Claudio Gonzales
Carleton College, MN·Mathematics & Statistics
Polynomials appear everywhere in mathematics and the sciences, from describing motion to designing cryptographic schemes. A basic and very old question is how hard it is to solve them: given a polynomial, how complicated must a formula for its roots be?
- Fields
Mathematics & Statistics
Education Research
- Topics
resolvent degree
Hilbert's 13th Problem
polynomials
- Grants
- 1 known grant, through Aug 2027
What they work on
- Advance the framework of "special points" using polar cones, geometric objects encoding convexity data, to sharpen bounds on the resolvent degree of finite groups and address related arithmetic questions.
- Investigate the topology of moduli problems, spaces whose points are themselves geometric objects, following phenomena uncovered by Farb, Kisin and Wolfson but not yet fully explained.
- Study resolvent filtrations of étale fundamental groups, algebraic invariants that record the symmetries of coverings of a space, to clarify their role as a measure of polynomial-solving complexity.
- Develop computational techniques for resolvent degree and formalize ideas from the classical literature into general frameworks, with undergraduate projects on data-scientific analyses and translations of historical mathematical texts.
More about the research
Classical results show that some equations cannot be solved by radicals at all, and Hilbert's 13th Problem asked how much nesting of auxiliary solutions is really needed. Resolvent degree is a recently formalized way to measure exactly this kind of difficulty. This project develops the theory of resolvent degree and extends it to new settings.
Why it matters
Resolvent degree gives a precise way to say how hard a polynomial is to solve, a question running from classical algebra through Hilbert's 13th Problem. Sharper bounds and general frameworks would deepen understanding of this complexity measure and its links to number theory and algebraic geometry. The project also produces concrete teaching materials and translations of classical texts, and involves undergraduates in research and in mathematics history curriculum development.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Ananth Shankar
Northwestern University, IL·Mathematics & Statistics·Early-career
Construct and study integral models of exceptional Shimura varieties, that is, descriptions by polynomial equations with whole-number coefficients for the varieties attached to exceptional symmetry groups.
Hannah Larson
University of California-Berkeley, CA·Mathematics & Statistics·Early-career
Develop new intersection-theory methods for moduli spaces of low-genus algebraic curves, spaces whose points each represent a curve, and connect them to closely related moduli spaces.
Jeremy Miller
Purdue University, IN·Mathematics & Statistics
Prove homological stability and its variants, meaning that the hole-counting homology settles down as spaces grow, for classifying spaces of arithmetic groups, using tools such as locally symmetric spaces and homological algebra.
The school
Carleton College: 4 labs on PI Pages, $5M a year on research.
Reach out
A start for your email. Fill in the brackets.