Claudiu Raicu
University of Notre Dame, IN·Mathematics & Statistics
Homology is a mathematical way of measuring how things differ. In algebra it compares a set of objects defined implicitly (cycles) with a subset defined explicitly (boundaries).
- Fields
Mathematics & Statistics
- Topics
commutative algebra
algebraic geometry
representation theory
- Grants
- 1 known grant, through Jun 2027
What they work on
- 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.
- Work out concrete cases, including projective space via local cohomology of polynomial rings, Grassmannians, complete flag varieties and the incidence correspondence, to reveal how algebra, geometry and symmetry interact.
- Apply the new results to homological questions about matrix determinantal varieties and their higher-tensor analogues, including their equations and syzygies (relations among equations), and Koszul modules.
More about the research
In geometry it is used to tell shapes apart. Homology captures an object's most important features and reflects its symmetries, and it has found uses in data analysis and computer science.
Why it matters
This is basic mathematics. Learning how the cohomology of flag varieties depends on the prime p would fill a gap left by the Borel–Weil–Bott theorem. It could inform the study of fundamental algebraic objects such as matrices and higher-dimensional tensors, at the meeting point of commutative algebra, algebraic geometry and representation theory.
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.
Emily Witt
University of Kansas, KS·Mathematics & Statistics·Early-career
Use the Frobenius map, which raises ring elements to a prime power, to study hypersurfaces in prime characteristic, sharpening the understanding of test ideals and Frobenius jumping exponents that quantify how singular a shape is.
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
University of Notre Dame: 19 labs on PI Pages, $346M a year on research, 277 research doctorates a year.
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