Subject Tags

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.

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.

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.

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.

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.

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.

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.

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.