What is the Jacobian of a Riemann surface with boundary?

We define the Jacobian of a Riemann surface with analytically parametrized boundary components. These Jacobians belong to a moduli space of "open abelian varieties" which satisfies gluing axioms similar to those of Riemann surfaces, and therefore allows a notion of "conformal field theory" to be defined on this space. We further prove that chiral conformal field theories corresponding to even lattices factor through this moduli space of open abelian varieties.

Singular Bott-Chern classes and the arithmetic Grothendieck-Riemann-Roch theorem for closed immersions

We study the singular Bott-Chern classes introduced by Bismut, Gillet and Soulé. Singular Bott-Chern classes are the main ingredient to define direct images for closed immersions in arithmetic K-theory. In this paper we give an axiomatic definition of a theory of singular Bott-Chern classes, study their properties, and classify all possible theories of this kind. We identify the theory defined by Bismut, Gillet and Soulé as the only one that satisfies the additional condition of being homogeneous. We include a proof of the arithmetic Grothendieck-Riemann-Roch theorem for closed immersions that generalizes a result of Bismut, Gillet and Soulé and was already proved by Zha. This result can be combined with the arithmetic Grothendieck-Riemann-Roch theorem for submersions to extend this theorem to arbitrary projective morphisms. As a byproduct of this study we obtain two results of independent interest. First, we prove a Poincaré lemma for the complex of currents with fixed wave front set, and second we prove that certain direct images of Bott-Chern classes are closed.

Interpolating and sampling sequences in finite Riemann surfaces

We provide a description of the interpolating and sampling sequences on a space of holomorphic functions on a finite Riemann surface, where a uniform growth restriction is imposed on the holomorphic functions.

La funció Z de Riemann

El novembre de 1859 Riemann envià un manuscrit de sis fulls a l’Acadèmia de Berlín titulat Sobre el nombre de primers menors que una quantitat donada, el qual seria l’única publicació dedicada a la teoria de nombres de tota la seva producció científica. Aquest treball, sens dubte una de les peces mestres de les matemàtiques de tots els temps, és pioner en l’aplicació de tècniques analítiques per a l’estudi de problemes aritmètics. En ell Riemann introdueix la funció Z i en dóna diverses propietats, de les quals en treu conseqüències sobre l’acumulació dels nombres primers. També hi enuncia la famosa conjectura sobre els seus zeros que ha passat a la història amb el nom d’hipòtesi de Riemann, i que, havent resistit els esforços de molts dels millors matemàtics del segle xx, és considerada avui dia el problema obert més important de les matemàtiques. L’objectiu d’aquestes notes és explicar el contingut del treball de Riemann i el paper fonamental que ha jugat en l’estudi de la distribució dels nombres primers.

Reduction, Linearization and stability of relative equilibria for mechanical systems on riemannian manifolds

Consider a Riemannian manifold equipped with an infinitesimal isometry. For this setup, a unified treatment is provided, solely in the language of Riemannian geometry, of techniques in reduction, linearization, and stability of relative equilibria. In particular, for mechanical control systems, an explicit characterization is given for the manner in which reduction by an infinitesimal isometry, and linearization along a controlled trajectory "commute." As part of the development, relationships are derived between the Jacobi equation of geodesic variation and concepts from reduction theory, such as the curvature of the mechanical connection and the effective potential. As an application of our techniques, fiber and base stability of relative equilibria are studied. The paper also serves as a tutorial of Riemannian geometric methods applicable in the intersection of mechanics and control theory.

Parallel spinors on pseudo-riemannian spin(q) manifolds

We study simply-connected irreducible non-locally symmetric pseudo-Riemannian Spin(q) manifolds admitting parallel quaternionic spinors.

Deux extensions de Théorèmes de Hamburger (portant sur l'équation fonctionnelle de la fonction dzêta)

We propose two types of extensions to Hamburger’s theorems on the Dirichlet series with functional equation like the one of the Riemann zeta function, under weaker hypotheses. This builds upon the dictionary betweeen the moderate meromorphic functions with functional equation and the tempered distributions with extended S-support condition.

Moduli of smoothness and growth properties of Fourier transforms: two-sided estimates

We prove two-sided inequalities between the integral moduli of smoothness of a function on R d[superscript] / T d[superscript] and the weighted tail-type integrals of its Fourier transform/series. Sharpness of obtained results in particular is given by the equivalence results for functions satisfying certain regular conditions. Applications include a quantitative form of the Riemann-Lebesgue lemma as well as several other questions in approximation theory and the theory of function spaces.

Parametrization of spectral surfaces of a class of periodic 5-diagonal matrices

The main result of this work is a parametric description of the spectral surfaces of a class of periodic 5-diagonal matrices, related to the strong moment problem. This class is a self-adjoint twin of the class of CMV matrices. Jointly they form the simplest possible classes of 5-diagonal matrices.

Holomorphic maps between moduli spaces

We prove that every non-constant holomorphic map&em&M&/em&&sub&g,p&/sub&→ &em&M&/em&&sub& g',p'&/sub& between moduli spaces of Riemann surfaces is a forgetful map, provided that g ≥ 6 and g' ≤ 2g-2.

Proposed gravitational wave observatory based on solid elastic spheres

Spherical gravitational wave (GW) detectors offer a wealth of so far unexplored possibilities to detect gravitational radiation. We find that a sphere can be used as a powerful testbed for any metric theory of gravity, not only general relativity as considered so far, by making use of a deconvolution procedure for all the electric components of the Riemann tensor. We also find that the spheres cross section is large at two frequencies, and advantageous at higher frequencies in the sense that a single antenna constitutes a real xylophone in its own. Proposed GW networks will greatly benefit from this. The main features of a two large sphere observatory are reported.

Cosmic strings and Einstein-Rosen soliton waves

We consider all generalized soliton solutions of the Einstein-Rosen form in the cylindrical context. They are Petrov type-I solutions which describe solitonlike waves interacting with a line source placed on the symmetry axis. Some of the solutions develop a curvature singularity on the axis which is typical of massive line sources, whereas others just have the conical singularity revealing the presence of a static cosmic string. The analysis is based on the asymptotic behavior of the Riemann and metric tensors, the deficit angle, and a C-velocity associated to Thornes C-energy. The C-energy is found to be radiated along the null directions.

Periodic points of holomorphic maps via Lefschetz numbers

In this paper we study the set of periods of holomorphic maps on compact manifolds, using the periodic Lefschetz numbers introduced by Dold and Llibre, which can be computed from the homology class of the map. We show that these numbers contain information about the existence of periodic points of a given period; and, if we assume the map to be transversal, then they give us the exact number of such periodic orbits. We apply this result to the complex projective space of dimension n and to some special type of Hopf surfaces, partially characterizing their set of periods. In the first case we also show that any holomorphic map of CP(n) of degree greater than one has infinitely many distinct periodic orbits, hence generalizing a theorem of Fornaess and Sibony. We then characterize the set of periods of a holomorphic map on the Riemann sphere, hence giving an alternative proof of Baker's theorem.

Pointwise estimates for the Bergman kernel of the weighted Fock space

We prove upper pointwise estimates for the Bergman kernel of the weighted Fock space of entire functions in $L^{2}(e^{-2\phi}) $ where $\phi$ is a subharmonic function with $\Delta\phi$ a doubling measure. We derive estimates for the canonical solution operator to the inhomogeneous Cauchy-Riemann equation and we characterize the compactness of this operator in terms of $\Delta\phi$.