598 Geometric Analysis and Integral Geometry

AMS Special Session on Radon Transforms and Geometric Analysis in Honor of Sigurdur Helgason’s 85th Birthday January 4–7, 2012 Boston, MA

Tufts University Workshop on Geometric Analysis on Euclidean and Homogeneous Spaces January 8–9, 2012 Medford, MA

Eric Todd Quinto Fulton Gonzalez Jens Gerlach Christensen Editors

American Mathematical Society Geometric Analysis and Integral Geometry

AMS Special Session on Radon Transforms and Geometric Analysis in Honor of Sigurdur Helgason’s 85th Birthday January 4–7, 2012 Boston, MA

Tufts University Workshop on Geometric Analysis on Euclidean and Homogeneous Spaces January 8–9, 2012 Medford, MA

Eric Todd Quinto Fulton Gonzalez Jens Gerlach Christensen Editors

598

Geometric Analysis and Integral Geometry

AMS Special Session on Radon Transforms and Geometric Analysis in Honor of Sigurdur Helgason’s 85th Birthday January 4–7, 2012 Boston, MA

Tufts University Workshop on Geometric Analysis on Euclidean and Homogeneous Spaces January 8–9, 2012 Medford, MA

Eric Todd Quinto Fulton Gonzalez Jens Gerlach Christensen Editors

American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Martin J. Strauss

2010 Mathematics Subject Classification. Primary 22E30, 43A85, 44A12, 45Q05, 92C55; Secondary 22E46, 32L25, 35S30, 65R32.

Library of Congress Cataloging-in-Publication Data AMS Special Session on Radon Transforms and Geometric Analysis (2012 : Boston, Mass.) Geometric analysis and integral geometry : AMS special session in honor of Sigurdur Helgason’s 85th birthday, radon transforms and geometric analysis, January 4-7, 2012, Boston, MA ; Tufts University Workshop on Geometric Analysis on Euclidean and Homogeneous Spaces, January 8-9, 2012, Medford, MA / Eric Todd Quinto, Fulton Gonzalez, Jens Gerlach Christensen, editors. pages cm. – (Contemporary mathematics ; volume 598) Includes bibliographical references. ISBN 978-0-8218-8738-7 (alk. paper) 1. Radon transforms–Congresses. 2. Integral geometry–Congresses. 3. Geometric analysis– Congresses. I. Quinto, Eric Todd, 1951- editor of compilation. II. Gonzalez, Fulton, 1956- editor of compilation. III. Christensen, Jens Gerlach, 1975- editor of compilation. IV. Tufts University. Workshop on Geometric Analysis on Euclidean and Homogeneous Spaces (2012 : Medford, Mass.) V. Title. QA672.A4726 2012 515.1–dc23 2013013624

Contemporary Mathematics ISSN: 0271-4132 (print); ISSN: 1098-3627 (online) DOI: http://dx.doi.org/10.1090/conm/598

Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) c 2013 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 131211100908 This volume is dedicated to Sigurdur Helgason whose mathematics has inspired many.

v

Contents

Preface ix List of Presenters xi

Historical Articles

Some Personal Remarks on the Radon Transform Sigurdur Helgason 3 On the Life and Work of S. Helgason G. Olafsson´ and R. J. Stanton 21

Research and Expository Articles

Microlocal Analysis of an Ultrasound Transform with Circular Source and Receiver Trajectories G. Ambartsoumian, J. Boman, V. P. Krishnan, and E. T. Quinto 45 Cuspidal discrete series for projective hyperbolic spaces Nils Byrial Andersen and Mogens Flensted-Jensen 59 The Radon transform on SO(3): motivations, generalizations, discretization Swanhild Bernstein and Isaac Z. Pesenson 77 Atomic decompositions of Besov spaces related to symmetric cones Jens Gerlach Christensen 97 A double fibration transform for complex projective space Michael Eastwood 111 Magnetic Schr¨odinger equation on compact symmetric spaces and the geodesic Radon transform of one forms Tomoyuki Kakehi 129 F -method for constructing equivariant differential operators Toshiyuki Kobayashi 139 Schiffer’s Conjecture, Interior Transmission Eigenvalues and Invisibility Cloaking: Singular Problem vs. Nonsingular Problem Hongyu Liu 147

