# Graduate Student Travel

We thank the National Science Foundation, the Simons Foundation, and the Office of the Vice-President for Research, for helping fund graduate student travel (listed below in reverse chronological order).

**Kenny Jacobs: An Equidistribution Result in Non-Archimedean Dynamics**

Venue: Joint AMS-MAA Meeting in San Antonio, Texas

Date: January 12, 2015

Abstract: Let K be an algebraically closed field that is complete with respect to a non-Archimedean absolute value. Let \phi\in K(z) have degree d\geq 2. Recently, Rumely introduced a measure \nu_{\phi} on the Berkovich line over K that carries information about the reduction of \phi. In particular, the measure \nu_{\phi} charges a single point if and only if $\phi$ has good reduction at that point. Otherwise, \nu_{\phi}$charges finitely many points, which can be thought of as having “spread out” the point of good reduction. In this talk, we will show that the family of measures \{\nu_{\phi^n}\} attached to the iterates of \phi equidistribute to the invariant measure \mu_\phi, a canonical object arising in the study of discrete dynamical systems.

**Allan Lacy: ****On the index of genus one curves over infinite, finitely generated fields.**

Venue: Joint AMS-MAA Meeting in San Antonio, Texas

Date: January 12, 2015

Abstract: We show that every infinite, finitely generated field admits genus one curves with index equal to any prescribed positive integer. The proof is by induction on the transcendence degree. This generalizes – and uses as the base case of an inductive argument – an older result on the number field case. There is a separate base case in every positive characteristic p, and these use work on the conjecture of Birch and Swinnerton-Dyer over function fields.

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**Adrian Brunyate: ****A Compact Moduli Space of Elliptic K3 Surfaces.**

Venue: Joint AMS-MAA Meeting in San Antonio, Texas

Date: January 12, 2015

Abstract: We will discuss recent results detailing a geometric (KSBA-type) compactication of the moduli of elliptic K3 surfaces,including how to explicitly compute limits and how the compactication relates to toroidal compactications of the period domain.

**Natalie LF Hobson (and Sayonita Ghosh Hajra): ****Studying students’ preferences and performance in a cooperative mathematics classroom**

Venue: Joint AMS-MAA Meeting in San Antonio, Texas

Date: January 10, 2015

Abstract: In this study, we discuss our experience with cooperative learning in a mathematics content course. Twenty undergraduate students from a southern public university participated in this study. The instructional method used in the classroom was cooperative. We rely on previous research and literature to guide the implementation of cooperative learning in the class. The goal of our study is to investigate the relationship between students’ preferences and performance in a cooperative learning setting. We collected data through assessments, surveys, and observations. Results show no significant difference in the comparison of students’ preferences and performance. Based on this study, we provide suggestions in teaching mathematics content courses for prospective teachers in a cooperative learning setting.

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**Lee Troupe: Bounded gaps between primes in \mathbb{F}_q[t] with a given primitive root**

Venue: 23rd Meeting of the Palmetto Number Theory Series, University of South Carolina at Columbia

Date: December 6-7, 2014

Abstract: A famous conjecture of Artin states that there are infinitely many prime numbers for which a fixed integer g is a primitive root, provided g \neq -1 and g is not a perfect square. Thanks to work of Hooley, we know that this conjecture is true, conditional on the truth of the Generalized Riemann Hypothesis. Using a combination of Hooley’s analysis and the techniques of Maynard-Tao used to prove the existence of bounded gaps between primes, Pollack has shown that (conditional on GRH) there are bounded gaps between primes with a prescribed primitive root. In this talk, we discuss the analogue of Pollack’s work in the function field case; namely, that given a monic polynomial g(t) which is not an \ellth power for any \ell dividing q-1, there are bounded gaps between monic irreducible polynomials P(t) in \mathbb{F}_q[t] for which g(t)$ is a primitive root (which is to say that g(t) generates the group of units modulo P(t)). In particular, we obtain bounded gaps between primitive polynomials, corresponding to the choice g(t) = t.

**Lee Troupe: ****The number of prime factors of s(n).**

Venue: Fall Southeastern Sectional Meeting of the AMS, University of North Carolina at Greensboro

Date: November 8-9, 2014

Abstract: Let ω(n) denote the number of distinct prime divisors of a natural number n. In 1917, Hardy and Ramanujan famously proved that the normal order of ω(n) is log log n; in other words, a typical natural number n has about log log n distinct prime factors. Erd˝os and Kac later generalized Hardy and Ramanujan’s result, showing (roughly speaking) that ω(n) is normally distributed and thereby giving rise to the field of probabilistic number theory. In this talk, we’ll discuss the normal order of ω(s(n)), where s(n) is the usual sum-of-proper-divisors function. This new result supports a conjecture of Erd˝os, Granville, Pomerance, and Spiro; namely, that if a set of natural numbers has asymptotic density zero, then so does its preimage under s.

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**Ziqing Xiang: Tight Block Designs**

Venue: Workshop on Sphere Packings, Lattices, and Designs, Erwin Schrodinger International Institute, Vienna, Austria

Date: October 27-31, 2014

**Lee Troupe: ****The Hardy-Ramanujan theorem and related results.**

Venue: Clemson University

Date: October 22, 2014

Abstract: Let ω(n) denote the number of distinct prime divisors of a natural number n. In 1917, Hardy and Ramanujan famously proved that the normal order of ω(n) is log log n; in other words, a typical natural number n has about log log n distinct prime factors. Erd˝os and Kac later generalized Hardy and Ramanujan’s result, showing (roughly speaking) that ω(n) is normally distributed and thereby giving rise to the field of probabilistic number theory. In this talk, we’ll discuss the normal order of ω(s(n)), where s(n) is the usual sum-of-proper-divisors function. This new result supports a conjecture of Erd˝os, Granville, Pomerance, and Spiro; namely, that if a set of natural numbers has asymptotic density zero, then so does its preimage under s.

**Kenny Jacobs: An Equidistribution Result in Non-Archimedean Dynamics**

Venue: Clemson University Number Theory Seminar

Date: October 8, 2014

Abstract: Let K be an algebraically closed field that is complete with respect to a non-Archimedean absolute value. Let \phi\in K(z) have degree d\geq 2. Recently, Rumely introduced a measure \nu_{\phi} on the Berkovich line over K that carries information about the reduction of \phi. In particular, the measure \nu_{\phi} charges a single point if and only if $\phi$ has good reduction at that point. Otherwise, \nu_{\phi}$charges finitely many points, which can be thought of as having “spread out” the point of good reduction. In this talk, we will show that the family of measures \{\nu_{\phi^n}\} attached to the iterates of \phi equidistribute to the invariant measure \mu_\phi, a canonical object arising in the study of discrete dynamical systems.

**Theresa Brons: Parabolic Subgroups and the Line-Bundle Cohomology over the Flag Variety**

Venue: Central Fall Sectional Meeting of the AMS, Eau Claire, Wisconsin

Date: September 19-21, 2014

Abstract: H.H. Andersen determined the socle of H1(λ), which is potentially non-zero only when there exists a unique simple root α such that ⟨λ, α∨⟩ < 0. In this work he did so by first determining the socle in the case when G is of type A1 where H1(λ) a Weyl module and λ an anti-dominant weight, and later extended this to the case when P(α) is a minimal parabolic subgroup. In this talk, this approach will be generalized, leading to some new vanishing results and some interesting avenues for further study.