GSS – Fall 2024

Abstract: In their 1984 work, Ash, Grayson, and Green used the cohomology of congruence subgroups of SL(3, Z) to identify cuspidal automorphic forms and determine local factors for their associated L-functions. In this talk, we will first provide an introduction to modular forms to motivate the intuition behind higher dimensional automorphic forms. Then, we will describe an extension of the Ash-Grayson-Green method that allows for more efficient computation. Finally, we will present data obtained using this method, which has allowed us to check all prime levels less than 2000 (previously computed for prime level less than 340).

Abstract: We are interested in studying the common properties that finite Von Neumann regular rings have and come up with an axiomatization for the first order theory for the class of finite cVNr rings. In order to do this, we need to study the infinite models of this theory a.k.a. pseudofinite cVNr rings.

Abstract: Let $\psi$ be a function from the natural numbers to the real numbers. Given $\gamma \in \mathbb{R}^n$, consider the inequality $$|qX-p-\gamma|<\psi(|q|),$$ where $|q|$ denotes the max norm, $X=(x_{ij})$ is a real $m \times n$ matrix and $(p,q) \in \mathbb{Z}^n \times \mathbb{Z}^n$. Let $W(\psi,\gamma)$ be the set of $X$ such that there are infinitely many $(p,q)$ satisfying the inequality. The Inhomogeneous Khintchine-Groshev theorem tells us that if $\psi$ is non-increasing, then the Lebesgue measure of $W(\psi,\gamma)$ is either zero or one depending on whether $\sum_{q=1}^\infty q^{m-1}\psi^n(q)$ converges or diverges. In this talk, we will build up this theorem and explore problems associated with this theorem.

Abstract: After a brief introduction to classification theory, we define the PM$_k$ hierarchy (as a natural weakening of positive maximality), and review some results about the classes of theories that it defines. Next, going through some central notions and results in model theory, we describe a plan to establish the robustness of these “dividing lines”.

Abstract: Most of us are familiar with the fundamental group as a tool in topology. We will define the higher homotopy groups of a space $X$ (a generalization of the fundamental group) to be the groups of homotopy classes of maps from higher dimensional spheres into $X$. While higher homotopy groups have many of the same properties as fundamental groups, they come with some interesting new properties, as well. We will consider these properties, the relationship between higher homotopy groups and homology groups, and give some examples of spaces whose homotopy and homology groups are widely different.

Abstract: Given a real vector space $V$ of finite dimension endowed with a positive definite symmetric linear form we can define a reflection with respect to any nonzero vector on $V$. The root system is then a finite set which is invariant under the set of reflections with respect to itself. The same principle applies to a finitely generated free $\mathbb{Z}$-module $L$ in which the roots are defined as the set of norm 1 or 2 vectors.

Abstract: Previously, we introduced the notion of a corridor in a van Kampen diagram, motivated by mapping tori. In the pursuit of understanding mapping tori of right-angled Artin groups, we’ll broadly sketch how corridors are used in the study of free-abelian-by-cyclic groups, free-by-cyclic groups, and central extensions.

Abstract: Given a disjoint union of Euclidean unit cubes of various dimensions, one can construct a cube complex by identifying faces of cubes via isometries. After defining cube complexes, we will learn what it means for a space to be CAT(0) (a generalization of non-positive curvature). Then, we will determine which cube complexes are CAT(0) and outline the conditions they must satisfy. If time permits, we will investigate the nature of geodesics in CAT(0) cube complexes.

Abstract: Consider the following optimization problem: Given a group, is there an optimal space that acts on? Minimal volume entropy is a number which provides an answer to this question. Volume entropy captures the exponential growth rate of a space, and among all trees with exponential growth that admit a cocompact group action, the trivalent tree achieves the smallest volume entropy. We completely characterize which virtually free groups act geometrically on the trivalent tree, and as a result can calculate the minimal volume entropy for such groups. We will discuss the characterization and, as application, will show that every virtually free right-angled Coxeter group acts geometrically on the trivalent tree.

Abstract: Given a space and a homeomorphism from that space to itself, we can construct a new space, called the mapping torus. Mapping tori and their fundamental groups have proven to be interesting classes of spaces and groups, respectively. We will walk through the construction and see how fundamental groups of mapping tori lead to a natural tool of study in Dehn functions, called corridors.

Abstract: The classical Livšic Theorem for smooth, transitive Anosov flows states that any Hölder continuous function which integrates to zero over all closed orbits of the flow must be a derivative. Lopes and Thieullen generalized this by showing that if the periodic integrals of a Hölder function are non-negative, the function can be bounded below by a derivative. We extend this result to the setting of geodesic flows on locally CAT(-1) spaces. This allows us to prove a volume rigidity result for certain locally CAT(-1) spaces. This is joint work with Dave Constantine and Yandi Wu.

Abstract: The word prime is often associated with integers that are not the product of two smaller numbers. However, the notion of an element being prime can be defined for any commutative ring. When we extend the rational numbers, we can study the behavior of rational primes and any new factors they may acquire. In this talk, we’ll discuss the notions of number fields, rings of integers, and the prime ideals that they contain. We will also define what it means for a prime to split, ramify, or remain inert.

Abstract: Diophantine Approximation constitutes the study of approximating irrational numbers by rational numbers. If one is given an approximating function $\psi$, then one may define the set $W(\psi)$ to be the set of $\psi$-well approximable numbers. Khintchine’s Theorem states that as long as $\psi$ is monotonic, then the Lebesgue measure of $W(\psi)$ will either be zero or full, depending on whether $\sum_{q=1}^{\infty} \psi(q)$ converges or diverges. In this talk we will explore the beginning of Diophantine Approximation as well as the proof of Khintchine’s Theorem.