Abstract. In preparation.

Abstract. In preparation.

Abstract. In preparation.

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

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, 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.

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.

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 \(R=D_4\).

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.

Abstract. We discuss basic facts about Algebraic Groups and their associated Lie algebras.

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.