Overview

The members of our group have varied interests that encompass several areas in Analysis and Number Theory. Roughly speaking, it can be said that the unifying topic is Fourier analysis and the aims are the applications of its methods to some problems that lie mainly in analytic number theory, combinatorics and PDEs.
Among the generic goals towards which our efforts are directed, we outline: to contribute to the analytic theory of automorphic forms; to exploit the arithmetic interpretations of spectral theory; to extend the current knowledge about lattice point problems; to establish properties of special Fourier series; to explore the application of number theoretical techniques in some problems in quantum physics; to apply the theory of singular integrals to the equations in fluid mechanics; to state pointwise inequalities for nonlocal operators; to understand the development of singularities in some partial differential equations; to contribute to advances in higher order Fourier analysis, especially by deepening our understanding of the relation between Gowers uniformity norms and nilmanifolds; to improve the applications of harmonic analysis in arithmetic combinatorics and to develop new such applications.

Key words

Exponential sums, automorphic forms, ergodic theory, arithmetics combinatorics, singular integrals, fluid mechanics.

Research lines

Keeping in mind the MSC2010, a more concrete list of our research lines is:
  1. Sequences and sets.
  2. Multiplicative number theory.
  3. Harmonic analysis on Euclidean spaces.
  4. Additive number theory.
  5. Discontinuous groups and automorphic forms.
  6. Fluid mechanics.
  7. Ergodic theory.
  8. Spectral theory.

Members

Research team (Equipo de investigación)
  1. Pablo Candela
  2. Fernando Chamizo (IP1)
  3. Antonio Córdoba (IP2)
Working team (Equipo de trabajo)
  1. Diego Alonso
  2. Carlos Alberto Catalá
  3. Diego González
  4. José Granados
  5. Eric Latorre
  6. Ángel David Martínez
  7. Jesús Ocáriz
  8. Carlos Pastor
  9. Dulcinea Raboso
  10. Adrián Ubis

Five years publication record


  1. [LINK] D. Alonso-Orán, A. Córdoba, and Á. D. Martínez. Continuity of weak solutions of the critical surface quasigeostrophic equation on S^2. Adv. Math., 328:264-299, 2018.

  2. [LINK] D. Alonso-Orán, A. Córdoba, and Á. D. Martínez. Global well-posedness of critical surface quasigeostrophic equation on the sphere. Adv. Math., 328:248-263, 2018.

  3. [LINK] D. Alonso-Orán, A. Córdoba, and Á. D. Martínez. Integral representation for fractional Laplace-Beltrami operators. Adv. Math., 328:436-445, 2018.

  4. [LINK] A. Córdoba, D. Córdoba, and F. Gancedo. Uniqueness for {SQG} patch solutions. Trans. Amer. Math. Soc. Ser. B, 5:1-31, 2018.

  5. P. Candela. Notes on compact nilspaces. Discrete Anal., pages Paper No. 16, 57, 2017.

  6. P. Candela. Notes on nilspaces: algebraic aspects. Discrete Anal., pages Paper No. 15, 59, 2017.

  7. [LINK] F. Chamizo and A. González-Arroyo. Tachyonic instabilities in {$2+1$} dimensional Yang-Mills theory and its connection to number theory. J. Phys. A, 50(26):265401, 17, 2017.

  8. [LINK] F. Chamizo, Izabela Petrykiewicz, and Seraf\'\i~n Ruiz-Cabello. The Hölder exponent of some Fourier series. J. Fourier Anal. Appl., 23(4):758-777, 2017.

  9. [LINK] A. Córdoba and Eric Latorre Crespo. Radial multipliers and restriction to surfaces of the Fourier transform in mixed-norm spaces. Math. Z., 286(3-4):1479-1493, 2017.

  10. A. Córdoba, Elias Stein, Terence Tao, Louis Nirenberg, Joseph J. Kohn, Sun-Yung Alice Chang, C. Robin Graham, D. Córdoba, Bo'az Klartag, J\"urg Fröhlich, L. Seco, and M. Weinstein. Ad honorem Charles Fefferman. Notices Amer. Math. Soc., 64(11):1254-1273, 2017.

