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 fibre bundles, and fluid dynamics on Riemannian three-manifolds. My interest centres on how geometric structure constrains analytic behaviour: 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.

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.

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.

My approach to stochastic modelling 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.

Analysis is the study of limits, and of the structures that survive them: continuity, differentiation, integration, and convergence. Its modern form treats the derivative as a linear map rather than a slope, which is what allows it to move off the line and onto manifolds. Writing here builds analysis in the direction of differential forms and Stokes' theorem, where the fundamental theorem of calculus turns out to have been one case of a statement about boundaries.

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 explicit 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.

A differential equation relates a function to its rates of change, and solving one means finding the functions that satisfy that relation. The linear case is the one that yields completely: the solution set of a linear system is the kernel of a linear operator, its dimension is the order of the equation, and the matrix exponential writes every solution down at once. Writing here treats an ordinary differential equation as a statement about an operator on a function space, which is what makes superposition a theorem rather than a convenience. Partial differential equations are filed separately.

Geometric algebra extends vector algebra with a single associative geometric product, uv = u·v + u∧v, built from a real vector space and an inner product by imposing the contraction v² = g(v,v). The resulting algebra contains scalars, vectors, and higher-grade blades uniformly, and its rotors give a coordinate-free, singularity-free description of rotations that replaces the ad hoc matrix and cross-product machinery of classical vector algebra. Specialising to Cl(3,0) recovers ordinary three-dimensional space, and to Cl(1,3) recovers the spacetime algebra used in relativistic physics.

A Clifford algebra Cl(V,g) is the associative algebra generated by a vector space V and a quadratic form g, subject to the single relation v² = g(v,v) for every vector v. It is constructed as a quotient of the tensor algebra on V, has dimension 2ⁿ for an n-dimensional V, and contains the exterior algebra as its underlying vector space. Clifford algebras furnish the double covers Spin(n) of the rotation groups SO(n) and the spinor representations used throughout quantum mechanics and general relativity.

Gauge theory is the mathematical framework underlying all fundamental forces in the Standard Model. Its central object is a principal fibre 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 generalise 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.

Systems programming concerns the construction of software that manages hardware resources directly: memory allocators, OS kernels, device drivers, network stacks, and the runtimes on which higher-level programmes run. It requires precise control over memory layout, execution order, and the interaction between software and hardware. Rust occupies the frontier of this space, providing the resource control of C with memory safety guaranteed by the type system. Key topics include memory models (ownership, lifetimes, the borrow checker), async runtimes (Tokio, async-std), SIMD and GPU compute, and the FFI bridge to C and Python.

A manifold is a space that looks locally like a vector space while being permitted to bend globally, which is the setting in which calculus continues to work once flatness is given up. Charts supply the local coordinates, transition functions record how those coordinates disagree, and the objects worth naming are the ones no chart can see. Writing here builds manifolds as the home of differential forms and integration, and keeps the distinction between an object and its coordinate description in the foreground.

Topology studies the properties of a space that survive continuous deformation: connectivity, compactness, the number of holes. It supplies the vocabulary in which convergence and continuity are stated without reference to distance, and the invariants that distinguish spaces no coordinate calculation can separate. Writing here meets topology through integration on manifolds, where orientation, boundary and compact support decide which integrals are defined.

Riemannian geometry equips a manifold with a metric, and with it lengths, angles, geodesics and curvature. The metric is also what turns a gradient into a vector, since a derivative is naturally a covector and needs a metric before it can be pointed anywhere. Writing here uses Riemannian structure where curvature carries physical content, in configuration spaces, in homogenisation, and wherever a shortest path is the object of interest.

Physics supplies most of the questions that the mathematics on this site was built to answer, and a good deal of the intuition for the answers. The recurring theme is that a physical law is usually a statement about a geometric object, and that choosing coordinates early hides it. Writing here covers mechanics, gauge theory, relativity and gravitation, and prefers the formulation in which the symmetry is visible.

Classical mechanics describes the motion of bodies under Newton's laws, from a single point mass to a rigid body rotating in three dimensions, together with its variational reformulation through the Lagrangian and Hamiltonian formalisms. Central objects include momentum, angular momentum, the inertia tensor, and the action functional, whose stationary points give the equations of motion via the Euler-Lagrange equations. My interest is in the two computational languages available for the same physics, matrix and vector calculus alongside geometric algebra, which replaces Euler angles and the cross product with rotors and bivectors.

