Christine Berkesch
University of Minnesota, MN·Mathematics & Statistics
Algebra and geometry are two central branches of mathematics. Algebraic geometry studies shapes defined by polynomial equations, while symplectic geometry is a kind of geometry that underlies physics.
- Fields
Mathematics & Statistics
- Topics
commutative algebra
multigraded polynomial rings
toric varieties
- Grants
- 1 known grant, through Jul 2028
What they work on
- 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.
- Extend Orlov's Theorem, a foundational result in homological algebra, to multigraded and toric settings, and construct new categorical structures that unify algebraic and geometric viewpoints.
- Apply these methods to virtual resolutions and other shortest-resolution questions, and extend them beyond toric varieties using toric degenerations and explicit links to Fukaya categories.
More about the research
In recent decades, researchers found unexpected links between the two through homological mirror symmetry, an idea that came from physics. This collaborative project uses those links to build new tools in algebra and geometry and to fill long-standing gaps in commutative algebra, the study of polynomial rings and their equations.
Why it matters
This is pure mathematics. It adds new tools connecting algebra, toric geometry and symplectic geometry, and answers open questions about multigraded polynomial rings. Because algebra and geometry underlie fields such as computer science, data analysis, robotics and theoretical physics, these tools and the planned public software could eventually inform those areas.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Daniel Freed
Harvard University, MA·Mathematics & Statistics
Investigate expected relationships between Fukaya categories of symplectic resolutions, which are algebraic structures built by counting pseudoholomorphic curves, and structures from representation theory, using microlocal sheaf theory.
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.
Mark Walker
University of Nebraska-Lincoln, NE·Mathematics & Statistics
Study non-commutative analogues, phrased for dg categories, of classical conjectures about smooth varieties, including the Hodge conjecture and Grothendieck's Standard Conjectures.
The school
University of Minnesota: 88 labs on PI Pages, $1.4B a year on research, 722 research doctorates a year.
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