Sumandeep Kaur, Shanghai University

• Title: On a conjecture of Lenny Jones about some monogenic polynomials
• Abstract: To know whether a monic irreducible polynomial is monogenic or not is one of the important problems in algebraic number theory. In an attempt to answer this problem for certain family of polynomials, L. Jones in [Bull. Aust. Math. Soc. 100 (2019), 239-244] conjectured that if $\gcd(n,mB)=1$ and $A$ is a prime number, then the polynomial $x^{n}+A(Bx+1)^{m} \in Z[x]$ with $n \ge 3$ and $1 \le m \le n-1$ is monogenic if and only if $n^n+(-1)^{n+m}B^n(n-m)^{n-m}m^mA$ is square-free. In this talk using Dedekind Criterion and classical results from algebraic number theory, we will see that this conjecture is true.

Alejandro Castillo, Universidad Nacional Autónoma de México

• Title: A Galois Theory for Algebras
• Abstract: An overview to a parallel of Galois Theory for algebras, and the analogue to the Fundamental Theorem.

Maddie Allen, Emory University

• Title: The Artin Springer Theorem for Algebras with Involution over Semi-Global Fields
• Abstract: Let $q$ be a quadratic form over a field $K$ and $L/K$ a field extension of odd degree, then it is a well-known result of Artin and Springer that $q$ is isotropic if and only if $q_L$ is anisotropic. More generally, it is an open question that if $(A,\sigma)$ is a central simple algebra with involution over $K$ and $L/K$ is a finite extension of degree relatively prime to $2\text{Ind} (A)$, then $(A,\sigma)$ is isotropic if and only if $(A,\sigma)_L$ is isotropic. Let $K$ be a complete discrete valued field with residue field $k$ such that $char(k) \neq 2$ and $F=K(X)$ the function field of a smooth, projective, geometrically integral curve $X$ over $K$. We give a positive answer to this question for any central simple algebra with involution over $K$ and for any central simple algebras with involution $(A,\sigma)$ over $F$ with $\text{Ind}(A)=2^n$ assuming a positive answer over finite extensions of $k$ and $k(t)$.

Chirag Singhal, University of Illinois at Chicago

• Title: Global Lifting of an infinite system of abelian varieties over finite field
• Abstract: Consider a number field $K$ and abelian varieties $A_P$ defined over finite fields $F_P$ for each prime $P$ outside a finite set of primes of the ring of integers $O_K$. When does there exist an abelian variety $A/K$ such that $A$ modulo $P$ is $F_P$-isogenous to $A_P$ for all such $P$? We prove that, under GRH and a mild growth condition on the minimal conductor among the partial matching classes up to primes of bounded norm, there exists a unique $K$-isogeny class that lifts the entire infinite system.

Hamza Abuel-Eneen, Tanta University

• Title: Arithmetic and Motivic Perspectives on $\mathrm{Spec}(\overline{\mathbb{Q}}(\zeta(3)))$
• Abstract: We investigate arithmetic and motivic aspects related to $\mathrm{Spec}\overline{\mathbb{Q}}(\zeta(3)))$, motivated by the role ofspecial values of zeta functions in number theory. The focus of this talk is on how the arithmetic structure of the field $\overline{\mathbb{Q}}(\zeta(3))$ is reflected in its spectrum and related categorical or geometric viewpoints. We present a concise observation illustrating connections between $\mathrm{Spec}(\overline{\mathbb{Q}}(\zeta(3)))$, motivic ideas, and Galois symmetries, highlighting how arithmetic information can manifest in geometric settings. This short talk aims to provide intuition and context rather than technical details, offering a glimpse into the broader themes addressed by the conference.

Anna Lowery, Rice University

• Title: Character Varieties of Knot and Link Complements
• Abstract: This talk will discuss the character varieties of hyperbolic 3-manifolds obtained from knot and link complements. The character variety is an affine algebraic set whose points correspond to characters of representations of the fundamental group of a manifold. Geometric and arithmetic information about the character variety often corresponds to topological information about the manifold, which makes them a useful object of study.