Tag
Geometric Analysis
Geometric analysis sits at the intersection of differential geometry and the analytic theory of PDEs. Much of my research studies how geometric constraints (confinement geometry, curvature, topology) shape the dynamics of physical fields. The active nematics work is the clearest example: topological defect trajectories encode braid-group invariants of the underlying flow.
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 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.