Abstract. Suppose that \(X\) is an integral scheme (quasi-)projective over a complete local ring of mixed characteristic. Using ideas of Takamatsu-Yoshikawa and Bhatt-Ma-et. al, we define a notion of a +-test ideal on \(X\), including for divisors and linear series. We obtain global generation results in this setting that generalize the well known global generation results obtained via multiplier ideal sheaf techniques in characteristic \(0\) and via test ideals in characteristic \(p>0\). We also obtain applications to the non-nef locus in mixed characteristic.

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.

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.

Abstract. To aid departments in growing and supporting the potential of novice instructors, this paper shares a peer-mentoring program that has been implemented at three universities over the last five years for graduate student instructors (GSIs, master and doctoral) within mathematics departments. This paper provides frameworks, curricular topics, necessary conditions, methods of implementation, successes, and anticipated challenges from five years of research to help establish and sustain a peer-mentoring program for GSIs. Positive outcomes from this peer-mentoring program included improving support for Novices (resulting in anticipating teaching concerns), lowering the rate at which students received a D, F, or Withdrew (DFW), having fewer complaints and concerns, and sustaining stewardship.

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 \(\mathbb{C}\).

Abstract. This document is roughly divided into four chapters. The first outlines basic preliminary material, definitions, and foundational theorems required throughout the text. The second chapter, which is joint work with Dr. Matthew Ballard, gives an example of a family of Fano arithmetic toric varieties in which the derived category is able to detect the existence of \(k\)-rational points. More succinctly, we show that if \(X\) is a generalized del Pezzo variety defined over a field \(k\), then \(X\) contains a \(k\)-rational point (and is in fact \(k\)-rational, that is, birational to \(\mathbb{P}^n_k\) ) if and only if \(D^b(X)\) admits a full étale exceptional collection.

In the third chapter, which is joint work with Dr. Matthew Ballard, Dr. Alexander Duncan, and Dr. Patrick McFaddin, we describe, using techniques from Galois cohomology, 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. In addition, we completely describe the cohomological invariants of a reductive algebraic group of degree 2 with values in a special torus, which generalizes a result of Blinstein and Merkurjev.

In the final chapter, which is joint work with Dr. Matthew Ballard, Dr. Alexander Duncan, and Dr. Patrick McFaddin, we develop tools to understand the effect that twisting by a torsor has on the derived category. Applying these to the setting of arithmetic toric varieties reveals a surprising dichotomy in behavior split along the fault line of retract rationality. As a consequence of the general theory, we give a negative answer to a question of Bernardara and Bolognesi relating a categorical notion of dimension to rationality. Moreover, we show that a smooth projective toric variety over a field \(k\) possessing a full \(k\)-exceptional collection is automatically \(k\)-rational.

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 \(G\).

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 \(\mathbb{Z}/n\mathbb{Z}\).

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 prove 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 \(D\neq\{3,6,7,9\}\).

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.

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, Erdös 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.