CCMS Colloquium: Frobenius Invariants — Austyn Simpson (Bates College)

Abstract: Rings containing a finite field \(\mathbb{F}_p\) (such as the ring of polynomials with coefficients in \(\mathbb{F}_p\)) admit a map called the Frobenius homomorphism which sends every element to its \(p\)-th power. This operation encodes delicate information about the geometry of the zero sets of polynomial equations in prime characteristic, such as whether they are smooth. In this talk, I will survey some numerical invariants defined using limits and asymptotic properties of this homomorphism which quantify how non-smooth a polynomial is. I will also discuss the current capabilities of large language models at solving problems related to these invariants, as well as a surprising appearance of the number \(\frac{4}{3}+\frac{5}{14\sqrt{7}}\).

For questions about the event, please contact your CCMS Colloquium chairs:

Prof. Robert Cass (Robert.Cass@ClaremontMcKenna.edu) or Prof. Qidi Peng (Qidi.Peng@cgu.edu)

For logistical questions, contact Maxwell Kooiker (Maxwell.Kooiker@cgu.edu), the CCMS Colloquium Assistant.