# 1986 ICM - Berkeley

## 1986 International Congress of Mathematicians - Berkeley, California

The International Congress of Mathematicians was held in Berkeley, California, USA, from 3 August to 11 August 1986. There were 3711 registered ordinary members from 82 countries.
All sessions of the Congress took place on the campus of the University of California, Berkeley.
The Opening Ceremony took place in the Greek Theater. All plenary lectures and the Closing Ceremony took place in the Zellerbach Auditorium. Closed circuit television relayed these to other large lecture theatres.
The Chancellor of the University of California, Berkeley hosted an outdoor reception in the Faculty Glade on Sunday 3 August.
A western-style barbecue and rodeo was held on the evening of Thursday 7 August in the Cow Palace in San Francisco.
A jazz concert took place on the evening of Saturday 9 August and a classical concert on the evening of Sunday 10 August.
We give below:
1.       Secretary's Report on Preparations.
The National Academy of Sciences of the United States, through the U.S. National Committee for Mathematics, invited the International Mathematical Union to hold the 1986 International Congress of Mathematicians in the United States at the University of California, Berkeley. The invitation was accepted by the International Mathematical Union at the Warsaw Congress in August, 1983. Final action was taken a few days later when the Warsaw Congress at its Closing Ceremonies accepted the invitation to Berkeley, which was presented by Professor Jack K Hale on behalf of the mathematical community of the United States of America.

The Academy, in turn, asked the American Mathematical Society to handle the organisational aspects of the Congress, The Society organized the Congress as a non-profit corporation, ICM-86, and Jill P Mesirov was appointed Executive Director. ICM-86 used the services of the American Mathematical Society's Meetings Department headed by H Hope Daly, who served as Congress Manager.

The scientific program was organized by a Program Committee appointed by the International Mathematical Union. Its members were Enrico Bombieri, Lennart Carleson, Friedrich E P Hirzebruch (Chairman), David Mumford, Louis Nirenberg, Michael O Rabin, J A Rozanov, David P Ruelle, and I M Singer. The committee divided the programme of the Congress into 19 sections and appointed a panel to nominate speakers in each section. The committee also approved the inclusion of sessions for ten-minute Short Communications in the Congress. In October of 1985 the Program Committee selected 16 mathematicians to give Plenary Addresses and 148 to give lectures in sections. Invitations were issued over the following weeks. Of the 164 invited, 14 plenary and 132 section speakers were present at the Congress.
The Fields Medals Committee, consisting of P Deligne, J Glimm, L Hörmander, K Ito, J Milnor, J Moser (Chairman), S Novikov, and C S Seshadri arrived at its decisions in early 1986. The Nevanlinna Prize Committee, whose members were S Cook, L D Faddeev (Chairman), and S Winograd, also completed its work early that year.

The Board of Directors of ICM-86 appointed a Steering Committee, chaired by Andrew M Gleason of Harvard University, to oversee the arrangements. There were several subcommittees of the Steering Committee. The Local Arrangements Committee was chaired by John W Addison Jr. of the University of California, Berkeley; the Scheduling and Abstracts Committee consisted of Kenneth A Ross of the University of Oregon and Hugo Rossi of the University of Utah; the Committee on Special Funds was chaired by Richard D Anderson, Louisiana State University; and the Public Information Committee was chaired by Yousef Alavi of Western Michigan University.

Late in 1984, preliminary announcements of the Congress were sent out to all countries where some mathematical organisation could be located, requesting information on the number of copies of the First Announcement they required. The First Announcement, prepared in English, French, German, and Russian, was sent to these mathematical organisations in early 1985. The Second Announcement contained detailed information about the Congress, instructions about submitting abstracts for Short Communications, and the preregistration and housing form. It was sent to over 8,000 individuals starting in December of 1985. The Third Announcement, which contained a preliminary version of the scientific program, was distributed with the acknowledgment of preregistration between March and June of 1986. In all, 3,586 Ordinary Members, 340 Accompanying Members, and 69 Child Members registered for the Congress. Of these 721 Ordinary Members, 24 Accompanying Members, and 7 Child Members registered on site.

The Congress had several sources of revenue. A subvention was received early on from the International Mathematical Union. Grants were later received from the Air Force Office of Scientific Research, Army Research Office, Office of Energy Research, Office of Naval Research, National Science Foundation, Alfred P Sloan Foundation, and the Vaughn Foundation Fund. Through the efforts of the staff and the Committee on Special Funds, donations in the form of cash, goods, and services were received from a number of both public and private individuals and corporations. It is especially interesting to note that nearly 9,000 members of the American Mathematical Society made contributions to the Congress totalling over $30,000. Registration fees were$125 for Ordinary Members registered before May 15, 1986, and $163 thereafter;$60 and $78 for Accompanying Members; and$30 for Child Members.

