Previous Talks (with notes if applicable)
Oct 12 Finn Goss
Title: Density of visible lattice points
Abstract: A lattice point is said to be visible if the gcd of its coordinates is 1. The distribution of such points in an n-dimensional lattice is a classical problem in the geometry of numbers and their natural density is known to be 1/ζ(n), with ζ referring to the Riemann zeta function. In this talk, we investigate how this density varies over the affine integral subspaces of ℤⁿ. In particular, we use elementary sieve methods to compute a general density formula related to Jordan's totient function.
Oct 5 Illia Hayes
Title: Discrete Curvature on Piecewise Flat Manifolds
Abstract: A piecewise flat manifold is a triangulated manifold given a geometric structure by a choice of edge lengths that determine non-degenerate Euclidean simplices. We will discuss one definition of discrete curvature on surfaces based on vertex angle deficits, and give an application to circle packings of maximal planar graphs. We then develop a notion of curvature in the general n dimensional case and establish its relationship with the scalar curvature measure of a Riemannian metric.
Sep 28 Tyler Lendman
Title: Vector-Valued Functions, Bochner Spaces, and Kinetic Equations
Abstract: Many problems in analysis and partial differential equations naturally lead to functions whose values are taken in Banach spaces. This perspective motivates the study of vector-valued functions and Bochner spaces, which provide a framework for integrating and analyzing these types of functions. This talk will begin with an introduction to vector-valued functions, including strong measurability, Bochner integration, and the spaces, Lᵖ(0,T;X). We will develop these ideas through concrete examples, emphasizing the connection between familiar scalar-valued function spaces and their Banach-valued counterparts. In particular, we will view a time-dependent function u(x,t) as a curve t ↦ u(⋅, t) in an appropriate function space. The second half of the talk will introduce linear kinetic transport equations and illustrate how this functional-analytic framework arises naturally in their study. We will discuss how Bochner-space estimates, weak convergence, and compactness provide tools for studying families of solutions. The goal is to give a gentle introduction to the role of vector-valued analysis in kinetic theory, with particular emphasis on ideas accessible to graduate students who may not have a background in functional analysis or partial differential equations.
Sep 21 Bhargav Kale
Title: A Brief Introduction to the Philosophy of Mathematics
Abstract: Positioning Theory is a discourse analysis methodology that conceptualizes social interaction as a dynamic, discursive process in which participants continuously negotiate positions/positioning, storylines, and speech acts, which together shape meaning, power, and participation within specific social episodes (Davies & Harre, 1999). Researchers in mathematics education have largely used Positioning Theory to understand how classroom interactions influence teachers’ positioning of their students (e.g., Herbel-Eisenmann & Wagner, 2010) and students' positioning of each other (e.g. Wood, 2013) that can (re)-create traditional mathematics logics. As teachers may enact these logics, they can engage in racialized and gendered deficit positioning that marginalize minoritized students in mathematics spaces in the US (e.g. Yamakawa et al., 2009). While some research has tried to understand how supportive classroom interactions can disrupt these logics (e.g. Vaidya & Battey, 2022), none of them have incorporated positioning theory as an analytical tool. In this talk, I will give a brief introduction of positioning theory and how it can be used as a tool to understand teaching practices that support Dalit and other caste-marginalized students in higher education spaces in India to form positive mathematical identities. Using literature on experiences of Dalit students and faculty in HE spaces, I will explain how discourses of systemic casteism in India engages in deficit positioning of Dalit students and faculty in mathematics (e.g. Sadafule & Berntsen, 2017) and higher education spaces (e.g. Padalia, 2024) that diminishes learning, teaching and research opportunities for several Dalit students and faculty (Subramanian & Visawanathan, 2023).
Sep 14 Charlie Moll
Title: A Brief Introduction to the Philosophy of Mathematics
Abstract: In this talk, I will lay out the basic positions/debates in the philosophy of mathematics (Platonism, Nominalism, Intuitionism, and Structuralism) with an emphasis on the ontology of mathematical objects. We will also look at how some of these philosophical positions motivate alternative foundations of mathematics. Throughout all of this, we will need some basic notions from mathematical logic which will be given an informal treatment in the interest of time.
Aug 31 James Lee
Title: p-invariants of Abelian varieties (and maybe the Torelli locus)
Abstract: Abelian varieties are higher-dimensional analogues of elliptic curves and play a central role in modern algebraic geometry and number theory. In layman's terms, they are projective varieties that have a group structure. When working over a field of positive characteristic p, the geometry of an abelian variety is influenced in interesting ways by the prime p. It is analogous to how fields of characteristic p > 0 admit inseparable field extensions and have a lot of exotic phenomena. Consequently, there is a class of invariants, called p-invariants, that can be assigned to abelian varieties over characteristic p, and a current research topic is studying which invariants are possible. In particular, one can look at the stratifications induced by the invariants and study whether they have nice geometric properties such as irreducibility, smoothness, etc.