Subject Tags

Differential geometry studies smooth manifolds, tensor fields, connections, and curvature using the tools of calculus and linear algebra. It provides the language for modern physics: general relativity lives on pseudo-Riemannian manifolds, gauge theories on principal fiber bundles, and fluid dynamics on Riemannian three-manifolds. My interest centers on how geometric structure constrains analytic behavior — when curvature forces singularities, when topology obstructs global constructions, and how coordinate-free formulations reveal invariances that coordinate-based methods obscure.

Rust is my primary language for systems that must be both fast and correct. I use it for the Polybius trading engine: async Tokio runtime, zero-copy WebSocket feeds, and a DuckDB-backed ledger.

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.

My approach to stochastic modeling is grounded in non-stationary SDE theory: the view that financial price dynamics are better described by a changing diffusion manifold than by a fixed parametric model. The Polybius engine models price dynamics as probability mass flow under a modified Fokker-Planck operator, using a Nyström diffusion map to track the evolving market manifold and random matrix theory to classify execution regimes in real time.

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.

Market microstructure is the study of how prices form from the mechanics of trading: order flow, liquidity, adverse selection, and information asymmetry. My engineering work on Polybius applies Kyle's lambda as a real-time adverse-selection estimator, and my lead-lag study measures sub-millisecond price discovery between Bitcoin and Ethereum on Binance spot.

Quantitative research, to me, means building models that are mathematically honest about their assumptions and empirically validated before deployment. The Polybius project applies this discipline to prediction market trading: cross-validated calibration, out-of-sample backtesting, and a live deployment with hard risk limits before any capital is at stake.

Numerical analysis studies the design, convergence, and stability of algorithms that approximate continuous mathematical objects with finite, computable structures. Classical methods (finite differences, finite elements, spectral methods) each preserve fragments of the underlying geometry by accident. Discrete exterior calculus (DEC) preserves it by construction, discretising differential forms on simplicial complexes so that topological identities like Stokes' theorem hold exactly at the discrete level. My work uses DEC through the cartan library for applications in fluid dynamics, electromagnetics, and quantum mechanics.

Active matter systems convert stored energy into mechanical work at the microscale, producing collective dynamics that have no equilibrium analogue. My work on confined active nematics focuses on how defect-vortex coupling drives spontaneous chaotic mixing, an emergent transport mechanism arising purely from the internal stress of the active fluid.

Gauge theory is the mathematical framework underlying all fundamental forces in the Standard Model. Its central object is a principal fiber bundle over spacetime with a Lie group as structure group; the gauge field is a connection on that bundle, and the physical field strength is its curvature. Electromagnetism is a U(1) gauge theory, the weak force is SU(2), and the strong force is SU(3). The Yang–Mills equations generalize Maxwell's equations to non-abelian gauge groups, and their quantum mass gap (whether pure SU(2) Yang–Mills on R⁴ has a positive lowest energy excitation) remains one of the Clay Millennium Prize Problems.

Computational biology tools where correctness and performance matter. inferCNAsc uses a Rust core for copy number variation and ascertainment bias inference, exposed to the single-cell genomics ecosystem via PyO3.

General relativity is Einstein's geometric theory of gravitation, in which spacetime is a four-dimensional pseudo-Riemannian manifold and the gravitational field is encoded in the metric tensor. The Einstein field equations relate the Einstein curvature tensor to the stress-energy content of spacetime, with the cosmological constant accounting for a uniform vacuum energy density. Its empirical successes span Mercury's perihelion precession, gravitational lensing, GPS clock correction, Shapiro time delay, gravitational waves from binary inspirals, and the shadow of M87*; its open problems include reconciliation with quantum theory, the singularity theorems of Penrose and Hawking, and the cosmic-censorship conjecture.

Black holes are general-relativistic solutions with an event horizon, a causal boundary from which no future-directed null geodesic reaches future null infinity. The canonical families are Schwarzschild (mass only), Reissner-Nordström (mass and charge), Kerr (mass and angular momentum), and Kerr-Newman (all three), with the no-hair theorem restricting stationary vacuum solutions to these parameters alone. Key structural results include the area theorem, Hawking radiation as a quantum consequence of the event horizon, the laws of black-hole thermodynamics, and the Penrose process for extracting rotational energy from the ergosphere of a Kerr black hole. The information paradox, concerning whether quantum information is preserved through gravitational collapse and evaporation, remains one of the deepest open questions in theoretical physics.

String theory replaces the point particles of quantum field theory with one-dimensional extended objects whose quantised vibrational modes reproduce the known spectrum of matter and force carriers, including a massless spin-two state identified with the graviton. Consistency at the perturbative level requires supersymmetry and either ten spacetime dimensions (superstring theories) or eleven (M-theory, which unifies the five ten-dimensional superstring theories through a web of dualities). Central structural results include the AdS/CFT correspondence, which identifies a gravitational theory on an anti-de Sitter background with a conformal gauge theory on its boundary, the Kawai-Lewellen-Tye relations expressing graviton amplitudes as squares of gauge amplitudes, and the realisation that brane configurations in higher dimensions can reproduce four-dimensional Standard Model physics in low-energy effective descriptions such as Randall-Sundrum.