The International Mathematical Union gave travel grants to 32 young mathematicians from developing countries or from countries with currency difficulties. The Congress waived their registration fees, and, thanks to funds received from the Sloan Foundation and GTE Laboratories Incorporated, was able to give them room and board in the residence halls.

All sessions of the Congress took place on the campus of the University of California, Berkeley. The Opening Ceremonies were held in the Greek Theater. All other plenary sessions were held in Zellerbach Auditorium. These sessions were simultaneously broadcast over closed circuit television to several large lecture halls. Video tapes of the plenary lectures were shown in the evenings. The lectures in the sections were given at a number of locations on the campus.

In addition to the invited lectures, about 700 ten-minute Short Communications were given and a large number of informal seminars were arranged by participating mathematicians. Abstracts of the Short Communications were published in a book distributed to all Ordinary Members. An educational materials exhibition consisting of 40 booths representing 26 exhibiting firms was open throughout the Congress.

The Steering Committee arranged various social events. The Chancellor of the University of California, Berkeley hosted an outdoor reception in the Faculty Glade on the first evening of the Congress, Sunday, August 3. A western-style barbecue and rodeo was held Thursday evening, August 7, in the Cow Palace in San Francisco, where 2,800 members attended. A jazz concert was presented on Saturday, August 9, and a classical concert on Sunday, August 10. Each of these concerts was attended by approximately 1,500 participants.
2.       Opening Ceremonies.

The opening session of the Congress was held in the Greek Amphitheatre of the University of California, Berkeley, at 9 a.m. on August 3, 1986.

The New Albion Brass Quintet played "Mini Overture for Brass Quintet" by Witold Lutoslawski.

2.1. Professor Jürgen Moser opens Congress.

Professor Jürgen Moser, President of the International Mathematical Union, opened the Congress with the following words:
On behalf of the International Mathematical Union I wish to welcome you to the International Congress of Mathematicians 1986. It is one of the primary functions of the IMU to support and promote the International Congress that is held every four years.

It may be good to recall that these Congresses have a long history going back to the last century. The first one was held in Zurich in 1897, the second in Paris in 1900. However, the sequence of Congresses was broken by the two World Wars. Now, since the Congress of 1950 in Cambridge, Massachusetts, at which the IMU was founded, we have been in the fortunate position to have had a long, uninterrupted sequence of Congresses. Today we are here to begin the tenth Congress in this series, and I am sure we are all united in the fervent hope that this series will continue uninterrupted well into the next century.

At the first Congress there were 216 participants; today more than 3000 mathematicians are attending this Congress. But regardless of this impressive increase, the Congress is still guided by the same principle: to foster personal relationships between mathematicians from different countries and to present a survey of the present state of science.

At a time of increasing specialisation and of rapid proliferation of mathematics into many subfields, these Congresses play a particularly important role in bringing together mathematicians of different interests and backgrounds. The danger of fragmentation of our science into many separate branches cannot be overemphasised. It is our hope that this Congress will help to counter this divisive tendency and give us a wide perspective of mathematics.

I am happy to greet mathematicians from about seventy countries here in Berkeley. I hope that the coming week will provide many occasions for fruitful exchanges of ideas as well as for lasting scientific and personal contacts.

The organisation of the Congress 1986 lies in the competent hands of the Organising Committee associated with the American Mathematical Society. The Chairman of its Steering Committee is Professor Andrew Gleason. I propose that we elect, here and now, by acclamation, Professor Gleason as President of the Berkeley Congress 1986.
2.2. Address by the President of the Congress.

Professor Gleason was elected and spoke as follows:
It is truly an honour to preside over this, the twentieth International Congress of Mathematicians. It is also a great pleasure to welcome you all to the City of Berkeley, a city made famous by the University of California and a city where the weather is almost always as pleasant as it is this morning. Speaking on behalf of the National Academy of Sciences and on behalf of the entire mathematical community I extend a special welcome to all those who have come from other countries. May your trip to the United States be a pleasant one, may you learn some new mathematics, make some new friends, and enjoy some of the marvellous sights in this vast country.

