Jay Jorgenson

Bio: Jay Jorgenson is an academic researcher from City College of New York. The author has contributed to research in topics: Eisenstein series & Heat kernel. The author has an hindex of 20, co-authored 100 publications receiving 1146 citations. Previous affiliations of Jay Jorgenson include Max Planck Society & Yale University.

Basic Analysis of Regularized Series and Products

Abstract: Some complex analytic properties of regularized products and series.- A Parseval formula for functions with a singular asymptotic expansion at the origin.

The Ubiquitous Heat Kernel

Abstract: Like others, we came to the heat kernel via one direction of mathematics. However, as we progressed in that direction, we realized that the heat kernel plays a central role in almost all directions we can think of. That it bears a name related to physics only indicates that it was originally discovered in connection with heat. But even in physics, its significance goes way beyond giving a mathematical model for heat distribution. The name cannot be changed — it’s too late for that — but the impression some people may have that an occurrence of a certain kernel called the heat kernel means that one is necessarily doing physics is a false impression. Maybe one is and may be one isn’t. Also one may be using the language of physics and heat distribution simply as a convenient backdrop for some theorems giving bounds or asymptotic estimates for the heat kernel, as we do below (“diffusion”). For a recent example, cf. Norris [Nor 97], with a notable bibliography. The issue of Acta where this paper appears actually contains two articles on the heat kernel (the other is [BiJ 97]), constituting 3/5th of the issue, which is one way of being ubiquitous. Less anecdotically, there is a universal gadget which is a dominant factor practically everywhere in mathematics, also in physics, and has very simple and powerful properties. We have no a priori explanation (psychological, philosophical, mathematical) for the phenomenon of the existence of such a universal gadget.

Bounding the sup-norm of automorphic forms

Abstract: No Abstract..

Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori

Abstract: By a discrete torus we mean the Cayley graph associated to a finite product of finite cycle groups with the generating set given by choosing a generator for each cyclic factor. In this article we examine the spectral theory of the combinatorial Laplacian for sequences of discrete tori when the orders of the cyclic factors tend to infinity at comparable rates. First, we show that the sequence of heat kernels corresponding to the degenerating family converges, after rescaling, to the heat kernel on an associated real torus. We then establish an asymptotic expansion, in the degeneration parameter, of the determinant of the combinatorial Laplacian. The zeta-regularized determinant of the Laplacian of the limiting real torus appears as the constant term in this expansion. On the other hand, using a classical theorem by Kirchhoff, the determinant of the combinatorial Laplacian of a finite graph divided by the number of vertices equals the number of spanning trees, called the complexity, of the graph. As a result, we establish a precise connection between the complexity of the Cayley graphs of finite abelian groups and heights of real tori. It is also known that spectral determinants on discrete tori can be expressed using trigonometric functions and that spectral determinants on real tori can be expressed using modular forms on general linear groups. Another interpretation of our analysis is thus to establish a link between limiting values of certain products of trigonometric functions and modular forms. The heat kernel analysis which we employ uses a careful study of I-Bessel functions. Our methods extend to prove the asymptotic behavior of other spectral invariants through degeneration, such as special values of spectral zeta functions and Epstein-Hurwitz–type zeta functions.

Bounds on Faltings's delta function through covers

Abstract: Let X be a compact Riemann surface of genus gX 1. In 1984, G. Faltings introduced a new invariant Fal(X) associated to X. In this paper we give explicit bounds for Fal(X) in terms of fundamental dierential geometric invariants arising from X, when gX > 1. As an application, we are able to give bounds for Faltings’s delta function for the family of modular curves X0(N) in terms of the genus only. In combination with work of A. Abbes, P. Michel and E. Ullmo, this leads to an asymptotic formula for the Faltings height of the Jacobian J0(N) associated to X0(N).

Table Of Integrals Series And Products

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Introduction to the arithmetic theory of automorphic functions

Abstract: * uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary quadratic fields and complex multiplication of elliptic curves * Modular functions of higher level * Zeta-functions of algebraic curves and abelian varieties * The cohomology group assoicated with cusp forms * Arithmetic Fuschian groups

Sex-related defferences in mathematics achievement, spatial visualization, and affective factors

Abstract: This study investigated (a) mathematics achievement (Test of Academic Progress) of 589 female and 644 male, predominantly white, 9th-12th grade students enrolled in mathematics courses from four schools, controlling for mathematics background and general ability (Quick Word Test); (b) relationships to mathematics achievement, and to sex-related differences in mathematics achievement, of spatial visualization (Differential Aptitude Test), eight attitudes measured by the Fennema-Sherman Mathematics Attitudes Scales, a measure of Mathematics Activities outside of school, and number of Mathematics Related Courses and Space Related Courses taken. Complex results were obtained. Few sex-related cognitive differences but many attitudinal differences were found. Analyses of variance, covariance, correlation, and principal components analysis techniques were used. The results showed important relationships between socio-cultural factors and sex-related cognitive differences.

Progress in Mathematics

Abstract: Let G be an inner form of a general linear group over a non-archimedean local field. We prove that the local Langlands correspondence for G preserves depths. We also show that the local Langlands correspondence for inner forms of special linear groups preserves the depths of essentially tame Langlands parameters.

The Laplace Transform

Abstract: THE theory of Fourier integrals arises out of the elegant pair of reciprocal formulae The Laplace Transform By David Vernon Widder. (Princeton Mathematical Series.) Pp. x + 406. (Princeton: Princeton University Press; London: Oxford University Press, 1941.) 36s. net.