The OSU Student Number Theory Seminar for Fall 2024 semester is held biweekly on Tuesday 9:10-10:05am, MW154. Anyone interested in number theory (broadly interpreted) is welcome to attend!
Tuesday 11/19: Jonathan Hales
Title: A Tale of Two Factoring Variants
Abstract: A new deterministic algorithm for finding square divisors, and finding $r$-power divisors in general, is presented. This algorithm is based on Lehman's method for integer factorization and is straightforward to implement. While the theoretical complexity of the new algorithm is far from best known, the algorithm becomes especially effective if even a loose bound on a square divisor is known. Additionally, we answer a question by D. Harvey and M. Hittmeir on whether their recent deterministic algorithm for integer factorization can be adapted to finding $r$-power divisors.
Tuesday 9/10: Tinghao Huang
Title: On Exponential Sums of the Form \sum_{r (q)}e(\frac{ar^k + br}{q})
Abstract: Non-trivial results on exponential forms of specific forms usually have deep implication to various problems in analytic number theory. In this expository talk, we focus on exponential sums of the form A_k(a,b;q) := \sum_{r (q)}e(\frac{ar^k + br}{q}), where a, b are integers and k, q are positive integers. We discuss some of the explicit evaluation or non-trivial estimates of the above A_k(a,b;q), with an emphasis to the case k = 2 and k = 3, and explain some of their applications to other number theoretic problems.
Tuesday 8/27: Jake Huryn
Title: Hilbert's Irreducibility Theorem
Abstract: A polynomial over k in variables T1,…,Tn, X can be interpreted as a family of one-variable polynomials by letting each Ti vary over k. This leads to several natural questions about the behavior of these one-variable polynomials as the Ti change. For example, how often do they have a root? How often are they irreducible? How are their Galois groups distributed? In this talk, I will prove a version of Hilbert's fundamental "irreducibility theorem", which answers some of these questions, and give applications to Galois theory.