Linear algebra is the study of vector spaces and the linear maps between them. A matrix is a coordinate representation of such a map, and almost every computational subject in this site's orbit is linear algebra wearing a costume: the solution set of a linear system of ODEs, the conservation laws of a reaction network, the attention scores of a transformer. Writing here treats the maps as the objects and the matrices as their shadows.

Probability theory measures uncertainty, built on a measure space of total mass one, with random variables as measurable functions and expectation as an integral. That framing is what lets limit theorems be proved rather than asserted, and what connects probability to analysis. Writing here uses probability where it meets geometry and estimation, and where a distribution is treated as a point on a manifold rather than a table of numbers.

Mathematical biology reconstructs biochemical networks as systems of ODEs via mass-action kinetics: each reaction species is a state variable, each reaction is a rate function, and the stoichiometric matrix encodes which species participate in which reactions. These networks are the substrate from which compartmental pharmacokinetic models are built; drug molecules diffuse between compartments, bind to receptors, are metabolised by enzymes, each process following mass-action laws. Quantitative systems pharmacology extends this framework to multi-compartment, multi-target reaction networks where we ask which species affect measurable output and what controls that mapping.

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.

Computer vision asks two things of a scene: what it is made of, and what each piece is. The first is a question about representation, running from a pixel lattice through point sets and triangulated surfaces to quotient manifolds of shapes. The second is a question about inference, running from energy minimisation through learned message passing to metric learning. Writing here treats both as geometry and works from measured data: terrestrial lidar, photogrammetry, and the meshes built from them.

Concurrency is the study of programmes in which multiple computations proceed simultaneously and interact. The central difficulty is that the set of possible executions grows combinatorially with the number of interacting components: n independent actions admit n! interleavings, and a bug may appear in only one of them. Mathematical tools for taming this complexity include labelled transition systems, bisimulation equivalence, Mazurkiewicz trace theory, event structures, and the happens-before partial order. Correctness conditions (linearizability, sequential consistency, causal consistency) are predicates over the partial orders of execution events. Logical clocks (Lamport scalar, vector, hybrid) make causal structure observable at runtime.

Formal methods apply mathematical logic and algebra to the specification, verification, and synthesis of software and hardware systems. The core tools include type systems (operational semantics, substructural logics, dependent types), model checking (LTL, CTL, bisimulation), theorem proving (Lean 4, Coq, Isabelle), and process algebras (CCS, CSP, the pi-calculus). In systems programming, formal methods appear as memory safety proofs (RustBelt, Iris), verified concurrent data structures (linearizability proofs), and certified compilers (CompCert). In mathematics, the Lean 4 Mathlib library provides a machine-checked foundation for undergraduate through research-level results.

Dynamical systems theory studies how systems evolve under differential equations, focusing on trajectories, fixed points, stability under linearisation, and bifurcations where qualitative behaviour changes. Classical tools include Lyapunov functions, invariant manifolds, and geometric structures that determine long-term behaviour. Compartmental models in pharmacology are low-dimensional dynamical systems; their fixed points represent steady state concentrations, their linearisations reveal how perturbations decay, and their GENERIC structure encodes the dissipative geometry underlying reversible and irreversible processes.

Control theory studies input-state-output systems: how external signals drive state evolution, which states we can reach (controllability) and observe (observability), and whether the system remains bounded under perturbation (stability). These questions have definitive algebraic answers, independent of the domain. We apply control-theoretic machinery to pharmacological compartmental models, recognising that compartments are states, drug inputs are control signals, and measurements are outputs; the theory then tells us what is possible and what is unidentifiable at arbitrary scale.

Information geometry treats a family of probability distributions as a manifold, with the Fisher information as its metric. Statistical questions then become geometric ones: an estimator is a point, its efficiency is a curvature statement, and the divergence between two models is a distance that is not symmetric. Writing here uses it where estimation and geometry meet, and where the shape of a model family explains what the model can learn.

Homogenisation replaces a medium with fine-scale structure by an effective medium that reproduces its large-scale behaviour, and makes precise the sense in which the replacement is valid. The effective coefficients are determined by cell problems that the microstructure poses, which is the whole subtlety, since averaging the original coefficients gives the wrong answer. Writing here meets homogenisation through mesh pipelines and representative volume elements, where the microstructure is a geometric object before it is a coefficient.

The quaternions are a four-dimensional division algebra with three anticommuting square roots of minus one, discovered by Hamilton in 1843 while trying to multiply triples. They compose rotations of three-dimensional space without the degeneracies of Euler angles, which is why graphics and robotics keep them. Writing here derives them as the even subalgebra of the geometric algebra of space, where the three imaginaries are bivectors and the half-angle is a record that a rotation is two reflections.

