On the Existence of Jenkins-Strebel Differentials Using Harmonic Maps from Surfaces to Graphs

We give a new proof of the existence (\cite{HM}, \cite{Ren}) of a Jenkins-Strebel differential $\Phi$ on a Riemann surface \SR with prescribed heights of cylinders by considering the harmonic map from \SR to the leaf space of the vertical foliation of $\Phi$, thought of as a Riemannian graph. The novelty of the argument is that it is essentially Riemannian as well as elementary; moreover, the harmonic maps existence theory on which it relies is classical, due mostly to Morrey (\cite{Mo}).Comment: 8 pages, 2 figures available upon reques

Dynamical Entropy Production in Spiking Neuron Networks in the Balanced State

We demonstrate deterministic extensive chaos in the dynamics of large sparse networks of theta neurons in the balanced state. The analysis is based on numerically exact calculations of the full spectrum of Lyapunov exponents, the entropy production rate and the attractor dimension. Extensive chaos is found in inhibitory networks and becomes more intense when an excitatory population is included. We find a strikingly high rate of entropy production that would limit information representation in cortical spike patterns to the immediate stimulus response.Comment: 4 pages, 4 figure

Spectrum Estimation: A Unified Framework for Covariance Matrix Estimation and PCA in Large Dimensions

Covariance matrix estimation and principal component analysis (PCA) are two cornerstones of multivariate analysis. Classic textbook solutions perform poorly when the dimension of the data is of a magnitude similar to the sample size, or even larger. In such settings, there is a common remedy for both statistical problems: nonlinear shrinkage of the eigenvalues of the sample covariance matrix. The optimal nonlinear shrinkage formula depends on unknown population quantities and is thus not available. It is, however, possible to consistently estimate an oracle nonlinear shrinkage, which is motivated on asymptotic grounds. A key tool to this end is consistent estimation of the set of eigenvalues of the population covariance matrix (also known as the spectrum), an interesting and challenging problem in its own right. Extensive Monte Carlo simulations demonstrate that our methods have desirable finite-sample properties and outperform previous proposals.Comment: 40 pages, 8 figures, 5 tables, University of Zurich, Department of Economics, Working Paper No. 105, Revised version, July 201

Polynomial cubic differentials and convex polygons in the projective plane

We construct and study a natural homeomorphism between the moduli space of polynomial cubic differentials of degree d on the complex plane and the space of projective equivalence classes of oriented convex polygons with d+3 vertices. This map arises from the construction of a complete hyperbolic affine sphere with prescribed Pick differential, and can be seen as an analogue of the Labourie-Loftin parameterization of convex RP^2 structures on a compact surface by the bundle of holomorphic cubic differentials over Teichmuller space.Comment: 64 pages, 5 figures. v3: Minor revisions according to referee report. v2: Corrections in section 5 and related new material in appendix

Non-Existence of Geometric Minimal Foliations in Hyperbolic Three-Manifolds

In this paper we show that every three-dimensional closed hyperbolic manifold admits no locally geometric $1$-parameter family of closed minimal surfaces.Comment: Commentarii Mathematici Helvetici, to appea