As you know, the Congress has been organised into nineteen sections. The Program Committee, under the able chairmanship of Professor Fritz Hirzebruch and with the aid of numerous subcommittees, has prepared an extraordinary scientific program with 16 plenary speakers and over 140 sectional speakers. In addition there are more than 400 contributed papers, and several specialised seminars have already been set up. We regret that a few of the invited speakers have been unable to come and that we have had to make some last-minute changes in the schedule. We will keep you informed of all the changes through the daily newsletter.

Mathematics has always been useful. Many of the oldest written records of human civilisation are accounting documents, and in fact today accounting still is the largest application of mathematics. But we are rapidly moving into a period in which more and more applications of mathematics are being found. New mathematical questions are being asked by scientists, engineers, and managers - often questions of an entirely different sort from those previously considered. New mathematical answers are being found often involving ideas previously thought to be entirely abstract and utterly non-utilitarian. As mathematicians we can justly be proud that the concepts we have worked so hard to develop are helping people to understand the real world just as they have helped us to understand our platonic world. There is a lesson in this, I think, and it is that, as we enter this new era dominated by computers, we should not fall into the trap of utilitarianism, but remember that the greatest progress in mathematics is always made by trying to understand the fundamental structures that underlie the subject rather than attempting to solve purely utilitarian problems.
2.3. Professor Mary Ellen Rudin addresses Congress.

He then recognised Professor Mary Ellen Rudin, Chairman of the United States National Committee for Mathematics, who said:
I would like to welcome all of you for the United States also. Exactly fifty years ago at the Congress held in Oslo, Norway, the first Fields Medals were awarded. The two 1936 Medals went to Jesse Douglas, who is no longer living, and to Lars Ahlfors, who was then a young man not yet thirty years of age. In special celebration of the fiftieth anniversary of the Fields Medal and of Professor Ahlfors' fifty years of continued contributions to mathematics, I would like to nominate Professor Lars Ahlfors to be Honorary President of the Congress.

Professor Ahlfors was elected by acclaim and came to the lectern and said:
I accept this great honour with a good conscience because I consider myself a link between this International Congress and the one in 1936, fifty years ago, the occasion on which the Fields Medals were given for the first time. I understand that my only duty here will be the pleasure of handing out the Fields Medals and the Nevanlinna Prize.

There are two traditions that go back to the very beginning. In the first place, the Committee to select the winners should consist of the top brass of contemporary mathematics. In 1936 the members of that Committee were G D Birkhoff, Caratheodory, Elie Cartan, Severi, and Takagi. Truly I would call that a panel of Olympian heroes. And I think that this tradition has been continued at subsequent Congresses. The other tradition is that the works of the winners should be commented on by prominent persons in the field. In 1936 both prizes were explained by Caratheodory.

As was mentioned there was no Congress until 1950, fourteen years later. On that occasion, which took place at Harvard, the medals were given to Atle Selberg and Laurent Schwartz, both known and admired by all mathematicians. From then on the Fields Medals have become more and more prestigious and it is a safe bet that many dream of getting it. Whether true or not that the existence of the medal has contributed to the phenomenal growth of mathematics both in quantity and quality during the last fifty years must remain anybody's guess.

Today it is safe to congratulate the winners in advance and I use this occasion to offer them my sincerest compliments to their success. I share their feeling of pride and accomplishment and I know that their continued success is guaranteed. I also share the disappointment of the many who may feel that they have been passed by. I wish them better luck next time or, if there is not a next time, that posterity will prove them right and the Committee wrong. Thank you.
The brass quintet then played selections from "A Brass Menagerie" by John Cheetham.

2.5. Professor Calvin C Moore addresses Congress.

Professor Gleason then introduced Professor Calvin C Moore, Associate Vice President for Academic Affairs of the University of California, who spoke as follows:
The President of the University of California, David Gardner, asked me to convey his regrets that he could not be here in person and to tell you how pleased he is that the University of California is hosting this distinguished International Congress. I am delighted to represent him on this occasion for at least two reasons. First of all, I am a mathematician myself, so it is a special pleasure to help with the official ceremony of a week of addresses, lectures, and opportunities for discussion with the most distinguished mathematicians in the world. As President Gardner noted in his message in the program of this opening ceremony, international cooperation has expanded in most disciplines, but nowhere has such cooperation flourished more than in the field of mathematics. Indeed, mathematics today is brimming with new ideas and developments; so this Congress promises more than its share of stimulation and excitement. Second, in my capacity as Associate Vice President for Academic Affairs of the University of California, I am pleased to welcome you to the University of California. I am also a faculty member here on the Berkeley campus and I can assure you that, even though the founders of the City of Berkeley named it after a philosopher rather than a mathematician, Berkeley has been a hospitable environment for mathematicians. The University is proud of its highly regarded Department of Mathematics and of the newly created Mathematical Sciences Research Institute, and I should add that Berkeley is situated in one of the most interesting and lively areas in the United States. I hope very much that you will take the time to explore some of the cultural and other high-points of the San Francisco Bay area during your stay. I think you will find it a stimulating and fascinating place. But my duty is not just to welcome you to the Berkeley campus, but to the entire University of California. Therefore, on behalf of the University and its Board of Regents, its Presidents, its Chancellors, its 9 campuses, and 150 Centers, Institutes, Laboratories, and Bureaus throughout California, its 148,000 students and 9,000 faculty members, including an aggregate of mathematics faculty of 450, I bring you greetings and warmest good wishes for a pleasant visit and a most successful conference.
2.6. Richard Johnson speaks on behalf of President Ronald Reagan.

Professor Gleason then read from a proclamation by Eugene Gus Newport, Mayor of the City of Berkeley, and a message from George Deukmejian, Governor of the State of California. He then introduced Richard Johnson, Acting Science Advisor to the President who spoke as follows on behalf of President Ronald Reagan:
On behalf of the American people, I am pleased to welcome you to the United States. For only the second time in the twentieth century the century that history will remember best for the remarkable discoveries and tremendous advances made in science and technology - this country is privileged to host the best mathematicians from some seventy nations around the world. I would like to use this occasion to congratulate, in advance, the winners of the Fields Medals and the Nevanlinna Prize. You recipients know who you are, but the suspense is beginning to wear on the rest of us, who are eager for the announcement and the opportunity to give you the accolades and recognition you so richly deserve.

Here in the United States, and I'm sure in your countries as well, my colleagues in science and technology engage in a certain friendly rivalry between our various disciplines. For example, I might say: "Physics is the most important of the sciences because it seeks to uncover the secrets of the tiniest particles of matter which make up our universe." The biologist might reply, "What could be more significant to humanity than understanding the building blocks of life itself." Or the chemist: "Don't underestimate the value of chemistry." But we scientists are not a contentious bunch by nature, and I can think of at least two points on which American scientists would echo resounding accord: First, science and technology are essential to improving the quality of life for all mankind, and second, mathematics is essential to every field of science and technology.

Of course, for me, it is ample honour to stand before this august gathering as a representative of the United States government, but I derive great personal pride and pleasure from the opportunity this event provides me, as a representative of the American scientific community, to express the gratitude of my fellow scientists to the mathematicians of the world, whose tireless efforts and manifold contributions form the very foundation upon which our work is built. We owe a large measure of our success to you. As science and technology progress at a breakneck pace - with mathematics as the fundamental driving force - we will continue to look to you for guidance, direction, and assistance. But today, I would also like to urge you to take on a role beyond solely enhancing your own research capabilities and contributions.

Throughout the world, science and technology increasingly are acknowledged to be essential to a nation's economic strength, industrial productivity, national security, and overall quality of life. While science is never static, right now we are in the midst of an unprecedented era of revolution in virtually every field of science and technology - a revolution that we created, and one which we have a responsibility to sustain and, indeed, to lead. The most important point, and the principal source of my concern, is that we must also possess the ability to respond to the opportunities afforded by this rapidly progressing scientific environment. To do so will require a strong base of talent - men and women, like you, who will lead the world's science and technology talent enterprise into the unforeseeable reaches of the twenty-first century. The pressing question is whether or not we will have the science and technology talent base adequate to meet the challenges of the future. Looking at current statistical indicators, the alarming answer, in most countries, is no.

We in the United States recognise that we have a serious problem. Fewer and fewer young people are studying science, mathematics, engineering, and technology; fewer and fewer young people are pursuing advanced degrees in these disciplines; and fewer and fewer young people are choosing careers in the scientific and technological fields. If this trend is allowed to continue, this country, and others similarly afflicted, will be unequipped to maintain the level of scientific leadership on which our improving standard of living, and the more general quality of life, depends.

To my mind, the most important aspect of this Congress is its international nature. Like mathematics itself, it transcends the barriers of language and ideology, the boundaries of geography, and the adversarial climate of competition. You are convened here this week, in the spirit of cooperation, to share your latest discoveries and to learn about new developments, and to pursue your mutual interests and the solutions to common problems. Perhaps that is what compels me to share this growing problem of an inadequate talent base with you, and to turn to you for help in solving it.

All of us who are not mathematicians feel a keen sense of envy at the abilities you possess. The intrinsic beauty (especially as now seen in 3-D colour), the coherence, the elegance of mathematics makes it an entirely worthwhile pursuit in its own right. You have the rare talent to appreciate fully the depth and power of your subject. But, with ability and talent come obligations and responsibilities, especially to those not so blessed. What you do in your laboratories, at your computer terminals, and in your studies is remarkable - indeed, it has changed the course of the world. But you must do more. It is no longer enough to be practitioners of science. You must also become citizens of science - leaders of a uniquely gifted community, bound by the common goal of assuring that the progress you have made and the improvements you have wrought for humanity will continue. Especially, you must help to assure that there is an expanding, well-qualified generation of scientists, engineers, and technologists to succeed you.

What the young students of the world need most, in my view, is someone to imbue them with the excitement and enthusiasm that brought you here today, someone to communicate to them the importance of science and technology to society, to demonstrate to them that mathematics, physics, biology, chemistry, and their sister disciplines are challenging pursuits, worthy of their highest interest and attention. Many young teachers, I hasten to add, need to hear this as well. No one is better qualified to carry those messages than the individuals who have made eminently successful careers out of the quest for knowledge. In short, the best spokesmen for mathematics, physics, biology, and chemistry are mathematicians, physicists, biologists, and chemists.

Unfortunately, too many students, especially at the grammar school level, still regard science and mathematics as "too hard," too difficult. To them, these subjects look ponderous, abstract, and impenetrable, and many children dismiss them before they've had a chance to appreciate the fun and excitement inherent in the challenge of discovery. At that moment, a potential scientist or mathematician is lost - for now, more than ever before, the attitudes and academic foundations developed in the early school years determine the future course of a student's education and career. In the current scientific and technological climate, with a growing demand for an expanded talent base, we cannot afford to lose these students in grammar school - well before there is a need for a carefully weighed career choice.

Therefore I believe it is incumbent upon you to become active participants in the community of science, to help others to grasp the broad importance of science and technology to the very fabric of our daily lives - to help them understand the momentous potential of science and technology to enhance the quality of life for all mankind. Certainly you can point with pride to the eradication of diseases, to the agricultural revolution, to hurricane forecasts, to the transportation, automation, electronics, and information revolutions, as examples. We must, of course, recognize and limit the risks associated with modern uses of science and technology. But, further, we must convey to our young students not only factual information on the present extensive benefits, but also the great challenges, personal opportunities, and potential future benefits associated with the scientific and technological disciplines.

I urge you to get involved in developing the talent bases needed in the twenty-first century for expanding the contributions of science and technology to humanity. On a small scale, you can, of course, accomplish a great deal simply by going to your local schools and talking to the teachers and the children about the contributions that they have the power to make through science. On a larger scale, you can work through existing mechanisms, like your professional societies, and enlist the help of your colleagues in devising innovative solutions to this set of perplexing problems. That, after all, has been your mission within the discipline of mathematics; now I challenge you with this added responsibility to mankind.

Individuals of the highest calibre must be sought out, trained, nurtured, and encouraged by every means possible to carry on the tradition of excellence you represent. I can think of none better suited to the task than you, who, by your example, your actions, and your words can express the merit and rewards of a life in science.

In closing, I have a personal message from a man who is himself excited by science and technology and firm in his commitment to their invaluable role in international health and prosperity.
[The text of President Reagan's message may be found below. See Section 3.]

2.7. The Nevanlinna Prize is awarded.

Professor Gleason thanked Dr Johnson and asked him to convey the thanks of the meeting to President Reagan. Then the brass quintet played "Fanfare" by David Amram. Professor Gleason announced that Professor Ahlfors would present the medals, and then called on Academician Ludwig Faddeev, the Chairman of the Committee to Award the Nevanlinna Prize, who responded as follows:
The Nevanlinna Prize has a much shorter history than the Fields Medal, as we have heard from Professor Ahlfors. It was established to a great extent due to the efforts of our past president, Lennart Carleson, with the aim of stressing the importance of the applied aspects of mathematics. Helsinki University generously provided necessary funds, so it is only appropriate that it is called the Nevanlinna Prize. More exactly, the prize is to be awarded to a young mathematician for outstanding work in the mathematical aspects of information science. "Young" means the same as for Fields Medals. The Prize is awarded for the second time. The first award was given in the Warsaw General Assembly. The Committee for the award for this year consisted of Professor Cook, University of Toronto; Professor Winograd, IBM; and myself. Let me inform you of our decision. The 1986 Nevanlinna Prize is awarded to Leslie Valiant, Harvard University, USA. I will ask Professor Ahlfors to present the Prize.
Professor Valiant then came forward to receive the Nevanlinna Prize from Professor Ahlfors.

