Robert C. Gunning

From Fourier Analysis and Number Theory to Radon Transforms and Geometry

Preface. Biographical Sketch of Leon Ehrenpreis (Yael Ehrenpreis Meyer).- Differences of Partition Functions - The Anti-Telescoping Method(G.E. Andrews).- The Extremal Plurisubharmonic Function for Linear Growth (D. Bainbridge).- Mahonian Partition Identities Via Polyhedral Geometry (M. Beck, B. Braun, N. Le).- Second Order Modular Forms with Characters (T. Blann, N. Diamantis).- Disjointness of Moebius From Horocycle Flows (J. Bourgain, P. Sarnak, T. Zeigler).- Duality and Differential Operators for Harmonic MAASS Forms (K. Bringmann, B. Kane, R.C. Rhoades).- Function Theory Related to the Group PSL2(R) (R. Bruggeman, J. Lewis, D. Zagier).- Analysis of Degenerate Diffusion Operators Arising in Population Biology (C.L. Epstein, R. Mazzeo).- A Matrix Related to the Theorem of Fermat and the Goldbach Conjecture (H.M. Farkas).- Continuous Solutions of Linear Equations (C. Fefferman, J. Kollar).- Recurrence for Stationary Group Actions (H. Furstenberg, E. Glasner).- On the Honda-Kaneko Congruences (P. Guerzhoy).- Some Intrinsic Constructions on Compact Riemann Surfaces (Robert C. Gunning).- The Parallel Refractor (C.E. Gutierrez, F. Tournier).- On a Theorem of N. Katz and Bases in Irreducible Representations (D. Kazhdan).- Vector-valued Modular Forms with an Unnatural Boundary (M. Knopp, G. Mason).- Loss of Derivatives (J.J. Kohn).- On an Oscillatory Result for the Coefficients of General Dirichlet Series (W. Kohnen, W. de Azevedo Pribitkin).- Representation Varieties of Fucsian Groups (M. Larsen, Alexander Lubotzky).- Two Embedding Theorems (G.A. Mendoza).- Cubature Formulas and Discrete Fourier Transform on Compact Manifolds (I. Z. Pesenson, D. Geller).- The Moment Zeta Function and Applications (I. Rivin).- A Transcendence Criterion for CM on Some Families of Calabi-Yau Manifolds (P. Tretkoff).- Ehrenpreis and the Fundamental Principle (F. Treves).- Minimal Entire Functions (B. Weiss).- A Conjecture by Leon Ehrenpreis about Zeroes of Exponential Polynomials (A. Yger).- The Discrete Analog of the Malgrange-Ehrenpreis Theorem (D. Zeilberger).- The Legacy of Leon Ehrenpreis.- PhD Students.- Publications of Leon Ehrenpreis.

On Projective Covariant Differentiation

On manifolds with the pseudogroup structure of the group of projective transformations there are linear differential operators between certain spaces of tensor fields, the analogue for projective structures of the affine covariant derivative; such operators have played an important role in connection with the Eichler cohomology groups on one-dimensional complex manifolds. This paper contains a discussion of such operators of first- or second-order on two-dimensional complex manifolds, indicating the invariance properties involved and the corresponding exact sequences leading to generalizations of the Eichler cohomology groups.

Generalized Theta Functions

If M is a marked compact Riemann surface of genus g>0 and ξ is a factor of automorphy with characteristic class c(ξ)=r then by the Riemann-Roch theorem γ(ξ)=γ(κξ-1)+r+1-g, where κ is the canonical factor of automorphy; and if r≥2g-1 then c(κξ)=2g-2-r<0 so that γ(κξ-1)=0 and γ(ξ)=r+1-g. Thus all factors of automorphy with characteristic class r admit equally many complex analytic relatively automorphic functions whenever r≥2g-1. Now the set of factors of automorphy with characteristic class r can be parametrized by the complex manifold ℂ g , by associating to any point t∊- g the factor of automorphy ρtζ r where ρ: ℂ g →Hom(Г, ℂ*) is the canonical homomorphism considered previously and ζ is a factor of automorphy representing the point bundle associated to the divisor 1·p0 for the base point p0 of the marked surface M; and it can be asked whether the set of all complex analytic relatively automorphic functions for these factors of automorphy can be parametrized accordingly by the complex manifold ℂg× ℂr+1-g if r≥2g-1. The answer is provided by the following result.

Some Intrinsic Constructions on Compact Riemann Surfaces

For any prescribed differential principal part on a compact Riemann surface, there are uniquely determined and intrinsically defined meromorphic abelian differentials with these principal parts, defined independently of any choice of a marking of the surface or of a basis for the holomorphic abelian differentials, and they are holomorphic functions of the singularities. They can be constructed explicitly in terms of intrinsically defined cross-ratio functions on the Riemann surfaces, the classical cross-ratio function for the Riemann sphere, and natural generalizations for surfaces of higher genus.

The Legacy of Leon Ehrenpreis

All those who knew Leon Ehrenpreis are well aware that he was a very multidimensional person. His interests went far beyond mathematics. Leon’s interests included Bible and Talmud studies, music, sports (handball and marathon running), philosophy, and more. In this volume, we have only concentrated on his mathematical interests.

Complex Manifolds and Vector Bundles

An n-dimensional topological manifold is a Hausdorff topological space M such that every point p∊M has an open neighborhood homeomorphic to an open subset of the n-dimensional number space ℝ n . A coordinate covering {U α , z α } of such a manifold M consists of a covering of M by open subsets U α together with homeomorphisms z α :U α →V α between the sets U α and open subsets V α ⊆ℝ n ; the sets U α are called coordinate neighborhoods and the mappings z α are called coordinate mappings. A topological manifold of course always admits coordinate coverings. Note that on each nonempty intersection U α ⋂U β of coordinate neighborhoods there are thus two homeomorphisms into ℝ n ; the compositions

Introduction to Holomorphic Functions of Several Variables

{f_{\alpha \beta }} = {z_\alpha }\,o\,z_\beta ^{ - 1}:{z_\beta }\left( {{U_\alpha } \cap {U_\beta }} \right) \to {z_\alpha }\left( {{U_\alpha } \cap {U_\beta }} \right)

Lectures on Riemann Surfaces: Jacobi Varieties

are called the coordinate transition functions of the coordinate covering, and for any point p∊U α ⋂U β the two coordinate mappings are related by z α (p)=f αβ (z β (p)). The manifold M is completely determined by the sets {V α } and the mappings {f αβ }; for M can be obtained from the disjoint union of all the sets V α by identifying a point z α ∊V α and a point z β ∊V β whenever z α =f αβ (z β ).