Computational fluid dynamics solves the equations of motion of a fluid numerically, and most of the difficulty lies in the fact that the interesting regimes are the ones where the nonlinearity dominates. The choices that matter are the discretisation, the treatment of the stiff linear part, and the dealiasing, since each decides which structures the simulation is capable of representing. Writing here covers pseudospectral Navier-Stokes solvers, where spectral accuracy is available and the cost is paid in geometry and in memory bandwidth.

Statistics infers properties of a distribution from samples drawn out of it, and quantifies how far the inference can be trusted. The questions worth asking are geometric as often as computational: which estimators are efficient, what a confidence set is a set of, and how much a model family can express. Writing here treats statistics alongside information geometry and estimation theory, where the answers take the form of a bound rather than a procedure.

Estimation theory asks how well a parameter can be recovered from noisy observations, and answers with bounds rather than recipes. The Cramér-Rao bound sets a floor on the variance of any unbiased estimator in terms of the Fisher information, which makes the limit a property of the model rather than of the analyst. Writing here uses estimation theory where it touches information geometry and filtering, and where the bound is the interesting object.

Pharmacology studies how chemical substances interact with living systems to produce physiological effects. Historically, the discipline fragmented into pharmacokinetics (concentration over time), pharmacodynamics (effect versus concentration), and quantitative systems pharmacology (multi-scale networks), each developing its own language and traditions. We reconstruct these as one control-theoretic object, applying classical linear systems theory to compartmental models so that controllability, observability, and BIBO stability become universal properties built once and specialised at will, rather than rediscovered in each subdiscipline.

Type theory is a foundation for mathematics and a design discipline for programming languages at once, in which a proposition is a type and a proof is an inhabitant of it. Its practical consequence is that a sufficiently expressive type system rules out whole classes of program error before the program runs. Writing here uses type theory where it explains what a language guarantees, and where a borrow checker turns out to be a proof system.

The study of programming languages asks what a program means, independently of the machine that runs it, and answers with operational and denotational semantics. Getting that right is what makes a compiler optimisation safe and a type system sound. Writing here covers formal semantics with an emphasis on ownership and lifetimes, where the language's guarantees are theorems about the semantics rather than promises about the implementation.

Python is the language most scientific and machine-learning work is written in, which makes it the interface most numerical libraries must present regardless of what they are written in. Writing here treats Python mainly as a boundary: what a native extension has to do to be usable from it, how arrays cross that boundary without being copied, and where the cost of the convenience is paid.

A graphics processor executes thousands of threads in lockstep, which suits problems with regular structure and wide parallelism and suits little else. Programming one is largely a matter of arranging memory so that the arrangement matches how the hardware wants to read it. Writing here covers GPU compute for numerical work, where the algorithm and the memory layout are chosen together rather than in sequence.

CUDA is NVIDIA's programming model for its GPUs, exposing the memory hierarchy and thread grouping of the hardware directly rather than hiding them. That directness is why it remains the default for numerical computing, and why its performance characteristics reward understanding the hardware. Writing here covers CUDA from Rust, including kernel compilation and the toolkit boundaries that a portable build has to respect.

Vulkan is a low-level, explicit graphics and compute API, portable across vendors, which requires the programmer to state what a driver would otherwise infer. For compute work it offers a vendor-neutral alternative to CUDA at the cost of considerably more setup. Writing here uses Vulkan compute where portability across hardware matters more than reaching the last increment of performance on one vendor's cards.

Medicine, treated mathematically, is the study of a small number of coupled dynamical systems: drug disposition, receptor binding, disease progression, all governed by the same state-space language as any other controlled system. Pharmacokinetics, pharmacodynamics, and quantitative systems pharmacology are not three separate disciplines but one object read at different points on a single dimension/structure ladder. My work here treats compartmental models, target-mediated drug disposition, and network pharmacology through control theory and thermodynamically consistent (GENERIC) dynamics.

Data infrastructure is the layer that moves data between storage and computation, and its quality is measured in what it makes possible rather than in what it does. Query engines, columnar formats and object-store clients decide whether a dataset is workable or merely present. Writing here covers contributions to distributed data engines, where the interesting constraints are build size, dependency weight and the cost of a scan.

Typesetting turns a description of mathematics into something legible, and the description and the rendering disagree more often than either party admits. Font selection, alphabet commands and spacing are semantic choices as much as visual ones, since a bold symbol and an upright one mean different things. Writing here covers work on terminal and web maths renderers, and the LaTeX standards that keep documents consistent across projects.