In preparation

On derived categories and rational points for a class of toric Fano varieties, (with Matthew Ballard). In preparation.
Abstract. In preparation.

Papers

new Consequences of the existence of exceptional collections in arithmetic and rationality, (with Matthew Ballard, Alexander Duncan, and Patrick McFaddin).
Abstract. A well-known conjecture of Orlov asks whether the existence of a full exceptional collection implies rationality of the underlying variety. We prove this conjecture for arithmetic toric varieties over general fields. We also investigate a slight generalization of this conjecture, where the endomorphism algebras of the exceptional objects are allowed to be separable field extensions of the base field. We show this generalization is false by exhibiting a smooth, projective threefold over the the field of rational numbers that possesses a full etale-exceptional collection but not a rational point. The counterexample comes from twisting a non-retract rational variety with a rational point and full etale-exceptional collection by a torsor that is invisible to Brauer invariants. Along the way, we develop some tools for linearizing objects, including a group that controls linearizations.
new A transcendental Brauer-Manin obstruction to weak approximation on a Calabi-Yau threefold, (with Sachi Hashimoto, Katrina Honigs, and Isabel Vogt, and an appendix by Nick Addington). Submitted.
Abstract. In this paper we investigate the \(\mathbb{Q}\)-rational points of a class of simply connected Calabi-Yau threefolds, originally studied by Hosono and Takagi in the context of mirror symmetry. These varieties are defined as a linear section of a double quintic symmetroid; their points correspond to rulings on quadric hypersurfaces. They come equipped with a natural 2-torsion Brauer class. Our main result shows that under certain conditions, this Brauer class gives rise to a transcendental Brauer-Manin obstruction to weak approximation. Hosono and Takagi also showed that over \(\mathbb{C}\) each of these Calabi-Yau threefolds \(Y\) is derived equivalent to a Reye congruence Calabi-Yau threefold \(X\). We show that these derived equivalences may also be constructed over \(\mathbb{Q}\) and give sufficient conditions for \(X\) to not satisfy weak approximation. In the appendix, N. Addington exhibits the Brauer groups of each class of Calabi-Yau variety over \(\mathbb{C}\).
Separable Algebras and Coflasque Resolutions, (with Matthew Ballard, Alexander Duncan, and Patrick McFaddin.) Submitted.
Abstract. Over a non-closed field, it is a common strategy to use separable algebras as invariants to distinguish algebraic and geometric objects. The most famous example is the deep connection between Severi-Brauer varieties and central simple algebras. For more general varieties, one might use endomorphism algebras of line bundles, of indecomposable vector bundles, or of exceptional objects in their derived categories.
Using Galois cohomology, we describe a new invariant of reductive algebraic groups that captures precisely when this strategy will fail. Our main result characterizes this invariant in terms of coflasque resolutions of linear algebraic groups introduced by Colliot-Thélène. We determine whether or not this invariant is trivial for many fields. For number fields, we show it agrees with the Tate-Shafarevich group of the linear algebraic group, up to behavior at real places.
Characterizing Finite Groups Using the Sum of the Orders of the Elements, (with Joshua Harrington and Lenny Jones). Int. J. Comb., vol. 2014.
Abstract. We give characterizations of various infinite sets of finite groups \(G\) under the assumption that \(G\) and the subgroups \(H\) of \(G\) satisfy certain properties involving the sum of the orders of the elements of \(G\) and \(H\). Additionally, we investigate the possible values for the sum of the orders of the elements of \(G\).
Representing Integers as the Sum of Two Squares in the Ring Zn, (with Joshua Harrington and Lenny Jones). J. Integer Seq. 17 (2014), no. 7, article 14.7.4.
Abstract. A classical theorem in number theory due to Euler states that a positive integer \(z\) can be written as the sum of two squares if and only if all prime factors \(q\) of \(z\), with \(q\equiv 3\pmod 4\), have even exponent in the prime factorization of \(z\). One can consider a variation of this theorem by not allowing the use of zero as a summand in the representation of \(z\) as the sum of two squares. Viewing each of these questions in \(\mathbb{Z}/n\mathbb{Z}\), the ring of integers modulo \(n\), we give a characterization of all integers \(n\geq 2\) such that every \(z\in\mathbb{Z}/n\mathbb{Z}\) can be written as the sum of two squares in \(\mathbb{Z}/n\mathbb{Z}\).
Generating d-Composite Sandwich Numbers, (with Lenny Jones). INT. 15A. (2015).
Abstract. Let \(d\in \mathcal{D} = \{1,\dots, 9\}\) and let \(k\) a positive integer with gcd(\(k,10d\))=1. Define a sequence \(\{s_n(k,d)\}^\infty_{n=1}\) by
\(s_n(k,d) := k dd\dots dk.\)
We say that \(k\) is a \(d\)-composite sandwich number if \(s_n(k,d)\) is composite for all \(n\geq 1\). For a \(d\)-composite sandwich number \(k\), we say \(k\) is trivial if \(s_n(k,d)\) is divisible by the same prime for all \(n\geq 1\), and nontrivial otherwise. In this paper, we develop a simple criterion to determine when a \(d\)-composite sandwich number is nontrivial, and we use it to establish many results concerning which types of integers can be \(d\)-composite sandwich numbers. For example, we proce that there exist infinitely many primes that are simultaneously trivial \(d\)-composite sandwich numbers for all \(d\in\mathcal{D}\). We also show that there exist infinitely many positive integers that are simultaneously nontrivial \(d\)-composite sandwich numbers for all \(d\in D\), where \(D\subset \mathcal{D}\) with \(|D| = 4\) and \(D\neq\{3,6,7,9\}\).
The irreducibility of polynomials related to a question of Schur, (with Lenny Jones). Involve 9 (2016), no. 3, 453-464.
Abstract. In 1908, Schur raised the question of the irreducibility over \(\mathbb{Q}\) of polynomials of the form \(f(x) = (x+a_1)(x+a_2)\cdots(x+a_m) + c\), where the \(a_i\) are distinct integers and \(c\in \{-1,1\}.\) Since then, many authors have addressed variations and generalizations of this question. In this article, we investigate the irreducibility of \(f(x)\) and \(f(x^2)\), where the integers \(a_i\) are consecutive terms of an arithmetic progression and \(c\) is a nonzero integer.
A Problem Related to a Conjecture of Polignac, (with Kellie Bresz, Lenny Jones, and Maria Markovich). INT. 16. (2016).
Abstract. In 1849, Polignac conjectured that every odd positive integer is of the form \(2^n + p\) for some integer \(n\geq 0\) and prime \(p\). Then, in 1950, Erdos provided infinitely many counterexamples to Polignac's conjecture. More recently, in 2012 the second author showed that there are infinitely many positive integers that are not of the form \(F_n + p\) or \(F_n-p\), where \(F_n\) denotes the \(n\)th Fibonacci number and \(p\) is prime. In this article, we consider a fusion of these problems and show that there exist infinitely many positive integers which cannot be written as \(2^n + F_n\pm p\). Additionally, we look at various results which follow from the main theorem concerning the construction of composite sequences.