vii viii CONTENTS

Approximate Reconstruction from Circular and Spherical Mean Radon Transform Data W. R. Madych 155 Analytic and Group-Theoretic Aspects of the Cosine Transform G. Olafsson,´ A. Pasquale, and B. Rubin 167 Quantization of linear algebra and its application to integral geometry Hiroshi Oda and Toshio Oshima 189 Mean value theorems on symmetric spaces Franc¸ois Rouviere` 209 Semyanistyi fractional integrals and Radon transforms B. Rubin 221 Radon-Penrose transform between symmetric spaces Hideko Sekiguchi 239 Principal series representations of infinite dimensional Lie groups, II: Construction of induced representations Joseph A. Wolf 257 Preface

Geometric analysis on Euclidean and homogeneous spaces encompasses parts of representation theory, integral geometry, harmonic analysis, microlocal analysis, and partial differential equations. It is used in a wide array of applications in fields as diverse as inverse problems, tomography, and signal and data analysis. This volume provides articles giving historical perspectives, overviews of current research in these interrelated areas, and new results. We hope this motivates beginning researchers in these fields, and we wish that readers will be left with a good sense of important past work as well as current research in these exciting and active fields of mathematics. One theme of the volume is the geometric analysis motivated by the work of Sigurdur Helgason. An historical perspective is provided in the first article by Prof. Helgason himself and in the second article by Profs. Olafsson´ and Stanton. This emphasis is natural, since the volume is based, in part, on the AMS Special Session on Radon Transforms and Geometric Analysis in honor of Sigurdur Helgason’s 85th birthday held in Boston during the 2012 AMS Annual Meeting. Research papers related to this viewpoint, in particular, on Radon transforms and related mathematics are presented by Bernstein & Pesenson, Kakehi, Madych, Olafsson,´ Pasquale & Rubin, Rouvi`ere, Rubin, and others. The workshop on Geometric analysis on Euclidean and homogeneous spaces, held at Tufts University immediately following the AMS annual meeting, sought to expand on the topics presented at the special session. It was broader in scope, as evidenced by the contributions to this volume. Among contributions in pure mathematics are articles on representation the- ory and equivariant differential operators (Kobayashi and Oda & Oshima), Pen- rose transforms (Eastwood and Sekiguchi), wavelets related to symmetric cones (Christensen), representation theory and inductive limits of Lie groups (Wolf), and noncommutative harmonic analysis (Andersen & Flensted-Jensen). The interplay between integral geometry and applications is explored in the more applied articles. These include developing an elliptical Radon transform for ultrasound (Ambartsoumian, Boman, Krishnan & Quinto), using Schiffer’s conjec- ture to understand partial cloaking (Liu), and Radon transforms in crystallography (Bernstein & Pesenson) and thermoacoustic tomography (Madych). We thank the U.S. National Science Foundation and the Tufts University Dean’s Discretionary Fund for their support of the Tufts workshop. We are grateful to Tufts University Staff Assistant Megan Monaghan for the work she did behind the scenes to make the workshop successful.

ix xPREFACE

We thank the American Mathematical Society for its support of the Special Session honoring Sigurdur Helgason, and finally, we are indebted to Christine M. Thivierge, Associate Editor for Proceedings, for her indispensable role in making these proceedings a success. List of Presenters

Radon Transforms and Geometric Analysis in Honor of Sigurdur Helgason’s 85th Birthday American Mathematical Society National Meeting January 4-7 2012, Boston, MA

