E-mail alex.haro@ub.edu
Position Associate Professor
Haro Provinciale, Alejandro

Àlex Haro is an assistant professor at the Departament de Matemàtiques i Informàtica de la Universitat de Barcelona (UB), where he is the coordinator of the PhD program in Mathematics and Computer Science.

He did his thesis (October 1998) under the supervision of professor Carles Simó. Funded by the Fulbright program, he did a postdoctoral stay at the University of Texas at Austin, where he started a fruitful collaboration with professor Rafael de la Llave, now at Georgia Institute of Technology (Atlanta). He got a permanent position at UB in July 2001. He was a member of the program "Hamiltonian systems, from topology to applications through analysis" at MSRI (Berkeley), during the second half of 2018.

His work with Jordi-Lluís Figueras and Alejandro Luque on computer-assisted proofs in KAM theory was awarded two international prizes: The Barcelona Dynamical Systems Prize (2017) and the R.E. Moore Prize for Applications of Interval Analysis (2018).

Other Research Interests


The research interests range in the area of Dynamical Systems, from rigorous results to applications, including computations and computer-assisted proofs, with a special interest in invariant manifolds such as invariant tori and their whiskers, and KAM theory. Applications spur his research.

Among his studies, one account:

  •  The development of the parameterization method for invariant tori and their whiskers for quasi-periodically forced systems, from rigorous results, algorithms, and implementations, (with Rafael de la Llave) to computer-assisted proofs (with Jordi-Lluís Figueras). The methods and results have been extended for normally hyperbolic invariant tori for general systems (with Marta Canadell).
  • The discovery of new mechanisms of the breakdown of invariant tori (with Rafael de la Llave), due to the bundle collapse of the invariant bundles and the sudden growth of the spectrum of transfer operators.
  • The introduction of a singularity theory for KAM tori (with Rafael de la Llave and Alejandra González), which involves tools from symplectic geometry and functional analysis, and leads to efficient algorithms of computation.
  • The analysis of the spectrum of long-range Schrödinger quasi-periodic skew-products (with Joaquim Puig), to obtain a Thouless formula relating the sum of the positive Lyapunov exponents and the logarithmic potential associated with the density of states of the corresponding operator.
  • A unified formulation (in book format) of the parameterization method for invariant manifolds (with Marta Canadell, Jordi-Lluís Figueras, Alejandro Luque, and Josep Maria Mondelo), which can be used to study different problems, such as invariant manifolds associated with fixed points, invariant tori in quasi-periodically forced systems, invariant tori in Hamiltonian systems, and normally hyperbolic invariant manifolds.
  • The application of new KAM strategies for constructing invariant tori (with Jordi-Lluís Figueras and Alejandro Luque), leading to new computational methods that produce stronger rigorous results and, in concrete examples, even almost optimal.
  • The design of new parameterization methods, based on flow maps, with an eye in the applications, especially in Celestial Mechanics and Astrodynamics (with Josep Maria Mondelo).
Selected publications

Haro, Àlex; de la Llave, Rafael. A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: rigorous results. J. Differential Equations 228 (2006), no. 2, 530–579.

Haro, Àlex; de la Llave, Rafael. Manifolds on the verge of a hyperbolicity breakdown. Chaos 16 (2006), no. 1, 013120, 8 pp.

González-Enríquez, Alejandra; Haro, Àlex; de la Llave, Rafael. Singularity theory for non-twist KAM tori. Mem. Amer. Math. Soc. 227 (2014), no. 1067, vi+115 pp. ISBN: 978-0-8218-9018-9

Haro, Àlex; Puig, Joaquim. A Thouless formula and Aubry duality for long-range Schrödinger skew-products. Nonlinearity 26 (2013), no. 5, 1163–1187.

Haro, Àlex; Canadell, Marta; Figueras, Jordi-Lluís; Luque, Alejandro; Mondelo, Josep-Maria. The parameterization method for invariant manifolds. From rigorous results to effective computations. Applied Mathematical Sciences, 195. Springer, [Cham], 2016. xvi+267 pp. ISBN: 978-3-319-29660-9; 978-3-319-29662-3

Figueras, Jordi-Lluís; Haro, Àlex; Luque, Alejandro. Rigorous computer-assisted application of KAM theory: a modern approach. Found. Comput. Math. 17 (2017), no. 5, 1123–1193.

Haro, Àlex; Mondelo, Josep Maria. Flow map parameterization methods for invariant tori in Hamiltonian systems. Commun. Nonlinear Sci. Numer. Simul. 101 (2021), Paper No. 105859, 34 pp.