2.8. The Fields Medal awards.

Professor Gleason then recognised Professor Moser in his capacity as Chairman of the Committee to Award the Fields Medals and Professor Moser reported as follows:
In the early 1930s the organisers of the Toronto Congress decided to award at each Congress two gold medals for outstanding achievements in mathematics. The driving force behind this resolution was John Charles Fields who expressed the wish that this prize be given, I quote, "in recognition of work already done and also as an encouragement for further achievements on the part of the recipients and as a stimulus to renewed efforts on the part of others." It is interesting to note that Fields insisted "that the medals be of a character as purely international and impersonal as possible." As a matter of fact, Fields' name does not appear on the medals. As we heard today already, the first medals were awarded at the 1936 Congress in Oslo, four years after Fields' death. Looking at the impressive list of Fields Medalists over the past fifty years, it is obvious that Fields' vision has been realised most successfully.

Since the Congress in Stockholm in 1962, the International Mathematical Union has been entrusted with the selection of the Fields Medal winners. The Committee for the Fields Medalists for this Congress consisted of P Deligne, J Glimm, L Hormander, K Ito, J Milnor, S Novikov, C S Seshadri, and myself as Chairman. The Committee had the difficult task of making a selection from an impressive list of brilliant mathematicians. It goes without saying that such a decision has no unique solution and that all the excellent mathematicians considered could be awarded the prize. On behalf of the International Mathematical Union, I want to thank the members of the Committee for their efforts to reach the decision in a spirit of responsibility and cooperation.

The Committee followed the tradition of limiting the awards to mathematicians under forty years of age. After long deliberation and extensive consultation it was decided that three young mathematicians be selected for this award. We are all deeply impressed by their exceptional achievements and I am happy to report that the Committee was unanimous in its support of the three Fields Medalists for 1986. Their names are

Simon Donaldson

Gerd Faltings

Michael Freedman
I offer them our warmest congratulations. I now ask the winners to come up to the podium.

The winners came forward and received their medals from Professor Ahlfors, and then the brass quintet played selections from "Three Pieces for Brass Quintet" by Minoru Fujishiro.

Professor Gleason congratulated the winners and announced that talks concerning their work would be given in Zellerbach Auditorium.

The session adjourned at 11:30 a.m.

3.       Ronald Reagan's message to the Congress
THE WHITE HOUSE
WASHINGTON

July 3, 1986

I extend a warm welcome to the thousands of mathematicians from around the globe who are attending the quadrennial International Congress of Mathematicians. All of us are pleased and, indeed, honoured that the United States was chosen to host this prestigious meeting.

I am gratified to note that this Congress will be the occasion for the awarding of Fields Medals to outstanding members in recognition of their contributions to mathematics. I wish to extend my congratulations to the winners. Mathematics is the enabling force for the revolutionary advances being made throughout the world in science and technology. The fundamental role of mathematics is becoming increasingly apparent in business, industry, and government. Modern mathematicians are giving new meaning to the famous tenet of the ancient Pythagoreans that "all is number."

It is appropriate that these honours be presented at an international meeting, for mathematics is intrinsically international, cutting across geographical and cultural boundaries with its own language. International competition is a concept alien to the study of mathematics. Indeed, cooperation between mathematicians of different nations has been a long-standing tradition.

I wish you the best for a successful meeting. God bless you.

Ronald Reagan
4.       Closing Ceremonies.

The final session of the Congress was held in Zellerbach Auditorium at 11 a.m. on August 9.

4.1. Professor Moser's Report.

Professor Gleason opened the meeting and called upon Professor Moser for a report on the International Mathematical Union. Professor Moser spoke as follows:
We have come to the end of this Congress with a rich and interesting program. At this occasion it is a tradition for the President of the International Mathematical Union to address the Congress and to report on the work and the decisions taken at the General Assembly.

Before doing so, it is my privilege to greet here Professor Marshall Stone, one of the past presidents of the International Mathematical Union. It was Professor Stone who played a decisive role in re-establishing the IMU in 1950 after it ceased to exist in 1932. Professor Stone, we are happy to have you with us at this Congress.