Speakers

Mark Agranovsky (Bar-Ilan University): Abel-Radon transform and CR functions. Nils Byrial Andersen, (Aarhus University, Denmark): Cusp Forms on hyperbolic spaces. Jan Boman (Stockholm University): Local injectivity of weighted Radon transforms. Jens Gerlach Christensen (Tufts University): Decomposition of spaces of distributions using G˚arding vectors. Susanna Dann (University of Missouri): Paley-Wiener Theorems on Rn with respect to the spectral parameter. Victor Guillemin (MIT): Characters of group representations and semi-classical analysis. Sigurdur Helgason (MIT): Orbital Integrals, applications and problems. Tomoyuki Kakehi (Okayama University): Schroedinger equation on certain compact symmetric spaces. Adam Koranyi (H. H. Lehman College, CUNY): Twisted Poisson integrals on bounded symmetric domains. Job J. Kuit (University of Copenhagen): Radon transformation on reductive symmetric spaces: support theorems. Gestur Olafsson´ (Louisiana State University): The cosλ-transform and intertwining operators for SL(n, F). Bent Ørsted (Aarhus University): Segal-Bargmann transforms: Old and new. Eyvindur Ari Palsson (University of Rochester): On multilinear generalized Radon transforms. Angela Pasquale (Universit´e Paul Verlaine - Metz): The bounded hyperge- ometric functions associated with root systems.

xi xii LIST OF PRESENTERS

Isaac Z. Pesenson (Temple University): Splines for Radon transform on compact Lie groups with application to SO(3). Francois Rouviere (Universit´edeNice): Mean value theorems on symmetric spaces. Boris Rubin (Louisiana State University): A Generalization of the Mader- Helgason Inversion Formulas for Radon Transforms. Henrik Schlichtkrull (University of Copenhagen): Counting lattice points on homogeneous spaces. Hideko Sekiguchi (The University of Tokyo): Penrose transforms between symmetric spaces. Robert J. Stanton (Ohio State University): Special geometries arising from some special symmetric spaces. Erik P. van den Ban (Utrecht University): Cusp forms for semisimple symmetric spaces. Joseph A. Wolf (University of California at Berkeley): Range of the Double Fibration Transform.

Workshop on Geometric Analysis on Euclidean and Homogeneous Spaces Tufts University January 8-9 2012

Speakers

Gaik Ambartsoumian (The University of Texas at Arlington): Exact In- version of Ultrasound Operators in the Spherical Geometry. Michael Eastwood (Australian National University): The Penrose trans- form for complex projective space. Suresh Eswarathasan (University of Rochester): Eigenfunction supremum bounds for deformations of Schr¨odinger operators. Fulton Gonzalez (Tufts University): Multitemporal Wave equations: Mean Value Solutions. Eric Grinberg (University of Massachusetts, Boston): Admissible and in- admissible complexes in integral geometry. Yulia Hristova (IMA University of Minnesota): Detection of low emission radiating sources using direction sensitive detectors. Alexander Iosevich (University of Rochester): Multi-linear generalized Radon transforms and applications to geometric measure theory and related areas. Hiroshi Isozaki (University of Tsukuba): Inverse scattering on a generalized arithmetic surface. Toshiyuki Kobayashi (IPMU and University of Tokyo): Conformally Equi- variant Differential Operators and Branching Problems of Verma Modules. LIST OF PRESENTERS xiii

Alexander Koldobsky (University of Missouri): Stability in volume compar- ison problems. Peter Kuchment (Texas A&M): Integral geometry and microlocal analysis in the hybrid imaging. Venkateswaran Krishnan (Tata Institute of Fundamental Research): A class of singular Fourier integral operators in synthetic aperture radar imaging. Hongyu Liu (University of California, Irvine): Enhanced Near-cloak by FSH Lining. Wolodymyr Madych (University of Connecticut): Approximate reconstruc- tion from circular and spherical mean Radon transform data. Yutaka Matsui (Kinki University): Topological Radon transforms and their applications. Tai Melcher (University of Virginia): A quasi-invariance result for heat kernel measures on infinite-dimensional Heisenberg groups. Linh Nguyen (University of Idaho): Range description for a spherical mean transform on spaces of constant curvatures. Hiroyuki Ochiai (Kyushu University): Positivity of an alpha determinant. Gestur Olafsson´ (Louisiana State University): Spherical functions on limits of compact symmetric spaces. Toshio Oshima (University of Tokyo): Generalizations of Radon transforms on compact homogeneous spaces. Angela Pasquale (University of Metz): Uncertainty principles for the Schr¨o- dinger equation on Riemannian symmetric spaces of the noncompact type. Isaac Z. Pesenson (Temple University): Band-limited Localized tight frames on Compact Homogeneous Manifolds. Mark A. Pinsky (Northwestern University): Can you feel the shape of a manifold with Brownian motion. Todd Quinto (Tufts University): Algorithms in bistatic ultrasound. Boris Rubin (Louisiana State University): Inversion Formulas for Spherical Means in Constant Curvature Spaces. Henrik Schlichtkrull (University of Copenhagen): A uniform bound on the matrix elements of the irreducible representations of SU(2). Plamen Stefanov (Purdue University): The Identification problem in SPECT: uniqueness, non-uniqueness and stability. Dustin Steinhauer (Texas A&M): Inverse Problems in Medical Imaging with Internal Information. Gunther Uhlmann (University of Washington): Travel Time Tomography and Tensor Tomography. Jim Vargo (Texas A&M): The Spherical Mean Problem. Ting Zhou (MIT): On approximate cloaking by nonsingular transformation me- dia. xiv LIST OF PRESENTERS

