Invariants of algebraic curves and topological expansion

For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their properties, in particular their variation under a variation of the curve, and their modular properties. We also study their limits when the curve becomes singular. In addition we find that they can be used to define a formal series, which satisfies formally an Hirota equation, and we thus obtain a new way of constructing a tau function attached to an algebraic curve. These invariants are constructed in order to coincide with the topological expansion of a matrix formal integral, when the algebraic curve is chosen as the large N limit of the matrix model's spectral curve. Surprisingly, we find that the same invariants also give the topological expansion of other models, in particular the matrix model with an external field, and the so-called double scaling limit of matrix models, i.e. the (p,q) minimal models of conformal field theory. As an example to illustrate the efficiency of our method, we apply it to the Kontsevitch integral, and we give a new and extremely easy proof that Kontsevitch integral depends only on odd times, and that it is a KdV tau-function.Comment: 92 pages, LaTex, 33 figures, many misprints corrected, small modifications, additional figure

Parallel algorithm with spectral convergence for nonlinear integro-differential equations

We discuss a numerical algorithm for solving nonlinear integro-differential equations, and illustrate our findings for the particular case of Volterra type equations. The algorithm combines a perturbation approach meant to render a linearized version of the problem and a spectral method where unknown functions are expanded in terms of Chebyshev polynomials (El-gendi's method). This approach is shown to be suitable for the calculation of two-point Green functions required in next to leading order studies of time-dependent quantum field theory.Comment: 15 pages, 9 figure

Gemini Planet Imager Observational Calibrations III: Empirical Measurement Methods and Applications of High-Resolution Microlens PSFs

The newly commissioned Gemini Planet Imager (GPI) combines extreme adaptive optics, an advanced coronagraph, precision wavefront control and a lenslet-based integral field spectrograph (IFS) to measure the spectra of young extrasolar giant planets between 0.9-2.5 um. Each GPI detector image, when in spectral model, consists of ~37,000 microspectra which are under or critically sampled in the spatial direction. This paper demonstrates how to obtain high-resolution microlens PSFs and discusses their use in enhancing the wavelength calibration, flexure compensation and spectral extraction. This method is generally applicable to any lenslet-based integral field spectrograph including proposed future instrument concepts for space missions.Comment: 10 pages, 6 figures. Proceedings of the SPIE, 9147-282 v2: reference adde

Billiards and boundary traces of eigenfunctions

This is a report for the 2003 Forges Les Eaux PDE conference on recent results with A. Hassell on quantum ergodicity of boundary traces of eigenfunctions on domains with ergodic billiards, and of work in progress with Hassell and Sogge on norms of boundary traces. Related work by Burq, Grieser and Smith-Sogge is also discussed.Comment: To appear in the proceedings of the 2003 Forges Les Eaux PDE conferenc

Gravitational parity anomaly with and without boundaries

In this paper we consider gravitational parity anomaly in three and four dimensions. We start with a re-computation of this anomaly on a 3D manifold without boundaries and with a critical comparison of our results to the previous calculations. Then we compute the anomaly on 4D manifolds with boundaries with local bag boundary conditions. We find, that gravitational parity anomaly is localized on the boundary and contains a gravitational Chern-Simons terms together with a term depending of the extrinsic curvature. We also discuss the main properties of the anomaly, as the conformal invariance, relations between 3D and 4D anomalies, etc.Comment: 16 pages, final version, accepted for publication in JHE

Adiabatic response for Lindblad dynamics

We study the adiabatic response of open systems governed by Lindblad evolutions. In such systems, there is an ambiguity in the assignment of observables to fluxes (rates) such as velocities and currents. For the appropriate notion of flux, the formulas for the transport coefficients are simple and explicit and are governed by the parallel transport on the manifold of instantaneous stationary states. Among our results we show that the response coefficients of open systems, whose stationary states are projections, is given by the adiabatic curvature.Comment: 33 pages, 4 figures, accepted versio