arXiv subjects

Cigole Thomas

Publications and source records attributed to Cigole Thomas.

4 recordsLinked to original sources

arXiv2025

Gen AI in Proof-based Math Courses: A Pilot Study

With the rapid rise of generative AI in higher education, understanding how students use AI is increasingly important. This exploratory study examines student use and perceptions of generative AI across three proof-based undergraduate mathematics courses: a first-semester abstract algebra course, a topology course, and a second-semester abstract algebra course. In each case, course policy permitted some use of generative AI. Drawing on survey responses and student interviews, we analyze how students engaged with AI tools as well as their perceptions of generative AI's usefulness, accuracy and limitations.

cs.AI↗

arXiv2025

Asymptotic Dynamics on Character Varieties over Finite Fields

In this paper, we prove the lack of asymptotic transitivity of the outer automorphism group action of $\mathbb{Z}^r$ on $\mathrm{SL}_n(\mathbb{F}_q)$-character varieties of $\mathbb{Z}^r$ for $n=2,3$ and $r\geq 2$. Along the way, we stratify the character varieties and compute the $E$-polynomial, also known as the Hodge-Deligne polynomial or Serre polynomial, of these character varieties.

math.AG↗

arXiv2024

Dynamical Irreducibility of Certain Families of Polynomials over Finite Fields

We determine necessary and sufficient conditions for unicritical polynomials to be dynamically irreducible over finite fields. This result extends the results of Boston-Jones and Hamblen-Jones-Madhu regarding the dynamical irreducibility of particular families of unicritical polynomials. We also investigate dynamical irreducibility conditions for cubic and shifted linearized polynomials.

math.NT↗

arXiv2019

Decomposition of complex hyperbolic isometries by involutions

A $k$-reflection of the $n$-dimensional complex hyperbolic space ${\rm H}_{\C}^n$ is an element in ${\rm U}(n,1)$ with negative type eigenvalue $λ$, $|λ|=1$, of multiplicity $k+1$ and positive type eigenvalue $1$ of multiplicity $n-k$. We prove that a holomorphic isometry of ${\rm H}_{\C}^n$ is a product of at most four involutions and a complex $k$-reflection, $k \leq 2$. Along the way, we prove that every element in ${\rm SU}(n)$ is a product of four or five involutions according as $n \neq 2 \mod 4$ or $n = 2 \mod 4$. We also give an easy proof of the result that every holomorphic isometry of ${\rm H}_{\C}^n$ is a product of two anti-holomorphic involutions.

math.GT↗