Talks

Caution: In most cases these are personal notes that I wrote for talks, there are likely errors in them.

Derived Categories, Arithmetic, and Rationality Questions
Algebraic Geometry and Number Theory Seminar @ Rice University.
Fall 2019.
Abstract. When trying to apply the machinery of derived categories in an arithmetic setting, a natural question is the following: for a smooth projective variety \(X\), to what extent can \(D^b(X)\) be used as an invariant to answer rationality questions? In particular, what properties of \(D^b(X)\) are implied by \(X\) being rational, stably rational, or having a rational point? On the other hand, is there a property of \(D^b(X)\) that implies that \(X\) is rational, stably rational, or has a rational point?
In this talk, we will examine a family of arithmetic toric varieties for which a member is rational if and only if its bounded derived category of coherent sheaves admits a full etale exceptional collection. Additionally, we will discuss the behavior of the derived category under twisting by a torsor, which is joint work with Matthew Ballard, Alexander Duncan, and Patrick McFaddin.
Exceptional collections of toric varieties associated to root systems
Algebraic Geometry Seminar @ USC. Fall 2018.
Abstract. Given a root system \(R\), one can construct a toric variety \(X(R)\) by taking the maximal cones of \(X(R)\) to be the Weyl chambers of \(R\). The automorphisms of \(R\) act on \(X(R)\), and a natural question arises: can one decompose the derived category of coherent sheaves on \(X(R)\) in a manner that is respected by Aut(\(R\))? Recently, Castravet and Tevelev constructed full exceptional collections for \(D^b(X(R))\) when \(R\) is of type \(A_n\). In this talk, we discuss progress towards answering this question in the case where \(R\) is of type \(D_n\), with emphasis on the interesting case of \(R=D_4\).
Examples of Spherical Varieties
Spherical Varieties Seminar @ USC. Summer 2018.
Abstract. We discuss various examples of spherical, horospherical, and wonderful varieties, as well as relatied definitions and theorems that have not yet been introduced in the seminar.
An introduction to Algebraic Groups
Spherical Varieties Seminar @ USC. Summer 2018.
Abstract. We discuss basic facts about Algebraic Groups and their associated Lie algebras.
Pure motives as a universal cohomology theory
Motives @ South Carolina. Spring 2018.
Abstract. In this talk, we attempt to answer the following questions: Why did we build the category of pure motives over \(k\)? What is a Weil cohomology theory? Does the category of pure motives over \(k\) give us what we want?
For more information on this seminar, see MaSC, organized by Patrick McFaddin.

Notes

Coming soon.