Mark Walker
University of Nebraska-Lincoln, NE·Mathematics & Statistics
Commutative algebra studies systems that follow the familiar rules of high-school algebra, such as the fact that the order of multiplication does not matter, but in a far more general setting. It sits at the heart of pure mathematics and connects to number theory and algebraic geometry.
- Fields
Mathematics & Statistics
- Topics
commutative algebra
homological algebra
dg categories
- Grants
- 2 known grants, through Jul 2029
What they work on
- Study non-commutative analogues, phrased for dg categories, of classical conjectures about smooth varieties, including the Hodge conjecture and Grothendieck's Standard Conjectures.
- Establish a version of Poincaré duality for the Hochschild homology and periodic cyclic homology of commutative rings and schemes, and explore its consequences.
- Probe homological properties of modules of finite projective dimension, including the length conjecture, which predicts that such a module's length is at least the multiplicity of the ring.
- Determine whether Ulrich sheaves exist on all projective varieties, and settle the Buchweitz-Greuel-Schreyer conjecture on the minimum size of matrix factorizations in certain cases.
- Train undergraduates, graduate students and postdoctoral researchers in commutative algebra through a first-year REU for first-generation students, an international REU in Guanajuato, Mexico, and an early-career workshop.
More about the research
Homological algebra, a related branch tied to the study of shapes (topology), offers tools for measuring the structure of these systems. This research aims to settle open conjectures where these two areas meet. A related training grant supports the commutative algebra group at Nebraska.
Why it matters
This is pure mathematics. Settling conjectures about dg categories, Hochschild homology, modules of finite projective dimension, Ulrich sheaves and matrix factorizations would deepen understanding of commutative algebra and its links to algebraic geometry and topology. The training grant offers concrete routes into algebra research for first-generation and Hispanic undergraduates, graduate students and postdoctoral researchers.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
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.
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.
The school
University of Nebraska-Lincoln: 17 labs on PI Pages, $657M a year on research, 261 research doctorates a year.
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