# 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).

**2017**

**Luca Schaffler: The KSBA compactification of the moduli space of $D_{1,6}$-polarized Enriques surfaces**

Venue: Workshop on Algebraic Varieties, Hodge Theory and Motives, Fields Institute, March 2017

Abstract: In this talk we describe the moduli compactification by stable pairs (also known as KSBA compactification) of a 4-dimensional family of Enriques surfaces, which arise as the $\mathbb{Z}_2^2$-covers of the blow up of $\mathbb{P}^2$ at three general points branched along a configuration of three pairs of lines. The chosen divisor is an appropriate multiple of the ramification locus. Using the theory of stable toric pairs we are able to study the degenerations parametrized by the boundary and its stratification. We relate this compactification to the Baily-Borel compactification of the same family of Enriques surfaces. Part of the boundary of this stable pairs compactification has a toroidal behavior, another part is isomorphic to the Baily-Borel compactification, and what remains is a mixture of these two.

**Luca Schaffler: The KSBA compactification of the moduli space of $D_{1,6}$-polarized Enriques surfaces**

Venue: Focused Research Group on Hodge Theory, Moduli and Representation Theory: Workshop VIII, Washington University in St. Louis, January 2017

**Luca Schaffler: The KSBA compactification of the moduli space of $D_{1,6}$-polarized Enriques surfaces**

Venue: AMS Contributed Paper Session in Algebraic Geometry, Joint Mathematics Meetings, Atlanta, January 2017

**Hans Parshall: Spherical configurations over finite fields**

Venue: Joint Mathematics Meetings, Atlanta, January 2017

Abstract: In their 1973 paper, Erdos, Graham, Montgomery, Rothschild, Spencer, and Straus proved that every Euclidean Ramsey set is contained in some sphere, and Graham conjectures that every finite spherical set is indeed Ramsey. This conjecture remains open (and contested) even in the case of a generic four point subset of a circle. We provide evidence for Graham’s conjecture by proving something stronger in the finite field setting: for any a in (0,1) every subset A of F_q^{10} with |A| > aq^{10} contains an isometric copy of every four point spherical set spanning two dimensions, provided q is taken sufficiently large with respect to a. For d > 2k + 5, comparable results are obtained in F_q^d for arbitrary (k + 2)-point spherical configurations spanning k dimensions.

**Natalie LF Hobson: Identities between first Chern class of vector bundles of conformal blocks**

Venue: AMS Contributed Paper Session in Algebraic Geometry, JMM January 2017

Abstract: Given a simple Lie algebra g, a positive integer `, and an n-tuple ~λ of dominant integral weights for g at level `, one can define a vector bundle on Mg,n known as a vector bundle of conformal blocks. These bundles are nef in genus g = 0 and so this family provides potentially an infinite number of elements in the nef cone of M0,n to analyze. Result relating these divisors with different data is thus significant in understanding these objects. In this talk, we use correspondences of these bundles with products in quantum cohomology in order to classify when a bundle with sl2 or sp2` is rank one. We show this is also a necessary and sufficient condition for when these divisors are equivalent.

**Natalie LF Hobson: Vector Bundles of Conformal Blocks- Rank one and finite generation**

Venue: AWM Poster Session, JMM January 2017

Abstract: The moduli space of curves, M0,n, parametrizes stable n-pointed rational curves. To understand this projective variety, we study vector bundles on it. Vector bundles of conformal blocks are an infinite family of such bundles. Since these bundles are all globally generated, they are especially interesting to analyze, as their first Chern classes, the conformal blocks divisors, are all nef. It is an open question as to whether the nef cone, Nef(M0,n), is finitely generated for n > 7. How does the infinite family of conformal blocks divisors live in Nef(M0,n)? Is the subcone generated by conformal blocks divisors polyhedral? In this report, I give several of my results to these questions for specific cases of interest.

**2016**

**Hans Parshall: Spherical configurations in dense sets**

Venue: The Ohio State University, November 2016

Abstract: We will discuss an arithmetic combinatorics perspective on how to locate geometric configurations. By controlling a counting operator with a uniformity norm, one can argue that uniform sets contain many configurations. In joint work with Neil Lyall and Akos Magyar, we further prove an inverse theorem and establish, for example, that all large subsets of vector spaces over finite fields contain isometric copies of all spherical quadrilaterals.