Since 1962 the IMU has been responsible for preparing the scientific program of the Congresses. This task lies in the hands of the Program Committee, which is appointed in part by the IMU and in part by the host country. For the present Congress this Program Committee consists of Fritz Hirzebruch (chairman), Enrico Bombieri, Lennart Carleson, David Mumford, Louis Nirenberg, Michael Rabin, Yuri Rozanov, David Ruelle, and Isadore Singer. I would like to express our thanks to Professor Hirzebruch and his Committee for presenting us with an excellent program of great diversity.

It was a great disappointment for all of us, however, that many of the invited speakers from the Soviet Union did not come to Berkeley; in fact, almost half of the Soviet speakers were not present. This is a serious loss for everybody concerned and defeats the purpose of the Congress. It is most important for any Congress that the invited speakers are able to attend in order to deliver their lecture in person and to take part in the exchange of ideas.

We are aware that our Soviet colleagues worked very hard at resolving this problem, and we appreciate their efforts. Also, most of the manuscripts of the absent speakers were made available and could be presented by other mathematicians.

Regardless of circumstances, it is always a disappointment if invited speakers from any country are unable to attend, and let me express our hope that at the Congress 1990 all invited speakers from all countries will be present.

As you may know, the IMU is a member of the International Council of Scientific Unions (ICSU) and as such is committed to the ICSU principle of free circulation of scientists. I am happy to report that to the best of my knowledge the host country has granted all visas which have been applied for. In some difficult cases the help from ICSU was indeed essential. This again demonstrates the importance of the ICSU principle for our Union. Let me add that two weeks ago at the General Assembly of the IMU a resolution was adopted reaffirming an ICSU article on non-discrimination. I will now give a brief report on the General Assembly of the IMU. On July 31 and August 1, the General Assembly met in Oakland, California, and I want to inform you about the main decisions taken there.

Two new members were accepted by the General Assembly, and Ivory Coast as well as the People's Republic of China belong now to the IMU. Let me mention that the question of the China membership has a long and complicated history. Here I want to thank especially the Secretary of the IMU, Professor Olli Lehto, for his untiring efforts to bring this problem to a successful solution. We also want to express our appreciation to China-Taiwan for its cooperation in helping to solve this problem.

The total number of IMU members is now fifty-three. Furthermore, the General Assembly elected new Committees and I want to report to you the outcome. I begin with the new IMU Executive Committee for 1987-1990. It consists of the President, Ludwig Faddeev; the Vice-Presidents, Walter Feit and Lars Hormander; the Secretary, Olli Lehto; and the members, John Coates, Hikosaburo Komatsu, László Lovász, Jacob Palis, Jr., and C S Seshadri. To this is to be added the Past President as ex officio member.

Now I come to the sub-commissions. The IMU has two sub-commissions: the International Commission on Mathematical Instruction (ICMI) and the Commission on Development and Exchange (CDE). The new Executive Committee of ICMI is as follows: the President, Jean Pierre Kahane; two Vice-Presidents, Peng-Yee Lee and Emilio Lluis Riera; the Secretary, A G Howson; and the three members are Hiroshi Fujita, Jeremy Kilpatrick, and Mogens Niss. The new CDE consists of the Chairman, M S Narasimhan, and the members Jean Pierre Bourguignon, Phillip Griffiths, M Immanaliev, A O Kuku, Lê Dung Tráng, Shingo Murakami, and Giovanni Vidossich. To the ICMI Executive Committee and the CDE, the IMU President and Secretary have to be added as ex officio members.

Finally, the Site Committee made its proposal to the General Assembly about the location and time of the next Congress.

Before giving the word to the next speaker, let me conclude with some personal remarks. In my work during the past four years as President, I was fortunate to have the advice and support from many colleagues throughout he world. I would like to thank all of them-in particular, the members of the Executive Committee and the Fields Medals Committee. The strongest support , however, came from Olli Lehto, who together with his secretary, Mrs Tuulikki Mäkeläinen, was always ready to help at any time in every difficulty. Without his encouragement and sensitive judgment I would not have carried out this task. To both of them go my warmest thanks.
4.2. Invitation to ICM 1990.

Professor Moser then yielded the floor to Professor Nagata who invited the audience to the next International Congress with these words:
On behalf of the Japanese Committee for Mathematics, I have the honour of inviting you to the next International Congress of Mathematicians in Kyoto. Kyoto had been the capital of Japan for about one thousand years and can show you some of the old Japanese culture.

