Emily Witt
University of Kansas, KS·Mathematics & Statistics·Early-career
Polynomial equations appear everywhere in science and technology, and the solution sets they define can have sharp corners, crossings, and other irregular points. Commutative algebra is the branch of mathematics that studies such equations and their solution sets, and it supplies tools used in areas from cryptography to genomics.
- Fields
Mathematics & Statistics
- Topics
commutative algebra
algebraic geometry
Frobenius map
- Grants
- 1 known grant, through Mar 2027
What they work on
- 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.
- Develop effective algorithms for computing the Bernstein-Sato polynomial over the complex numbers, a quantity that encodes the severity of a singularity attached to a polynomial.
- Build a theory of differential operators, algebraic analogues of derivatives, and of modules over them in non-regular (singular) rings, then apply this theory to objects such as multiplier ideals.
- Determine new geometric and topological properties detected by local cohomology, focusing on how the irreducible components of a ring's spectrum are connected.
- Create computer algebra software supporting research and teaching in commutative algebra, and run an undergraduate research training program for students underrepresented in STEM.
More about the research
This project works on the basic theory of commutative rings and algebraic varieties, the algebraic and geometric objects built from polynomial equations, with special attention to singularities, the points where a solution set fails to be smooth.
Why it matters
Work here is foundational mathematics: it adds to the understanding of how singular solution sets of polynomial equations behave and how their complexity can be measured and computed. Because commutative algebra underpins tools used in cryptography and genomics, better theory and new algorithms could eventually support those applications, while the planned computer algebra software and undergraduate research program would give students usable computational training.
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.
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.
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 Kansas: 26 labs on PI Pages, $546M a year on research, 351 research doctorates a year.
Reach out
A start for your email. Fill in the brackets.