arXiv subjects

Matthew McMillan

Publications and source records attributed to Matthew McMillan.

5 recordsLinked to original sources

arXiv2026

Tooling for digital accessibility in mathematics: Quickly build compliant course websites that benefit all students

Public universities in the US must now meet digital accessibility (DA) standards under 2024 updates to Title II of the ADA. For math instructors, course materials must be screen-reader parsable, which standard LaTeX-to-PDF workflows historically do not achieve. Despite MathML's availability as a web standard for accessible math, instructor adoption of DA-compliant workflows remains very low, creating a gap between available technology and classroom practice. This paper makes three contributions. First, we present a taxonomy of existing approaches to DA-compliant math content, organized by print (PDF) versus web (HTML) output targets, analyzing tradeoffs for instructor adoption. Second, we describe a free workflow using Obsidian (Markdown-based content management), Quartz (static site generator), Git (collaboration and version control), and Cloudflare Pages (free hosting, private source files) that enables math instructors to create and publish DA-compliant course websites with MathML from TeX-based syntax. Setup takes approximately 1-2 hours; thereafter, site updates occur in minutes via a single command. A public setup tutorial is made available. Third, we present an empirical study of student outcomes across 31 sections of Calculus II over 6 semesters. Sections using the proposed system outperformed controls, with the treatment group reaching 2.46 standard deviations above the control mean in the final semester. Although all treatment sections were taught by one instructor, evidence such as acclimation trajectories of other new instructors suggests the system itself meaningfully contributes to performance gains. A student experience survey shows no statistically significant difference between groups, indicating no negative effect on experience. A proposed second study phase will assess barriers to adoption at other institutions.

math.HO↗

arXiv2024

Tensor $2$-Product for $\mathfrak{sl}_{2}$: Extensions to the Negative Half

In a recent paper, the author defined an operation of tensor product for a large class of $2$-representations of $\mathcal{U}^{+}$, the positive half of the $2$-category associated to $\mathfrak{sl}_{2}$. In this paper, we prove that the operation extends to give an operation of tensor product for $2$-representations of the full $2$-category $\mathcal{U}$: when the inputs are $2$-representations of the full $\mathcal{U}$, the $2$-product is also a $2$-representation of the full $\mathcal{U}$. As in the previous paper, the $2$-product is given for a simple $2$-representation $\mathcal{L}(1)$ and an abelian $2$-representation $\mathcal{V}$ taken from the $2$-category of algebras. This is the first construction of an operation of tensor product for higher representations of a full Lie algebra in the abelian setting.

math.RT↗

arXiv2023

A tensor 2-product of 2-representations of $\mathfrak{sl}_{2}^{+}$

We construct an explicit abelian model for the operation of tensor $2$-product of $2$-representations of $\mathfrak{sl}_{2}^{+}$, specifically the product of a simple $2$-representation $\mathcal{L}(1)$ with a given abelian $2$-representation $\mathcal{V}$ taken from the $2$-category of algebras. We study the case $\mathcal{V}=\mathcal{L}(1)$ in detail, and we show that the $2$-product in this case recovers the expected structure. Our construction partially verifies a conjecture of Rouquier from 2008.

math.RT↗

arXiv2014

On symplectic capacities of toric domains

A toric domain is a subset of $(\mathbb{C}^n,ω_{\text{std}})$ which is invariant under the standard rotation action of $\mathbb{T}^n$ on $\mathbb{C}^n$. For a toric domain $U$ from a certain large class for which this action is not free, we find a corresponding toric domain $V$ where the standard action is free, and for which $c(U)=c(V)$ for any symplectic capacity $c$. Michael Hutchings gives a combinatorial formula for calculating his embedded contact homology symplectic capacities for certain toric four-manifolds on which the $\mathbb{T}^2$-action is free. Our theorem allows one to extend this formula to a class of toric domains where the action is not free. We apply our theorem to compute ECH capacities for certain intersections of ellipsoids, and find that these capacities give sharp obstructions to symplectically embedding these ellipsoid intersections into balls.

math.SG↗

arXiv2013

Radial Forcing and Edgar Allan Poe's Lengthening Pendulum

Inspired by Edgar Allan Poe's The Pit and the Pendulum, we investigate a radially driven, lengthening pendulum. We first show that increasing the length of an undriven pendulum at a uniform rate does not amplify the oscillations in a manner consistent with the behavior of the scythe in Poe's story. We discuss parametric amplification and the transfer of energy (through the parameter of the pendulum's length) to the oscillating part of the system. In this manner radial driving may easily and intuitively be understood, and the fundamental concept applied in many other areas. We propose and show by a numerical model that appropriately timed radial forcing can increase the oscillation amplitude in a manner consistent with Poe's story. Our analysis contributes a computational exploration of the complex harmonic motion that can result from radially driving a pendulum, and sheds light on a mechanism by which oscillations can be amplified parametrically. These insights should prove especially valuable in the undergraduate physics classroom, where investigations into pendulums and oscillations are commonplace.

physics.class-ph↗