Graduate Student Posters Matthew Dawson (Louisiana State University): Direct Systems of Spherical Representations and Compact Riemannian Symmetric Spaces. Daniel Fresen (University of Missouri): Concentration inequalities for ran- dom polytopes. Vivian Ho (Louisiana State University): Paley-Wiener Theorem for Line Bundles over Compact Symmetric Spaces. Koichi Kaizuka (University of Tsukuba): Uniform resolvent estimates on symmetric spaces of noncompact type. Toshihisa Kubo (Oklahoma State University): Systems of second-order invariant differential operators of non-Heisenberg parabolic type. Kyung-Taek Lim (Tufts University): Surjectivity and range description of the single radius spherical mean on Euclidean space. Carlos Montalto (Purdue University): Stable determination of generic simple metrics, vector field and potential from the hyperbolic Dirichlet-to-Neumann map. Vignon Oussa (Saint Louis University): Bandlimited Spaces on Some 2-step Nilpotent Lie Groups With One Parseval Frame Generator. Patrick Spencer (University of Missouri): Lorentz Balls Are Not Intersection Bodies.

Abstracts and coauthors, if any, can be found at the following URLs The AMS special session: http://jointmathematicsmeetings.org/ meetings/national/jmm2012/2138 program ss17.html#title The Tufts University workshop: http://equinto.math.tufts.edu/workshop2012/at.pdf or from the proceedings editors. Published Titles in This Series

