Jean-Michel Bismut is professor of mathematics at the Universite Paris-Sud, Orsay.

• *FrontMatter, pg. i*Contents, pg. vii*Acknowledgments, pg. xi*Introduction, pg. 1*Chapter One. Clifford and Heisenberg algebras, pg. 12*Chapter Two. The hypoelliptic Laplacian on X = G/K, pg. 22*Chapter Three. The displacement function and the return map, pg. 48*Chapter Four. Elliptic and hypoelliptic orbital integrals, pg. 76*Chapter Five. Evaluation of supertraces for a model operator, pg. 92*Chapter Six. A formula for semisimple orbital integrals, pg. 113*Chapter Seven. An application to local index theory, pg. 120*Chapter Eight. The case where [k (gamma); p0] = 0, pg. 138*Chapter Nine. A proof of the main identity, pg. 142*Chapter Ten. The action functional and the harmonic oscillator, pg. 161*Chapter Eleven. The analysis of the hypoelliptic Laplacian, pg. 187*Chapter Twelve. Rough estimates on the scalar heat kernel, pg. 212*Chapter Thirteen. Refined estimates on the scalar heat kernel for bounded b, pg. 248*Chapter Fourteen. The heat kernel qXb;t for bounded b, pg. 262*Chapter Fifteen. The heat kernel qXb;t for b large, pg. 290*Bibliography, pg. 317*Subject Index, pg. 323*Index of Notation, pg. 325

Jean-Michel Bismut is professor of mathematics at the Universite Paris-Sud, Orsay.

• *FrontMatter, pg. i*Contents, pg. vii*Acknowledgments, pg. xi*Introduction, pg. 1*Chapter One. Clifford and Heisenberg algebras, pg. 12*Chapter Two. The hypoelliptic Laplacian on X = G/K, pg. 22*Chapter Three. The displacement function and the return map, pg. 48*Chapter Four. Elliptic and hypoelliptic orbital integrals, pg. 76*Chapter Five. Evaluation of supertraces for a model operator, pg. 92*Chapter Six. A formula for semisimple orbital integrals, pg. 113*Chapter Seven. An application to local index theory, pg. 120*Chapter Eight. The case where [k (gamma); p0] = 0, pg. 138*Chapter Nine. A proof of the main identity, pg. 142*Chapter Ten. The action functional and the harmonic oscillator, pg. 161*Chapter Eleven. The analysis of the hypoelliptic Laplacian, pg. 187*Chapter Twelve. Rough estimates on the scalar heat kernel, pg. 212*Chapter Thirteen. Refined estimates on the scalar heat kernel for bounded b, pg. 248*Chapter Fourteen. The heat kernel qXb;t for bounded b, pg. 262*Chapter Fifteen. The heat kernel qXb;t for b large, pg. 290*Bibliography, pg. 317*Subject Index, pg. 323*Index of Notation, pg. 325

Jean-Michel Bismut is professor of mathematics at the Universite Paris-Sud, Orsay.

• *FrontMatter, pg. i*Contents, pg. vii*Acknowledgments, pg. xi*Introduction, pg. 1*Chapter One. Clifford and Heisenberg algebras, pg. 12*Chapter Two. The hypoelliptic Laplacian on X = G/K, pg. 22*Chapter Three. The displacement function and the return map, pg. 48*Chapter Four. Elliptic and hypoelliptic orbital integrals, pg. 76*Chapter Five. Evaluation of supertraces for a model operator, pg. 92*Chapter Six. A formula for semisimple orbital integrals, pg. 113*Chapter Seven. An application to local index theory, pg. 120*Chapter Eight. The case where [k (gamma); p0] = 0, pg. 138*Chapter Nine. A proof of the main identity, pg. 142*Chapter Ten. The action functional and the harmonic oscillator, pg. 161*Chapter Eleven. The analysis of the hypoelliptic Laplacian, pg. 187*Chapter Twelve. Rough estimates on the scalar heat kernel, pg. 212*Chapter Thirteen. Refined estimates on the scalar heat kernel for bounded b, pg. 248*Chapter Fourteen. The heat kernel qXb;t for bounded b, pg. 262*Chapter Fifteen. The heat kernel qXb;t for b large, pg. 290*Bibliography, pg. 317*Subject Index, pg. 323*Index of Notation, pg. 325