The Ubiquitous Heat Kernel

Preface v

Positivity of zeta distributions and small unitary representations

L. Barchini, M. Sepanski, and R. Zierau 1

The heat equation and representations of the Jacobi group

R. Berndt 47

Kato’s inequality and asymptotic spectral properties for discrete

magnetic Laplacians

J. Dodziuk and V. Mathai 69

The heat kernel in low-dimensional quantum theories

D. Fine 83

Heat kernels on weighted manifolds and applications

A. Grigor’yan 93

Heat kernels in geometric evolution equations

J. Grotowski 193

The range of the heat operator

B. Hall 203

Heat kernels and cycles

B. Harris 233

Green currents on K¨ahler manifolds

G. Hein 245

Heat kernels and Riesz transforms

S. Hofmann 257

The contest between the kernels in the Selberg trace formula for the

(q + 1)-regular tree

M. Horton, D. Newland, and A. Terras 265

Expressing Arakelov invariants using hyperbolic heat kernels

J. Jorgenson and J. Kramer 295

Grassmanians of higher local fields and multivariable tau functions

M. Lee and E. Previato 311

iii

iv CONTENTS

Heat kernels and the range of the trace on completions of twisted

group algebras

V. Mathai, with an appendix by I. Chatterji 321

Theta functions, old and new

E. Previato 347

The heat kernel on the symmetric space SL(n,F)/SU(n,F)

P. Sawyer 369

Incidence structure

B. Wang 393