598 Eric Todd Quinto, Fulton Gonzalez, and Jens Gerlach Christensen, Editors, Geometric Analysis and Integral Geometry, 2013 592 Arkady Berenstein and Vladimir Retakh, Editors, Noncommutative Birational Geometry, Representations and Combinatorics, 2013 590 Ursula Hamenst¨adt, Alan W. Reid, Rub´ıRodr´ıguez, Steffen Rohde, and Michael Wolf, Editors, In the Tradition of Ahlfors-Bers, VI, 2013 588 David A. Bader, Henning Meyerhenke, Peter Sanders, and Dorothea Wagner, Editors, Graph Partitioning and Graph Clustering, 2013 587 Wai Kiu Chan, Lenny Fukshansky, Rainer Schulze-Pillot, and Jeffrey D. Vaaler, Editors, Diophantine Methods, Lattices, and Arithmetic Theory of Quadratic Forms, 2013 586 Jichun Li, Hongtao Yang, and Eric Machorro, Editors, Recent Advances in Scientific Computing and Applications, 2013 585 Nicol´as Andruskiewitsch, Juan Cuadra, and Blas Torrecillas, Editors, Hopf Algebras and Tensor Categories, 2013 584 Clara L. Aldana, Maxim Braverman, Bruno Iochum, and Carolina Neira Jim´enez, Editors, Analysis, Geometry and Quantum Field Theory, 2012 583 Sam Evens, Michael Gekhtman, Brian C. Hall, Xiaobo Liu, and Claudia Polini, Editors, Mathematical Aspects of Quantization, 2012 582 Benjamin Fine, Delaram Kahrobaei, and Gerhard Rosenberger, Editors, Computational and Combinatorial Group Theory and Cryptography, 2012 581 Andrea R. Nahmod, Christopher D. Sogge, Xiaoyi Zhang, and Shijun Zheng, Editors, Recent Advances in Harmonic Analysis and Partial Differential Equations, 2012 580 Chris Athorne, Diane Maclagan, and Ian Strachan, Editors, Tropical Geometry and Integrable Systems, 2012 579 Michel Lavrauw, Gary L. Mullen, Svetla Nikova, Daniel Panario, and Leo Storme, Editors, Theory and Applications of Finite Fields, 2012 578 G. L´opez Lagomasino, Recent Advances in Orthogonal Polynomials, Special Functions, and Their Applications, 2012 577 Habib Ammari, Yves Capdeboscq, and Hyeonbae Kang, Editors, Multi-Scale and High-Contrast PDE, 2012 576 Lutz Str¨ungmann, Manfred Droste, L´aszl´o Fuchs, and Katrin Tent, Editors, Groups and Model Theory, 2012 575 Yunping Jiang and Sudeb Mitra, Editors, Quasiconformal Mappings, Riemann Surfaces, and Teichm¨uller Spaces, 2012 574 Yves Aubry, Christophe Ritzenthaler, and Alexey Zykin, Editors, Arithmetic, Geometry, Cryptography and Coding Theory, 2012 573 Francis Bonahon, Robert L. Devaney, Frederick P. Gardiner, and Dragomir Sari´ˇ c, Editors, Conformal Dynamics and Hyperbolic Geometry, 2012 572 Mika Sepp¨al¨a and Emil Volcheck, Editors, Computational Algebraic and Analytic Geometry, 2012 571 Jos´e Ignacio Burgos Gil, Rob de Jeu, James D. Lewis, Juan Carlos Naranjo, Wayne Raskind, and Xavier Xarles, Editors, Regulators, 2012 570 Joaqu´ın P´erez and Jos´eA.G´alvez, Editors, Geometric Analysis, 2012 569 Victor Goryunov, Kevin Houston, and Roberta Wik-Atique, Editors, Real and Complex Singularities, 2012 568 Simeon Reich and Alexander J. Zaslavski, Editors, Optimization Theory and Related Topics, 2012 567 Lewis Bowen, Rostislav Grigorchuk, and Yaroslav Vorobets, Editors, Dynamical Systems and Group Actions, 2012 566 Antonio Campillo, Gabriel Cardona, Alejandro Melle-Hern´andez, Wim Veys, and Wilson A. Z´u˜niga-Galindo, Editors, Zeta Functions in Algebra and Geometry, 2012 CONM 598 emti nlssadItga Geometry Integral and Analysis Geometric

This volume contains the proceedings of the AMS Special Session on Radon Transforms and Geometric Analysis, in honor of Sigurdur Helgason’s 85th Birthday, held from January 4–7, 2012, in Boston, MA, and the Tufts University Workshop on Geometric Analysis on Euclidean and Homogeneous Spaces, held from January 8–9, 2012, in Medford, MA. This volume provides an historical overview of several decades in integral geometry and geometric analysis as well as recent advances in these fields and closely related areas. It contains several articles focusing on the mathematical work of Sigurdur Helgason, includ- ing an overview of his research by Gestur Olafsson´ and Robert Stanton. The first article in the volume contains Helgason’s own reminiscences about the development of the group- theoretical aspects of the Radon transform and its relation to geometric analysis. Other contributions cover Radon transforms, harmonic analysis, Penrose transforms, represen- tation theory, wavelets, partial differential operators on groups, and inverse problems in • tomography and cloaking that are related to integral geometry. Editors Christensen, and Gonzalez, Quinto, Many articles contain both an overview of their respective fields as well as new research results. The volume will therefore appeal to experienced researchers as well as a younger generation of mathematicians. With a good blend of pure and applied topics the volume will be a valuable source for interdisciplinary research.

ISBN 978-0-8218-8738-7

9 780821 887387

CONM/598 AMS