Dear Colleagues, Ladies and Gentleman:

I am greatly honored and pleased to welcome you all to ICM 2002, the 24th International Congress of Mathematicians.

This is in many ways a very special Congress. Indeed, it is the first in the new Millennium and, therefore, we are bringing the dreams of Cantor and Felix Klein, dreamed in the late 1900s, into the 21st Century. They realized, then, that mathematics was becoming too large and diversified a subject and that was almost impossible for one person to embrace, like probably was the case of Monge, Laplace, Lagrange and Gauss, among others, at the turn of the 19th Century. Thus, interaction among mathematicians both at a national and international level was the clear road for its development. Their dream was not only robust in time, but has grown in dimension; mathematics has become more and more international, and solidarity across countries has been increasing at a fast pace. This is occurring not only at a world basis, particularly through the activities of IMU, among which the ICM is a major event, but also in regional scenarios, as indicated by the rather recent creations of the European Mathematical Society and the Latin American and Caribbean Mathematical Union, following that of the African Mathematical Union and of the International Council for Industrial and Applied Mathematics. The first two organizations are affiliated to IMU, and we have solid relations with the last ones.

The 24th ICM is also unique because for the first time it is taking place in a developing country, and in fact in the fastest growing country in the world at present, with a population which is about a fourth of humanity. Per se this makes the ICM more inclusive and being inclusive is a basic principle of our Union, as also shown by our joint efforts with the Local Organizing Committee in providing the opportunity to more than 400 colleagues, young and senior, from less affluent parts of the world, to participate in the Congress. By having the Congress here, we are giving our trust to China for its commitment to mathematics and in particular to its young talents. But China is also paying a precious tribute to the Union, by the presence among us, for the first time in our history, of the highest authority of the host country, President Jiang Ze-min. About a year and a half ago, he accepted our invitation to be in this Opening Ceremony and jointly with us award the Fields Medals. In doing so, the President is showing his appreciation for our science and its importance to the world of today. We are very confident that the Congress here in China will mark a formidable change in the level and scope of activities of mathematics in this country: a tree that was planted by S.S. Chern, L. K. Hua and K. Feng, as well as by C.H. Gu, W. T. Wu and S. Liao, and more recently S. T. Yau and G. Tian, among others.

This Congress is also a culmination of an intense period of activities in mathematics throughout the world, as well as for achieving a certain maturity concerning the perspective for its future development. In this respect, besides fundamental research, the importance of the interaction of mathematics with other areas of science, beyond the classical case of physics, is now largely accepted. Also, more emphasis in applications is to be given. Moreover, there should be no division between pure and applied mathematics in accordance with Pasteur's beautiful sentence that there is no applied science, but applications of science. In terms of activities, we had an intense celebration of the Year 2000 as the World Mathematical Year: IMU published a book "Mathematics: Frontiers and Perspectives"; co-sponsored major conferences in Europe, Latin America, Africa and Asia, one of them through its Commission on Mathematical Instruction, and promoted many mathematical exhibitions and events directed to the general public. Such a celebration was part of a Declaration made by Jacques-Louis Lions, in Rio de Janeiro, in 1992.

Unfortunately, I have to register that he, Jurgen Moser and Lion's former adviser, the Fields Medallist Laurent Schwartz passed away in the last years. Of prime importance in this period, has been the activity of the Union's Committee on Electronic Information and Communication and the work of the IMU Commissions on Development and Exchanges (CDE), Instruction (ICMI) and History (ICHM).

The present Congress is also special in other ways. For the first time, the IMU General Assembly has elected a woman to its Executive Committee and also a Chinese. Furthermore, at this occasion, the mathematical community can commemorate the creation of two new prizes. The first, called the Gauss Prize for Applications of Mathematics is to be jointly awarded once every four years by IMU and the German Mathematical Society. The second, in honor of Abel, shall be awarded every year by the Norwegian Academy of Sciences: similar to the Nobel Prize, it has the potential to change, in years to come, the landscape of mathematics in the world scenario of sciences.

Finally, on behalf of all of us, I wish to express our sincere gratitude to the Chinese Institutions that made the Congress possible and most especially to our colleagues Zhi-ming Ma, K.C. Chang, Daqian Li, Weiyue Ding and Ya-xiang Yuan for their warm reception and excellent organization.

Thank you very much.

Today, we are particularly overjoyed at the grand opening of the 24th International Congress of Mathematicians. On behalf of the China Association for Science and Technology and the Chinese scientific community, I would like to express our warmest welcome to participants from all over the world and our sincere congratulations to the newly awarded Fields medallists and the winner of Nevalinna Prize.

The reason of our being particularly overjoyed lies primarily on the fact that the subject of this Congress is mathematics, which has been respected as "the queen of sciences" for its brilliant intellectual accomplishments, as suggested by the examples of the discovery of Goedel's theorem and the proof of the Fermat Last theorem in the last century. Mathematics is also "the servant of sciences" as explained by the great German mathematician Gauss when he spoke of "the queen of sciences". In the past century the application of mathematics witnessed rapid and more exciting development. The highly abstract languages, structures, methods and ideas created by mathematicians have been repeatedly proven to be universal instruments useful to other fields of science and technology and to economic and social development. This truly reflects the marvelous and close relations between mathematical theories and the objective world. Just by mentioning Riemann geometry and the theory of Relativity, Turing machine and the real computers, Radon integral and the CT scanners, we can see that mathematics is exerting more and more important influence on the modern civilization and social progress.

China had created glorious scientific and technological achievements in ancient times before a decline set in some three or four centuries ago. In 1915, the first Chinese comprehensive scientific society -"the Chinese Society for Science" was founded. Its founders were a group of students studying abroad, including a mathematician who was the first Chinese Ph.D. in mathematics. Starting with only 180 members at the beginning, the seeds it sowed are blossoming and bearing fruits in China today. The reform and opening up policy that China has adopted since 1978 has given tremendous impetus to the country's science and education. We have built up a well distributed system of research and a network of academic societies. Our scientists are working on many frontier projects in various fields. In the past 20 years Chinese scientists succeeded in constructing the electron-positron collider, developing large computers and strong laser light sources, breeding hybrid rice and determining genetic codes, developing sophisticated word processing systems for Chinese characters, and setting up terrestrial stations for satellite remote sensing and nation-wide network for ecology observations. In mathematics, Chinese scholars have achieved important results in fields such as number theory, theorem-proving by computer, differential geometry, topology, complex analysis, probability and mathematical statistics, PDE, functional analysis, numerical analysis and control theory and so on.

Today, we have entered a new age, in which the social development is more dependent than ever before on the advancement of knowledge. This situation has brought about both opportunities and challenges to the development of science and technology in China. We have to work hard to keep pace with science and technology development in the world and strive to make greater contributions to the progress of human society. Science is an international endeavor, and no nation could be successful in isolation. International exchanges and cooperation in mathematics is of greater significance. As a universal language of science, mathematics plays a unique role in merging diverse cultures on the Earth. A typical example is the transmission of the oriental decimal numeration and the Greek geometry in history. I hope sincerely that the first International Congress of Mathematicians in the 21st century will open a new page in the history of world cultural exchanges. We will continue our efforts to promote international cooperation in science and technology.

In conclusion, I wish the Congress a great success, and hope that you all enjoy your stay in Beijing.

Thank you very much.

It is my great pleasure to welcome you to this gathering. We are in an ancient country that is very different from Western Europe where modern mathematics started. In 2000, we had a mathematics year, an effort to attract more people to math. We now have a vast field and a large number of professional mathematicians whose major work is mathematics. Mathematics used to be individual work. But now we have a public. In such a situation a prime duty seems to be to make our progress available to the people. There is clearly considerable room for popular expositions. I also wonder if it is possible for research articles to be produced by a historical and popular introduction. The net phenomenon could be described as a globalization. It is more than geographical. In recent studies different fields were not only found to have contacts, but were merging. We might even foresee a unification of mathematics, including both pure and applied, and even the possibility of the emergence of a new Gauss.

China has a long way to go in modern mathematics. In recent contests of the international mathematical Olympiad China has consistently done very well. Thus China has begun from the roots and China has the advantage of "number" (of people). