Since September 2022 I am a Ramon y Cajal fellow at the Universitat de Barcelona in the department of Mathematics. I work in between the areas of number theory, geometry, dynamics, graph theory. The Fields Medalists Margulis and Lindenstrauss studied connections between ergodic theory and number theory in their breakthroughs proving the Oppenheim and the Littlewood conjectures respectively, I study a new connection between diophantine approximation and number theory. I obtained in 2017-2022 my own Ambizione Grant (517 120 CHF) from the Swiss National Foundation (SNF) to conduct my research at ETH Zurich and became a Senior Research Assistant for five years. Ambizione grants are aimed at young researchers who wish to conduct, manage and lead an independent project; they cover the grantee's salary and the funds needed to carry out the project. The novelty of my proposed approach is establishing bridges between modularity, diophantine approximation and hyperbolic geometry. My background in algebraic and analytic number theory goes back to my PhD (Paris VI, 2013, supervised by Don Zagier, director of the Max Planck Institute for Mathematics and professor at Collège de France at that time). I acquired my expertise in diophantine approximation during my first postdoctoral position, working in the EPSRC research project New Frameworks in Metric Number Theory, at the University of York, UK. The publications 1),2),6),7),8) are results of my SNF project, but also two more preprints and new work is in progress with O. Imamoglu (ETH Zurich).
Besides this research project, I also work on more classical number theory, and published new results on diophantine equations, publications 3), 4), 5). My work 4) proves a conjecture of Mueller and Schmidt (Acta Mathematica, 1988) for almost all binary forms of fixed degree. My work 3) improves on a previous paper by Akthari and Bhargava on average of solutions to a certain family of Thue equations.
I am a single author of several of my papers, and I have good collaborators as well. I am regularly invited for mathematical exchanges and discussions by one of the best research centers and research groups in Europe, the Max Planck Institute for Mathematics, EPFL CV date 21/05/2023 Lausanne (invited by Philippe Michel and Maryna Viazovska) and ETH Zurich (invited by Ozlem Imamoglu and Emmanuel Kowalski).
1) P. Bengoechea, On the distribution of cycle integrals of modular functions, Proceedings of
the American Mathematical Society (2022), in press. DOI: https://doi.org/10.1090/proc/16410
2) P. Bengoechea, Cycle integrals of the j-function on Markov geodesics, Commentarii
Mathematici Helvetici 97 (2022), no. 4, pp. 611–633.
3) P. Bengoechea, Thue equations that simultaneously fail the Hasse principle, Mathematical
Proceedings of the Cambridge Philosophical Society Volume 172, Issue 3 pp. 617-626
4) P. Bengoechea, Thue inequalities with few coefficients, International Mathematics
Research Notices, IMRN , Volume 2022, Issue 2 (2022) 1217-1244
5) S. Akhtari, P. Bengoechea, Representation of integers by sparse binary forms,
Transactions of the American Mathematical Society 374(3) (2021) 1687 - 1709.
6) P. Bengoechea, O. Imamoglu, Values of modular functions at real quadratics and
conjectures of Kaneko, Mathematische Annalen 377(1) (2020), 249-266.
7) P. Bengoechea, O. Imamoglu, Cycle integrals of modular functions, Markov geodesics
and conjectures of Kaneko, Algebra Number Theory 13-4 (2019), 943-962.
8) P. Bengoechea, Metric number theory of Fourier coefficients of modular forms,
Proceedings of the American Mathematical Society 147-07 (2018), 2835-2845.
9) P. Bengoechea, N. Moshchevitin and N. Stepanova, A note on badly approximable forms
on manifolds, Mathematika 63 (2) (2017) 587-601.