In general terms, our project considers various questions of harmonic analysis and their interactions with areas of number theory, combinatorics, ergodic theory, and mathematical physics. More precisely, we consider three lines of work, which are represented in a certain way by the members of the research team: generalized Fourier analysis and arithmetic combinatorics; analytic number theory; and applications of singular integrals.

All these lines are unified by methods related, in one way or another, to Fourier analysis. However, this interaction should not be interpreted in hierarchical terms, since the group precedents show that the interaction yields results and questions back within harmonic analysis itself, as a consequence of (and motivated by) the application of techniques in other areas.

Among the generic objectives to which we will dedicate our efforts are the following: contribute to the development of higher-order Fourier analysis, especially by deepening the understanding of the relationship between Gowers uniformity norms and nilspaces; improve and expand the applications and interactions of harmonic analysis with arithmetic combinatorics and ergodic theory; contribute to the analytical theory of automorphic forms; exploit arithmetic interpretations of spectral theory; expand present knowledge on lattice point problems; establish properties of special Fourier series; explore the application of arithmetic techniques in some problems in quantum physics; apply the theory of singular integrals to quantum-mechanical problems; formulate specific inequalities for non-local operators; understand the formation of singularities in some partial differential equations.

These are the concrete goals we propose ourselves for the current period of our grant:

- Further the unification of the theory of Gowers norms by showing that the inverse theorem for finite vector spaces (Tao-Ziegler) can be deduced from the structure theorem of cubic couplings and the nilspace approach (Candela-Szegedy).
- Find a proof of the inverse theorem for Gowers norms via nilspaces with good bounds, ideally for finite abelian groups in general.
- Extend the applications of the tools of ergodic theory (e.g. Rokhlin Lemma) in combinatorics and number theory, in particular for Motzkin's problem.
- Get point estimates of spectral means of Maass forms and study their arithmetic implications.
- Characterize the convergence of some special Fourier series and some other of Diophantine nature in terms of continued fractions.
- Use the factorization theory of Hurwitz quaternions in certain model in quantum computation.
- Develop new trigonometric sum estimates to get some advances in lattice point problems under symmetries and explore the possible connections to mathematical physics.
- Explore the application of arithmetic arguments in the study of partial differential equations and in equations coming from mathematical physics.
- Extend some recent results about Fourier series having k-powers as frequencies.
- Extend the characterization of minimal surfaces to the more general setting of Riemannian manifolds.
- Treat the case of the tight-binding model in which the Hamiltonian includes a strong magnetic field.