Alex Haro did his PhD thesis under the supervision of professor Carles Simó (october 1998). 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. to whom has been visiting at Austin and last years at the Georgia Institute of Technology (Atlanta). He got a permanent position at UB in july 2001. He has been regularly visiting Rafael de la Llave at Austin and last years at the Georgia Institute of Technology (Atlanta). He have been member of the program "Hamiltonian systems, from topology to applications through analysis" at MSRI (Berkeley), during the second half of 2018.

The research interests range the area of Dynamical Systems, from rigorous results to applications, including computations, with special interest in invariant manifolds such as invariant tori and their whiskers, and KAM theory. Several examples of his research are: (1) The development 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. (2) The discovery of new mechanisms of breakdown of invariant tori (with Rafael de la Llave), that lead to several conjectures that have been studied succesfully by researchers such as Kristian Bjerklöv, Masha Saprykina, Jordi-Lluís Figueras and Thomas Ohlson Timoudas. (3) The proof of a Thouless formula for long-range Schrödinger quasi-periodic skew products (with Joaquim Puig), relating the sum of the positive Lyapunov exponents and the logarithmic potential associated with the density of states of the corresponding operator, and concluding relations between the Lyapunov exponents of long-range Schrödinger quasi-periodic skew products and their duals. (4) The introduction 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. This work was awarded with two international prizes: The Barcelona Dynamical Systems Prize (2017) and the R.E. Moore Prize for Applications of Interval Analysis (2018). In collaboration with Marta Canadell, Jordi-Lluís Figueras, Alejandro Luque and Josep Maria Mondelo, Alex Haro published the book "The parameterization method for invariant manifolds: from rigorous results to effective computations" (2016), the series volume 195 of Springer Verlag's Applied Mathematical Sciences. Webgrec:https://webgrec.ub.edu/webpages/000006/cat/alex.haro.ub.edu.html

Grup de recerca

- Dynamical Systems Group

Línies de recerca

- Invariant manifolds

- KAM theory

- Sistemes Dinàmics

- Sistemes dinàmics computacionals

Activitats docents

- Equacions diferencials

- Sistemes Dinàmics