We are quite aware that it must be a difficult task to organise such a big meeting. However, taking into account the help of the International Mathematical Union and also the cooperation of the mathematical community of the world, I believe that we will be able to overcome the difficulties.

Hoping that you will accept our invitation, I welcome all of you to the next International Congress of Mathematicians to be held in August 1990 in Kyoto.

Professor Gleason then recognised Professor Adams who said:
Ladies and gentlemen, on such occasions it is right and proper for guests to thank their hosts.

We, the mathematicians of the world, have come to Berkeley from our many countries. We have had a most successful conference, and we wish to express our appreciation. We have had the chance to meet and speak to each other; we have heard many splendid lectures and thanked the speakers with our applause. To arrange all this has taken many people much work. As Professor Moser has explained, to choose the speakers was the task of a committee structure set up by the IMU, and presided over by Professor Hirzebruch. We know that mathematicians of all countries give their services in this most gladly; we still thank them again for making an informed choice on our behalf.

Almost all the work, however, falls upon the hosts. For this purpose the American Mathematical Society set up a committee structure presided over by Professor Gleason, and we thank all those involved. We also thank the permanent staff of the American Mathematical Society, whom you may have seen at the Congress Bureau. These people have had to book accommodation, arrange all the things that need arranging, and assume the responsibility of running a fair-sized business. (By the way, there are still a few cowboy hats left for sale ....) The praise and gratitude due to them will best be estimated by those who have tried anything of the sort before, for example Professors Czeslaw Olech, Olli Lehto, and Maurice Sion. If the rest of us would regard smaller exercises of the kind with apprehension, our applause will start on the right foot.

I ask all the mathematicians of other countries to join me in thanking our American hosts, and all of us to thank those who have done the work.

After thanking Professor Adams, Professor Gleason spoke as follows:
There are two principal ingredients in a successful Congress, the scientific program and the physical arrangements. The Program Committee chose an excellent roster of speakers, a very distinguished roster indeed, and I thank them for doing so. And I thank the speakers, many of whom came from afar to share their insights with us. The lectures were exceptionally fine. The program was further enriched by several special seminars and many contributed papers. The organisers and speakers deserve our thanks. The success of a scientific meeting depends on the participants. Over 3500 mathematicians attended this Congress. The total registration was 3970. Thank you all for coming. Speaking for the American mathematical community, I want to thank particularly those who came from other countries. We were honoured by your presence and we wish you great success in solving the problems that we have heard about in the lectures.

There are many members of the American mathematical community who worked hard for this Congress during the past two years - the list is so long that I cannot mention them all - but I would like to express my personal thanks and also to relay the generous words of Professor Adams, which I believe truly represent the feelings of the membership. I do want to mention the Steering Committee and, in particular, Kenneth Ross and Hugo Rossi, who were in charge of scheduling; the Board of Directors, under the chairmanship of Ronald Graham; the Committee on Special Funds, chaired by Richard Anderson; the Public Information Committee, chaired by Yousef Alavi; and the Local Arrangements Committee, chaired by John Addison. All of these Committees have pulled more than their weight and the results are manifest.

Among the mathematicians who worked for this Congress is one who deserves very special mention, Jill Mesirov, the Executive Director of ICM-86, who has been in daily contact with the preparations for the Congress since the beginning; we owe her a special vote of thanks.

I want to acknowledge here the outstanding contribution of the organisers of the Warsaw Congress. They received our Congress Manager most cordially in 1983 and shared their experiences with her. Their advice was invaluable.

Finally, I want to thank the Congress Manager, Hope Daly. She has been our field marshal for the past four years, directing all of the preparations, yet never shrinking from the minor tasks when an extra hand was needed.

Professor Gleason then presented Ms Daly with a silver pendant and a bouquet. There was much applause, after which he continued:

This Congress is part of a long tradition of internationalism. At least since the days of Archimedes, mathematicians have corresponded with one another and travelled great distances to study, teach, and confer. As the expense of printing and traveling has declined, the tradition has strengthened. Now hundreds of mathematical books and journals are published every year. These pass freely over international boundaries and propagate new mathematical ideas throughout the world. Mathematicians travel ever more frequently from one university or institute to another. As we think of this Congress, let us resolve to maintain and expand our great tradition of freedom to study, travel, and confer so that the Kyoto Congress will be even more truly international.
Professor Gleason then declared the Congress closed.