Research
My research lies at the interface of dynamical systems, nonlinear analysis and scientific computing. I develop computer-assisted proofs: rigorous, constructive methods that turn numerical simulations into mathematical theorems about infinite dimensional dynamical systems arising from partial, delay and ordinary differential equations.
Partial differential equations
Computer-assisted proofs of steady states, periodic orbits, traveling waves and localized patterns in PDEs, including Navier–Stokes, Swift–Hohenberg and ill-posed problems.
Ordinary differential equations
Periodic orbits, invariant manifolds, connecting orbits and bifurcations in ODEs via the parameterization method, Chebyshev series and radii polynomials.
Delay differential equations
Periodic orbits, eigenvalues, unstable manifolds and Hopf bifurcations in delay equations, including work towards Jones’ conjecture for Wright’s equation.
Bifurcations in dynamical systems
Computer-assisted proofs of saddle-node, pitchfork, Hopf and cusp bifurcations in ODEs, PDEs and delay equations.
Topological methods
Combining computer-assisted proofs with Morse–Conley–Floer theory to obtain forcing results for chaos and connecting orbits.
The N-body problem
Choreographies, relative equilibria and periodic orbits in n-body and vortex problems, including a proof of Marchal’s conjecture.
Rigorous continuation
Validated one- and multi-parameter continuation of branches and manifolds of solutions, and global bifurcation diagrams.
Codes accompanying publications
Most computer-assisted proofs come with the code needed to reproduce them.
- Spatially inhomogeneous two-cycles in an integrodifference equation [80]
- From the Lagrange Triangle to the Figure Eight Choreography: Proof of Marchal’s Conjecture [79]
- From heteroclinic loops to homoclinic snaking in reversible systems: rigorous forcing through computer-assisted proofs [78]
- Recent advances about the rigorous integration of parabolic PDEs via fully spectral Fourier-Chebyshev expansions [77]
- Computer-Assisted Proofs of Solitons in Bose-Einstein Condensates [76]
- Periodic localized traveling waves in the two-dimensional suspension bridge equation [75]
- Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs [74]
- Determination of stable branches of relative equilibria of the N-vortex problem on the sphere [72]
- A rigorous integrator and global existence for higher-dimensional semilinear parabolic PDEs via semigroup theory [71]
- Stationary non-radial localized patterns in the planar Swift-Hohenberg PDE: constructive proofs of existence [70]
- Rigorous computation of solutions of semi-linear PDEs on unbounded domains via spectral methods [69]
- Towards computational Morse-Floer homology: forcing results for connecting orbits by computing relative indices of critical points [68]
- Worrisome properties of neural network controllers and their symbolic representations [65]
- Saddle-type blow-up solutions with computer-assisted proofs: validation and extraction of global nature [63]
- Periodic orbits in Hořava-Lifshitz cosmologies [62]
- Rigorous numerics for nonlinear heat equations in the complex plane of time [61]
- Rotation invariant patterns for a nonlinear Laplace-Beltrami equation: a Taylor-Chebyshev series approach [60]
- Global dynamics in nonconservative nonlinear Schrödinger equations [59]
- Singularities and heteroclinic connections in complex-valued evolutionary equations with a quadratic nonlinearity [56]
- Rigorous verification of Hopf bifurcations in functional differential equations [55]
- Parameterization of unstable manifolds for DDEs: formal series solutions and validated error bounds [53]
- A general method for computer-assisted proofs of periodic solutions in delay differential problems [52]
- Rigorous verification of Hopf bifurcations via desingularization and continuation [51]
- A functional analytic approach to validated numerics for eigenvalues of delay equations [45]
- Traveling wave oscillatory patterns in a signed Kuramoto-Sivashinsky equation with absorption [44]
- Continuation of solutions and studying delay differential equations via rigorous numerics [41]
- Polynomial interpolation and a priori bootstrap for computer-assisted proofs in nonlinear ODEs [38]
- Rigorous numerics for ill-posed PDEs: periodic orbits in the Boussinesq equation [37]
- Rigorous continuation of bifurcation points in the diblock copolymer equation [35]
- Rapidly and slowly oscillating periodic oscillations of a delayed van der Pol oscillator [34]
- Stability and uniqueness of slowly oscillating periodic solutions to Wright's equation [33]
- A posteriori verification of invariant objects of evolution equations: periodic orbits in the Kuramoto-Sivashinsky PDE [31]
- Computer assisted Fourier analysis in sequence spaces of varying regularity [30]
- Rigorous verification of saddle-node bifurcations in ODEs [29]
- Automatic differentiation for Fourier series and the radii polynomial approach [28]
- Computation of smooth manifolds via rigorous multi-parameter continuation in infinite dimensions [26]
- Computation of maximal local (un)stable manifold patches by the parameterization method [25]
- Stationary coexistence of hexagons and rolls via rigorous computations [21]