  11. [LINK] Stéphane Seuret and Adrián Ubis. Local {$L^2$}-regularity of Riemann's Fourier series. Ann. Inst. Fourier (Grenoble), 67(5):2237-2264, 2017.

  12. [LINK] Adrián Ubis. Effective equidistribution of translates of large submanifolds in semisimple homogeneous spaces. Int. Math. Res. Not. IMRN, (18):5629-5666, 2017.

  13. [LINK] A. Avila and P. Candela. Towers for commuting endomorphisms, and combinatorial applications. Ann. Inst. Fourier (Grenoble), 66(4):1529-1544, 2016.

  14. [LINK] P. Candela, Balázs Szegedy, and Llu\'\i~s Vena. On linear configurations in subsets of compact abelian groups, and invariant measurable hypergraphs. Ann. Comb., 20(3):487-524, 2016.

  15. F. Chamizo. A theorem of Javier Cilleruelo. Gac. R. Soc. Mat. Esp., 19(3):607-614, 2016.

  16. A. Córdoba. J. Cilleruelo: the art of telling. Gac. R. Soc. Mat. Esp., 19(3):498-509, 2016.

  17. [LINK] A. Córdoba. Singular integrals, maximal functions and Fourier restriction to spheres: the disk multiplier revisited. Adv. Math., 290:208-235, 2016.

  18. [LINK] P. Candela and H. A. Helfgott. On the dimension of additive sets. Acta Arith., 167(1):91-100, 2015.

  19. F. Chamizo and C. Pastor. Rowland's sequence. Gac. R. Soc. Mat. Esp., 18(2):300, 2015.

  20. [LINK] F. Chamizo and D. Raboso. Lattice points in the 3-dimensional torus. J. Math. Anal. Appl., 429(2):733-743, 2015.

  21. [LINK] F. Chamizo and D. Raboso. On the Kuznetsov formula. J. Funct. Anal., 268(4):869-886, 2015.

  22. [LINK] F. Chamizo and D. Raboso. Van der Corput method and optical illusions. Indag. Math. (N.S.), 26(5):723-735, 2015.

  23. [LINK] A. Córdoba and Á. D. Martínez. A pointwise inequality for fractional Laplacians. Adv. Math., 280:79-85, 2015.

  24. [LINK] D. Raboso. When the modular world becomes non-holomorphic. In Trends in number theory, volume 649 of Contemp. Math., pages 221-244. Amer. Math. Soc., Providence, RI, 2015.

  25. [LINK] P. Sarnak and Adrián Ubis. The horocycle flow at prime times. J. Math. Pures Appl. (9), 103(2):575-618, 2015.

  26. [LINK] P. Candela and O. Sisask. Convergence results for systems of linear forms on cyclic groups and periodic nilsequences. SIAM J. Discrete Math., 28(2):786-810, 2014.

  27. [LINK] F. Chamizo and D. Raboso. Distributional properties of powers of matrices. Czechoslovak Math. J., 64(139)(3):801-817, 2014.

  28. [LINK] F. Chamizo and Adrián Ubis. Multifractal behavior of polynomial Fourier series. Adv. Math., 250:1-34, 2014.

  29. [LINK] A. Córdoba and K. M. Rogers. Weighted estimates for conic Fourier multipliers. Math. Z., 278(1-2):431-440, 2014.

Images

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The circle problem

The standard fundamental domain

A fundamental domain for Γ0(5)

A Maass form

Caustic in a cup

Impossible tower

Pencil line drawing effect with convolutions

Moiré effect

Avoidance of crossing

A problem in statics

Riemann's example

The function eπt/2Kit(x)

Modern art in Rutgers University

Modern art in the IAS

Group photo in "The poetry of analysis"

Poster of "The poetry of analysis"

Family tree

Wordle: titles and keywords of recent papers

Interference of 1D waves

Diffraction: rectangular slit

Diffraction: circular aperture

Kaczmarz algorithm

Colored Moebius band

Missing pencil puzzle

Wave equation on the sphere

Highest weights for SO(3) and SU(2)

Positive roots and Weyl chamber for SU(3)

An spectral sum for the flat torus

Skeletal dodecahedron

Planar circular restricted orbits