Torossian conjecture that the Kashiwara-Vergne group (KRV) equals the Grothendieck-Teichmueller group (GRT), by studying how both act on explicit geometric and topological objects.
Christine Berkesch
University of Minnesota, MN·Mathematics & Statistics
Develop algebraic versions of symplectic geometry techniques such as "stop manipulation" to uncover structure in multigraded polynomial rings, rings whose variables carry several degrees at once, that parallels classical results for standard graded rings.
The school
Florida State University: 15 labs on PI Pages, $461M a year on research, 481 research doctorates a year.
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