**Contemporary Mathematics**

Volume: 398;
2006;
402 pp;
Softcover

MSC: Primary 35; 22; 11; 14; 47; 58; 53; 32;

Print ISBN: 978-0-8218-3698-9

Product Code: CONM/398

List Price: $105.00

Individual Member Price: $84.00

**Electronic ISBN: 978-0-8218-7988-7
Product Code: CONM/398.E**

List Price: $105.00

Individual Member Price: $84.00

# The Ubiquitous Heat Kernel

Share this page *Edited by *
*Jay Jorgenson; Lynne Walling*

The aim of this volume is to bring together research ideas from various fields of mathematics which utilize the heat kernel or heat kernel techniques in their research. The intention of this collection of papers is to broaden productive communication across mathematical sub-disciplines and to provide a vehicle which would allow experts in one field to initiate research with individuals in another field, as well as to give non-experts a resource which can facilitate expanding their research and connecting with others.

#### Table of Contents

# Table of Contents

## The Ubiquitous Heat Kernel

- Contents iii4 free
- Preface v6 free
- Positivity of zeta distributions and small unitary representations 18 free
- The heat equation and representations of the Jacobi group 4754
- Kato's inequality and asymptotic spectral properties for discrete magnetic Laplacians 6976
- The heat kernel in low-dimensional quantum theories 8390
- Heat kernels on weighted manifolds and applications 93100
- 1. Introduction 94101
- 2. The Laplace operator 96103
- 3. The heat kernel 101108
- 4. Relations between different heat kernels 109116
- 5. Heat kernel estimates 116123
- 5.1 Uniform Faber-Krahn inequality 116123
- 5.2 Gaussian upper bounds 120127
- 5.3 On-diagonal lower bounds 124131
- 5.4 Relative Faber-Krahn inequality 125132
- 5.5 On-diagonal estimates on model manifolds 128135
- 5.6 Estimates with the mean curvature function 130137
- 5.7 Green function and Green operator 135142

- 6. Harnack inequality 138145
- 7. Eigenvalues of Schrödinger operators 149156
- 8. The Brownian motion 153160
- 9. Path properties of stochastic processes 161168
- 10. Heat kernels of Schrödinger operators 167174
- References 183190

- Heat kernels in geometric evolution equations 193200
- The range of the heat operator 203210
- Heat kernels and cycles 233240
- Green currents on Kähler manifolds 245252
- Heat kernels and Riesz transforms 257264
- The contest between the kernels in the Selberg trace formula for the (q + 1)-regular tree 265272
- Expressing Arakelov invariants using hyperbolic heat kernels 295302
- Grassmanians of higher local fields and multivariable tau functions 311318
- Heat kernels and the range of the trace on completions of twisted group algebras 321328
- Theta functions, old and new 347354
- The heat kernel on the symmetric space SL(n, F)/SU(n, F) 369376
- Incidence structure 393400

#### Readership

Graduate students and research mathematicians interested in analysis, representation theory, algebraic geometry, partial differential equations, mathematical physics.