Frank Thorne
University of South Carolina-Columbia, SC·Mathematics & Statistics
Number theory asks questions about whole numbers, and arithmetic statistics asks them in bulk: given a family of mathematical objects built from numbers, how many are there of a given size, and how large can certain features get? The objects in play here include number fields, which extend ordinary rational numbers by adding roots of polynomials, and ideal class groups, which measure how badly unique factorization fails inside such a field.
- Fields
Mathematics & Statistics
- Topics
arithmetic statistics
number theory
Fourier analysis
- Grants
- 1 known grant, through Aug 2028
What they work on
- Carry out finite Fourier transform computations for families of arithmetic objects, extending approaches the PI and collaborators have previously developed in arithmetic statistics.
- Improve the error terms in number field counting results, the quantity that limits how precisely such counts can be stated, bypassing a known obstacle to sharper bounds.
- Study Sato-Shintani zeta functions, generating functions encoding counts of arithmetic objects, outside their regions of absolute convergence, using Fourier analysis to establish that they are defined there.
- Develop the Fourier-analytic connections as reusable building blocks for other questions in arithmetic statistics, such as counting number fields and bounding ideal class groups.
More about the research
This project pursues these counting questions through Fourier analysis, the technique of breaking a function apart into waves, which connects the subject to many other active areas of mathematics.
Why it matters
This is basic mathematics: it sharpens how accurately mathematicians can count number fields and related arithmetic objects, and it extends the range in which certain zeta functions can be analyzed at all. Because Fourier-analytic tools in arithmetic statistics serve as building blocks for other problems in number theory, the results could support later work across the field, and the accompanying coursework and book would make the techniques teachable.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Ken Ono
University of Virginia, VA·Mathematics & Statistics
Develop algebraic, analytic and combinatorial tools for harmonic Maass forms, generalizations of modular forms that underlie Ramanujan's mock theta functions, for attacking open problems in number theory.
Aaron Landesman
Harvard University, MA·Mathematics & Statistics
Make progress on the Cohen-Lenstra conjectures about class groups of quadratic fields over function fields by computing relevant homology groups with new topological tools.
Melanie Wood
Harvard University, MA·Mathematics & Statistics
Develop a nonabelian analogue of the Cohen-Lenstra heuristics, replacing class groups with Galois groups that encode symmetries of nonabelian extensions of a varying number field.
The school
University of South Carolina-Columbia: 22 labs on PI Pages, $277M a year on research, 425 research doctorates a year.
Reach out
A start for your email. Fill in the brackets.