**Hans Parshall: Triangles and quadrilaterals over finite fields**

Venue: Missouri State University, November 2016

Abstract: We will discuss an arithmetic combinatorics approach to locating geometric patterns over finite fields. By defining a "counting operator" and a "uniformity norm", we will argue that "uniform" dense sets contain geometric configurations. This approach recently led to an improvement on the 2008 result of Hart and Iosevich on triangles (3-point configurations) and new results on quadrilaterals (4-point configurations), joint with Neil Lyall and Akos Magyar.

**Hans Parshall: Spherical configurations over finite fields**

Venue: University of West Georgia, October 2016

Abstract:

In the 1970s, it was shown by Erdos, Graham, Montgomery, Rothschild Spencer, and Straus that every Euclidean Ramsey set is spherical, and the converse remains an open conjecture. We provide evidence for this conjecture in the finite field setting. Following the setup of Hart and Iosevich, we show that every d-simplex appears isometrically in every sufficiently large subset of F_q^d, improving the necessary relationship between d and d. We will further discuss comparable results for spherical configurations over finite fields whose Euclidean analogues are not known to be Ramsey.

**Natalie LF Hobson: Vector Bundles of Conformal Blocks-- Rank One and Finite Generation**

Venues: University of Utah (September 2016), UPenn (September 2016), University of Illinois at Chicago (September 2016)

Abstract:

Given a simple Lie algebra \g, a positive integer l and an n-tuple of dominant integral weights for \g at level l, one can define a vector bundle on the moduli space of curves known as a vector bundle of conformal blocks. These bundles are nef in the case that the genus is zero and so this family provides potentially an infinite number of elements in Nef(M_0,n\bar) to analyze.

It is natural to ask how this infinite family of conformal blocks divisors lives in Nef(M_0,n\bar). Is the subcone generated by conformal blocks divisors polyhedral? In this talk, we give several results to this question for specific cases of interest. To show our results, we use a correspondence of the ranks of these bundles with computations in the quantum cohomology of the Grassmannian.

**Natalie LF Hobson: Quantum Kostka and the rank on problem for \sl_2m**

Venues: Rutgers University (October 2016), Univesity of Ilinois at Urbana-Champain (November 2016), Ohio State University (November 2016), University of British Colombia (November 2016)

Abstract: In this talk we will define and explore an infinite family of vector bundles, known as vector bundles of conformal blocks, on the moduli space M0,n of marked curves. These bundles arise from data associated to a simple Lie algebra. We will show a correspondence (in certain cases) of the rank of these bundles with coefficients in the cohomology of the Grassmannian. This correspondence allows us to use a formula for computing "quantum Kostka" numbers and explicitly characterize families of bundles of rank one by enumerating Young tableaux. We will show these results and illuminate the methods involved.

**Natalie LF Hobson: Vector bundles of conformal blocks with $\mathfrak{sp}_{2\ell}$ at level one**

Venue: 2016 AMS Spring Central Sectional Meeting, Fargo, North Dakota, April 2016

Abstract

**William Hardesty: On Support Varieties and the Humphreys Conjecture in type A **

Venue: AMS special session on Lie Theory, Representation Theory and Geometry, Athens, Georgia, March 2016

**Patrick K. McFaddin: Chow groups with coefficients and generalized Severi-Brauer varieties**

Venue: Emory University, February 2, 2016

Abstract: The theory of algebraic cycles on homogenous varieties has seen many useful applications to the study of central simple algebras, quadratic forms, and Galois cohomology. Significant results include the Merkurjev-Suslin Theorem and Suslin's Conjecture, recently proved by Merkurjev. Despite these successes, a general description of Chow groups and Chow groups with coefficients remains elusive, and computations of these groups are done in various cases. In this talk, I will give some background on K-cohomology groups of Severi-Brauer varieties and discuss some recent work on computing these groups for algebras of index 4.

