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FOREWORD This special issue of the American Journal of Mathematics is dedicated to the late Jun-ichi Igusa (1924–2013) in honor of his distinguished career as a mathematician , his professorship at the Johns Hopkins University, and his role as a former Editor-in-Chief of the AJM. Let me start with some personal reflections on Igusa’s life and career. I came to know Professor Igusa when I started my graduate studies at Hopkins in 1971. He was very much liked and respected by all around him and that immediately made a positive impression on me as a new student. This was all confirmed soon after I took courses with him, such as our qualifier Lie group course to which he gave a beautiful algebraic treatment, as well as a number of courses in number theory. From our small entry class of 1971, three studied number theory—one with Igusa, one with Takashi Ono, and myself with Joseph Shalika—and being such a small group, we were all aware of each other’s work and advisors. Igusa probably had one of the largest groups of students and graduates, and that continued through his career at Johns Hopkins. His remarkable contributions to algebraic geometry and number theory are well-known to many of us, and are of high quality and significance. I learned about theta functions first through his book, one of many he wrote throughout his career, the last of which was published by the AMS just a few years before he passed. One of his major contributions to number theory and algebraic geometry, among others, is the theory of Igusa zeta functions. In fact, the latter part of his career was devoted mainly to the study of these functions, and many of his later students worked on this subject. By their definition these zeta functions bring in interesting and hard questions in algebraic geometry, many of which he masterfully answered, either by himself, or through his students. The subject has been studied within, and with applications to, different disciplines, including the theory of pre-homogeneous vector spaces and motific integration: many of the contributions to this special issue are from the experts on Igusa zeta functions and motific integration. One should not forget the significant role that Igusa played as the Editor-inChief of the American Journal for a good number of years (from 1978 to 1993, having already been an associate editor starting in 1964), managing the backlogs accumulated in earlier years and bringing the journal to one of its highest standards. Similarly one should mention his leadership role in The Japan-U.S. Mathematics iii Institute (JAMI), which led to a wonderful collaboration between the U.S. and Japanese mathematical communities and lasted throughout the latter part of his career at Hopkins. I was very fortunate to meet with Professor Igusa and his wife a number of times during the last few years of his life, with visits which were arranged by two of his former students, Leon Strauss and Haluk Aritürk, during my trips to the Baltimore-Washington area, and from which I have many fine memories. As explained, this special issue contains a good number of articles by experts working on Igusa zeta functions and their consequences, providing us with state of the art results on the subject, as well as an informative and well-written survey by Diane Meuser. We are very thankful to all of our contributors. We are also grateful to Bảo Châu Ngô and his collaborators who accepted our invitation to submit a manuscript to this volume, particularly for its suitable content which touches upon the crossroads of the state of the art in algebraic geometry and number theory, the two passions of Igusa. Finally, we would like to thank all the referees for delivering thorough and careful reports within the short time spans available to us to meet our publication deadlines. Freydoon Shahidi, Managing Editor (Special Issue) December, 2015 iv ...

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