Emma Pickard, University of Kentucky

• Title: Degeneracy Loci Dimensions from Permutation Diagrams
• Abstract: Any permutation $\sigma \in S_{n}$ determines $n^{2}$ rank conditions on any $n \times n$ matrix. In doing so, $\sigma$ determines a degeneracy locus for a flagged vector bundle on a variety by imposing rank conditions on each fiber. The dimension of such loci and values of the rank can be obtained from diagrams constructed based on only the permutation itself. We will briefly discuss previously established versions of these diagrams, their connections to varieties we know and love, and discuss additions to our collection of types of diagrams.

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: TBA
• Abstract: TBA