**Natalie LF Hobson: Quantum kostka and the rank one problem for $\mathfrak{sl}_{2m}$**

Venue: 2016 Joint AMS and MAA Mathematics Meetings, Seattle, WA, January 2016

Abstract

**William Hardesty: On Support Varieties and the Humphreys Conjecture in type A**

Venue: AMS special session on Categorical and Geometric Methods in Representation Theory, Seattle, Washington, January 2016

**Kenneth Jacobs: Lyapunov Exponents in non-Archimedean Dynamics**

Venue: Joint AMS-MAA Meeting in Seattle, WA, January 6, 2016

Abstract: The Lyapunov exponent of a rational map f measures the rate of growth of a point in a generic orbit. It is related to the orbits of the critical points of f, and when f is defined over the complex numbers, a sharp lower bound is log(d)/2, where d is the degree of the map. Much less is known about Lyapunov exponents for maps defined over non-Archimedean fields. In this talk, we will give an explicit lower bound similar to the one over the complex numbers which is sharp for maps of good reduction. We will also give a formula relating Lyapunov exponents to Silverman's critical height.

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**2015**

**Kenneth Jacobs: ****Lower Bounds for non-Archimedean Lyapunov Exponents**

Venue: RTG Workshop in Arithmetic Dynamics (at the University of Michigan, Ann Arbor), December 5, 2015

Abstract: Lyapunov exponents measure the rate of expansion of a dynamical system. In classical complex dynamics, the Lyapunov exponent of a rational map is known to be bounded below by (log d)/2, where d is the degree of the map, and this bound is known to be sharp. In this talk, we will present a lower bound for rational maps defined over non-Archimedean valued fields which is sharp for maps of potential good reduction and for maps whose Berkovich Julia set satisfies a certain finiteness condition.

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**William Hardesty: On Support Varieties and the Humphreys Conjecture in type A**

Venue: 8th Southeastern Lie Theory Workshop on Algebraic and Combinatorial Representation Theory, Raleigh, North Carolina, October 2015

**Lee Troupe: Orders of reductions of elliptic curves with many and few prime factors**

Venue: Illinois Number Theory Conference 2015, August 13-14, 2015

Abstract

**Ziqing Xiang: Spherical Designs Over a Number Field**

Venue: 2015 Workshop on Combinatorics and Applications, at Shanghai Jiao Tong University, April 21-27, 2015

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**Ziqing Xiang: The Lit-Only σ-Game**

Venue: 2015 Workshop on Combinatorics and Applications, at Shanghai Jiao Tong University**, **April 21-27, 2015

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**William Hardesty: Support varieties of line bundle cohomology groups for SL(3)**

Venue: Southwest Group Theory Day 2015, Tucson, Arizona, March 2015

**Kenneth Jacobs: An Equidistribution Result in non-Archimedean Dynamics**

Venue: Algebra Seminar, Georgia Institute of Technology, January 26th, 2015

Abstract: Let K be a complete, algebraically closed, non-Archimedean field, and let f be a rational function defined over K with degree at least 2. Recently, Robert Rumely introduced two objects that carry information about the arithmetic and the dynamics of f. The first is a function ordRes_f, which describes the behavior of the resultant of f under coordinate changes on the projective line. The second is a discrete probability measure \nu_f supported on the Berkovich half space that carries arithmetic information about f and its action on the Berkovich line. In this talk, we will show that the functions ordRes_f converge locally uniformly to the Arakelov-Green's function attached to f, and that the family of measures \nu_{f^n} attached to the iterates of f converge to the equilibrium measure of f.

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**Kenneth 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.

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

Venue: Joint AMS-MAA Meeting in San Antonio, Texas, 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, 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, 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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**2014**

**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, 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, 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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**Kenneth Jacobs: A New Type of Equidistribution Result in non-Archimedean Dynamics**

Venue: Northwestern University Dynamics Seminar, November 4th, 2014

Abstract: Let K be an algebraically closed field that is complete with respect to a non-Archimedean absolute value. We study the dynamics of rational functions with coefficients in K. In this non-Archimedean setting, there is an associated rational map, called the reduction map, which is defined over the residue field of K and carries information about the dynamics. Recently, Rumely introduced a measure nu on the Berkovich line over K that carries information about the reduction of the conjugates of the map. In this talk, we will show that the sequence of measures {nu_n}, associated to the iterates of the map, equidistribute to a natural invariant measure on the Berkovich line. As time permits, we will also discuss recent work of a VIGRE group in which the crucial measures have been shown to give information about the location of the map as a point in moduli space.

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**Ziqing Xiang: Tight Block Designs**

Venue: Workshop on Sphere Packings, Lattices, and Designs, Erwin Schrodinger International Institute, Vienna, Austria, October 27-31, 2014

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

Venue: Clemson University, 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.

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

Venue: Clemson University Number Theory Seminar, 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.

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

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

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