Tag
Partial Differential Equations
PDEs have been the primary mathematical language across all of my research. The incompressible Navier-Stokes system underlies the fluid mechanics work; the Q-tensor PDE governs defect dynamics in the active nematics paper. I am particularly interested in the interplay between geometric structure and long-time behaviour of solutions to evolution equations.
Research
Submitted to Proc. Nat. Acad. Sci. | arXiv:2503.10880 · April 2025
Chaos-generating periodic orbits of topological defects in confined active nematics
Brandon Klein, Alejandro J. Soto Franco, Md Mainul Hasan Sabbir, Matthew J. Deutsch, Ross Kliegman, Robin L. B. Selinger, Kevin A. Mitchell, Daniel A. Beller
Work in progress, v1 · 2025
Introduction to Incompressible Fluid Mechanics
Alejandro J. Soto Franco
Blog
March 25, 2026
Reservoir Geometry: Riemannian Manifolds in Oil and Gas
Darcy's law recast as geodesic flow on a Riemannian manifold, pressure diffusion as the Laplace-Beltrami equation, and permeability tensor interpolation via SPD geodesics, with no_std Rust for embedded well-site monitoring.
March 22, 2026
Maxwell's Equations and Gauge Theory: Electromagnetism as a Principal Bundle
Four languages for one theory: vector calculus, differential forms, spacetime algebra, and principal fiber bundles. From the classical field equations to gauge invariance, the Aharonov-Bohm effect, Yang-Mills theory, and Dirac monopoles.
March 19, 2026
Navier–Stokes: Derivation in ℝ³ and on a Riemannian Manifold
An end-to-end derivation of the incompressible Navier–Stokes equations from continuum mechanics axioms, geometric reformulation via differential forms, coordinate-free lift to a Riemannian manifold, the Millennium Prize problem, functional analysis, and geometric algebra.
March 17, 2026
Price as Geometry: Resolution, Coarse-Graining, and the Structure of Market Noise
A rigorous tour through stationary and non-stationary models of price evolution, with geometric analysis at the forefront. From the random walk null and Black-Scholes as flat geometry, through mean reversion as curved Riemannian diffusion, wavelets, geometric harmonics, and information geometry, anchored throughout by empirical evidence from BTC/ETH millisecond data.
March 12, 2026
Diffusion on Curved Spaces
From the Gaussian heat kernel on R^n to the Laplace-Beltrami operator on Riemannian manifolds, with the short-time heat kernel expansion and spectral theory.
March 10, 2026
Manifolds: The Language of Modern Geometry
A rigorous construction of smooth manifolds from first principles: charts, tangent spaces, Riemannian metrics, curvature tensors, and geometric flows.
March 10, 2026
Kuramoto: How Order Emerges from Chaos
Fireflies, neurons, power grids: all governed by the same equation. A tour through the Kuramoto model, its order parameter, and the phase transition that turns noise into rhythm.