arXiv subjects

Mehrzad Ajoodanian

Publications and source records attributed to Mehrzad Ajoodanian.

4 recordsLinked to original sources

arXiv2026

A Non-Abelian Approach to Riemann Surfaces

We develop a non-abelian, gauge-theoretic framework for the Schwarzian derivative and for second-order differential equations on Riemann surfaces. As applications, we extend Dedekind's Schwarzian approach to elliptic periods to generic one-parameter families of curves of genus $g$ by replacing the non-canonical scalar Picard--Fuchs equation of order $2g$ with a canonical second-order equation with $g\times g$ matrix coefficients on the Hodge bundle. In higher dimensions, we discuss periods of a one-parameter family of cubic threefolds via the intermediate Jacobian. Finally, we discuss mass--spring systems in mechanics as a natural testing ground for the non-abelian Schwarzian viewpoint.

math.AG↗

arXiv2025

A Non-Abelian Approach to Riemann Surfaces Part I: Wronskian Geometry

We study projectively flat holomorphic vector bundles over Riemann surfaces. To each such bundle, we naturally assign a Wronskian line bundle. The main idea is a notion of the division of two meromorphic sections. Abel's identity is interpreted as the first Chern class of the Wronskian line bundle.

math.AG↗

arXiv2025

On Abel's Identity

We provide a natural duality that matches, in reverse order, the coefficients of the characteristic polynomial of the Maurer-Cartan of the Wronskian matrix with the coefficients of the original differential equation. Abel's identity is recovered as a corollary.

math.CO↗

arXiv2018

Gauge theory and foliations I; germ cords versus quantum cords

We apply gauge theory to study the space $F_k(M)$ of smooth codimension-$k$ framed foliations on a smooth manifold $M$. The quotient of Maurer-Cartan elements by the action of an infinite dimensional non-abelian gauge groupoid forms a moduli space, which contains $F_k(M)$ as a subspace. The notion of holonomy is naturally extended to this moduli space and the cohomology theory associated with points of this moduli space which correspond to non-singular foliations coincides with Bott cohomology. The quotient of the moduli space under concordance is identified as the space of homotopy classes of maps to the classifying spaces $BΓ^g_k$ and $BΓ^q_k$. While $BΓ^g$ is a classic and has been studied since Haefliger, $BΓ^q$ (which is a quotient of $BΓ^g$) carries a simpler topology and offers a rival theory.

math.DG↗