Isoparametric foliations on complex projective spaces

Irreducible isoparametric foliations of arbitrary codimension q on complex projective spaces CP^n are classified, except if n=15 and q=1. Remarkably, there are noncongruent examples that pull back under the Hopf map to congruent foliations on the sphere. Moreover, there exist many inhomogeneous isoparametric foliations, even of higher codimension. In fact, every irreducible isoparametric foliation on the complex projective n-space is homogeneous if and only if n+1 is prime. The main tool developed in this work is a method to study singular Riemannian foliations with closed leaves on complex projective spaces. This method is based on certain graph that generalizes extended Vogan diagrams of inner symmetric spaces.Comment: 39 pages, minor revision, to appear in Trans. Amer. Math. So

Complex Osserman Kaehler Manifolds

Let H be a 4 dimensional almost Hermitian manifold which satisfies the Kaehler identity. Then H is complex Osserman if and only if H has constant holomorphic sectional curvature. We also classify in arbitrary dimensions all the complex Osserman Kaehler manifolds which do not have 3 eigenvalues

Polar foliations on quaternionic projective spaces

We classify irreducible polar foliations of codimension $q$ on quaternionic projective spaces $\mathbb H P^n$, for all $(n,q)\neq(7,1)$. We prove that all irreducible polar foliations of any codimension (resp. of codimension one) on $\mathbb H P^n$ are homogeneous if and only if $n+1$ is a prime number (resp. $n$ is even or $n=1$). This shows the existence of inhomogeneous examples of codimension one and higher

Non-Hopf real hypersurfaces with constant principal curvatures in complex space forms

We classify real hypersurfaces in complex space forms with constant principal curvatures and whose Hopf vector field has two nontrivial projections onto the principal curvature spaces. In complex projective spaces such real hypersurfaces do not exist. In complex hyperbolic spaces these are holomorphically congruent to open parts of tubes around the ruled minimal submanifolds with totally real normal bundle introduced by Berndt and Bruck. In particular, they are open parts of homogenous ones

Inhomogeneous isoparametric hypersurfaces in complex hyperbolic spaces

We construct examples of inhomogeneous isoparametric real hypersurfaces in complex hyperbolic spaces

Introductory Lectures on Quantum Field Theory

In these lectures we present a few topics in Quantum Field Theory in detail. Some of them are conceptual and some more practical. They have been selected because they appear frequently in current applications to Particle Physics and String Theory.Comment: 112 pages, 18 figures, LaTeX, cernrep style, feynmf. v4 typos corrected and references added. v3 includes a new section on Feynman diagrams and an expanded discussion on RG fixed point

Isoparametric hypersurfaces in Damek-Ricci spaces

We construct uncountably many isoparametric families of hypersurfaces in Damek-Ricci spaces. We characterize those of them that have constant principal curvatures by means of the new concept of generalized Kahler angle. It follows that, in general, these examples are inhomogeneous and have nonconstant principal curvatures. We also find new cohomogeneity one actions on quaternionic hyperbolic spaces, and an isoparametric family of inhomogeneous hypersurfaces with constant principal curvatures in the Cayley hyperbolic plane.Comment: Some references update

Rescue of endemic states in interconnected networks with adaptive coupling

We study the Susceptible-Infected-Susceptible model of epidemic spreading on two layers of networks interconnected by adaptive links, which are rewired at random to avoid contacts between infected and susceptible nodes at the interlayer. We find that the rewiring reduces the effective connectivity for the transmission of the disease between layers, and may even totally decouple the networks. Weak endemic states, in which the epidemics spreads only if the two layers are interconnected, show a transition from the endemic to the healthy phase when the rewiring overcomes a threshold value that depends on the infection rate, the strength of the coupling and the mean connectivity of the networks. In the strong endemic scenario, in which the epidemics is able to spread on each separate network, the prevalence in each layer decreases when increasing the rewiring, arriving to single network values only in the limit of infinitely fast rewiring. We also find that finite-size effects are amplified by the rewiring, as there is a finite probability that the epidemics stays confined in only one network during its lifetime.Comment: 15 pages, 11 figure

Solutions to the overdetermined boundary problem for semilinear equations with position-dependent nonlinearities

We show that a wide range of overdetermined boundary problems for semilinear equations with position-dependent nonlinearities admits nontrivial solutions. The result holds true both on the Euclidean space and on compact Riemannian manifolds. As a byproduct of the proofs we also obtain some rigidity, or partial symmetry, results for solutions to overdetermined problems on Riemannian manifolds of nonconstant curvature.Comment: 36 page

Fitness for Synchronization of Network Motifs

We study the phase synchronization of Kuramoto's oscillators in small parts of networks known as motifs. We first report on the system dynamics for the case of a scale-free network and show the existence of a non-trivial critical point. We compute the probability that network motifs synchronize, and find that the fitness for synchronization correlates well with motif's interconnectedness and structural complexity. Possible implications for present debates about network evolution in biological and other systems are discussed.Comment: 6 pages. To be published in Physica A (2004