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Developments in Mathematics VOLUME 28 Series Editors: Krishnaswami Alladi, University of Florida Hershel M. Farkas, Hebrew University of Jerusalem Robert Guralnick, University of Southern California For further volumes: http://www.springer.com/series/5834

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Hershel M. Farkas • Robert C. Gunning Marvin I. Knopp • B.A. Taylor Editors From Fourier Analysis and Number Theory to Radon Transforms and Geometry In Memory of Leon Ehrenpreis 123

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Editors Hershel M. Farkas Robert C. Gunning Einstein Institute of Mathematics Department of Mathematics The Hebrew University of Jerusalem Princeton University Givat Ram, Jerusalem Princeton, NJ Israel USA Marvin I. Knopp B.A. Taylor Department of Mathematics Department of Mathematics Temple University University of Michigan Philadelphia, PA Ann Arbor, MI USA USA ISSN 1389-2177 ISBN 978-1-4614-4074-1 ISBN 978-1-4614-4075-8 (eBook) DOI 10.1007/978-1-4614-4075-8 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2012944371 © Springer Science+Business Media New York 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied speciﬁcally for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

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Leon Ehrenpreis (1930–2010)

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Preface This is a volume of papers dedicated to the memory of Leon Ehrenpreis. Although Leon was primarily an analyst, whose best known results deal with partial differen- tial equations, he was also very interested in and made signiﬁcant contributions to the ﬁelds of Riemann surfaces (both the algebraic and geometric theories), number theory (both analytic and combinatorial), and geometry in general. The contributors to this volume are mathematicians who appreciated Leon’s unique view of mathematics; most knew him well and admired his work, character, and unbounded energy. For the most part the papers are original contributions to areas of mathematics in which Leon worked; so this volume may convey a sense of the breadth of his interests. The papers cover topics in number theory and modular forms, combinato- rial number theory, representation theory, pure analysis, and topics in applied mathematics such as population biology and parallel refractors. Almost any mathe- matician will ﬁnd articles of professional interest here. Leon had interests that extended far beyond just mathematics. He was a student of Jewish Law and Talmud, a handball player, a pianist, a marathon runner, and above all a scholar and a gentleman. Since we would like the readers of this volume to have a better picture of the person to whom it is dedicated, we have included a biographical sketch of Leon Ehrenpreis, written by his daughter, a professional scientiﬁc journalist. We hope that all readers will ﬁnd this chapter fascinating and inspirational. Jerusalem, Israel H.M. Farkas Princeton, NJ R.C. Gunning Philadelphia, PA M. Knopp Ann Arbor, MI B.A. Taylor Marvin Knopp (of blessed memory) passed away on December 24, 2011, after almost the entire volume was edited by the four of us. Without him, this volume would not have appeared. vii

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Contents A Biography of Leon Ehrenpreis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Yael Nachama (Ehrenpreis) Meyer Differences of Partition Functions: The Anti-telescoping Method . . . . . . . . . 1 George E. Andrews The Extremal Plurisubharmonic Function for Linear Growth .. . . . . . . . . . . . 21 David Bainbridge Mahonian Partition Identities via Polyhedral Geometry . . . . . . . . . . . . . . . . . . . . 41 Matthias Beck, Benjamin Braun, and Nguyen Le Second-Order Modular Forms with Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Thomas Blann and Nikolaos Diamantis Disjointness of Moebius from Horocycle Flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 J. Bourgain, P. Sarnak, and T. Ziegler Duality and Differential Operators for Harmonic Maass Forms . . . . . . . . . . . 85 Kathrin Bringmann, Ben Kane, and Robert C. Rhoades Function Theory Related to the Group PSL2.R/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 R. Bruggeman, J. Lewis, and D. Zagier Analysis of Degenerate Diffusion Operators Arising in Population Biology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Charles L. Epstein and Rafe Mazzeo A Matrix Related to the Theorem of Fermat and the Goldbach Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Hershel M. Farkas Continuous Solutions of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Charles Fefferman and Ja´nos Kolla´r ix

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x Contents Recurrence for Stationary Group Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Hillel Furstenberg and Eli Glasner On the Honda - Kaneko Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 P. Guerzhoy Some Intrinsic Constructions on Compact Riemann Surfaces. . . . . . . . . . . . . . 303 Robert C. Gunning The Parallel Refractor .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Cristian E. Gutie´rrez and Federico Tournier On a Theorem of N. Katz and Bases in Irreducible Representations . . . . . . 335 David Kazhdan Vector-Valued Modular Forms with an Unnatural Boundary . . . . . . . . . . . . . . 341 Marvin Knopp and Geoffrey Mason Loss of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 J.J. Kohn On an Oscillatory Result for the Coefﬁcients of General Dirichlet Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 Winfried Kohnen and Wladimir de Azevedo Pribitkin Representation Varieties of Fuchsian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 Michael Larsen and Alexander Lubotzky Two Embedding Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 Gerardo A. Mendoza Cubature Formulas and Discrete Fourier Transform on Compact Manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 Isaac Z. Pesenson and Daryl Geller The Moment Zeta Function and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 Igor Rivin A Transcendence Criterion for CM on Some Families of Calabi–Yau Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 Paula Tretkoff and Marvin D. Tretkoff Ehrenpreis and the Fundamental Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 Franc¸ois Treves Minimal Entire Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 Benjamin Weiss A Conjecture by Leon Ehrenpreis About Zeroes of Exponential Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 Alain Yger

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Contents xi The Discrete Analog of the Malgrange–Ehrenpreis Theorem . . . . . . . . . . . . . . 537 Doron Zeilberger The Legacy of Leon Ehrenpreis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 Hershel M. Farkas, Robert C. Gunning, and B.A. Taylor

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A Biography of Leon Ehrenpreis By: Yael Nachama (Ehrenpreis) Meyer Dr. Leon Ehrenpreis (b. May 22, 1930; d. August 16, 2010), a leading mathematician of the twentieth century, proved the Fundamental Principle that became known as the Malgrange–Ehrenpreis theorem, a foundation of the modern theory of differential equations that became the basis for many subsequent theoretical and technological developments. He was a native New Yorker who taught and lectured throughout the USA, as well as in academic institutions in France, Israel, and Japan. Ehrenpreis made signiﬁcant and novel contributions to a number of other areas of modern mathe- matics including differential equations, Fourier analysis, Radon transforms, integral geometry, and number theory. He was known in the mathematical community for his commitment to religious principles and to his large family, as well as for his contributions to the essence of modern mathematics. Leon Ehrenpreis published two major works: Fourier Analysis in Several Complex Variables (1970) and The Universality of the Radon Transform (2003), authored many papers, and mentored 12 Ph.D. students in New York, Yeshiva, and Temple Universities over the course of a mathematical career that spanned over half a century. What follows is his story. Leon Ehrenpreis was born on May 22, 1930. His mother, Ethel, ne´e Balk, was born in Lithuania; his father, William, a native of Austria, had changed his last name from that of his own father (Kalb) to that of his mother, in order to escape the Russian draft. And so “Ehrenpreis,” the German word for “prize of honor,” became the family surname. Leon, whose parents also gave him the Hebrew name “Eliezer,” was born just at the close of the era during which millions of Eastern European Jews had left behind the homes where their families had lived for generations and survived eras of persecution, in order to reach the land that promised to take in all of “your tired, your poor, your huddled masses yearning to breathe free. . . ” and give their children the opportunity to become Americans. Coming ashore in New York City, many of these new immigrants settled in Manhattan’s Lower East Side, in Brooklyn, and the xiii

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xiv A Biography of Leon Ehrenpreis Bronx. Leon’s family was no exception; over the course of his childhood, he lived in all three of these boroughs. Initially, Ethel and William, their baby Leon, and his older brother Seymour, settled in a home in the Marine Park neighborhood of Brooklyn. When Leon was 10, the family moved to the Lower East Side, a neighborhood with a large Jewish community. Leon’s home was one in which the kitchen was kosher, and the Sabbath recognized, and their Jewishness the deﬁning personal, family, and communal identity, though without knowledge or emphasis on the subtle details of religious observance. So it was only there that Leon came into contact with boys of his own age whose families were strictly observant, an introduction to religious life that started Leon on his trajectory towards full-scale observance. He also attended a Jewish studies after-school program to prepare for his bar mitzvah, his entry into Jewish adulthood. Soon after his bar mitzvah, Leon stopped attending his after-school studies, though he continued to attend Sabbath services at the local synagogue as a result of his friends’ inﬂuence. The majority of New York’s Jews at that time were aiming to raise their children to be successful, high-achieving Americans, with academic success and intellectual pursuits an important priority for many, including the Ehrenpreis family. So it was that soon after his bar mitzvah, Leon followed his brother into the prestigious Stuyvesant High School in Manhattan. Leon had skipped two grades in elementary school and then skipped his initial year of high school, beginning Stuyvesant in the tenth grade. When Leon was 16, the family moved to the Bronx. Now more interested in learning about his Jewish heritage, Leon attended the Young Israel of Clay Avenue and joined Hashomer Hadati, a youth group that would be the forerunner of the religious Zionist Bnei Akiva movement. He now traveled downtown each day to Stuyvesant, where he continued to excel in his studies, though not in his class conduct! He recalled having the highest grades in French, but failing to be awarded the French medal because of his poor behavior. He also scored the highest on the chemistry medal qualifying exam (though a teacher’s error meant that he never actually received it), and he was also awarded the mathematics medal—though most of these were won by his new best friend Donald Newman, whom Leon credited with inﬂuencing him to become a mathematician. His mother, he recalled, considered the choice of mathematics a “cop out” to avoid having to do the serious lab work that a physics major would require. Leon initially met Donald Newman on his ﬁrst day at Stuyvesant, where his classmate was seated just on the other side of the aisle in their ﬁrst class of the day. Almost immediately, Donald handed a clipboard to Leon with the order to “solve this problem.” The board read “Sierpnerhe”—Ehrenpreis backwards. Already in ninth grade, Leon said, Donald’s reputation foreshadowed his greatness. The same was said about Leon from the tenth grade onward. “He was the great man of Stuyvesant—we already knew he would be a mathematical star.” The two created lifelong nicknames for each other, and “Flotzo-Flip” (Donald) and “Glockenshpiel” (Leon) formed a friendship that would last forever. “I felt like a real mathematician when Flotz and I discussed mathematics together,” Leon recalled. The two friends,

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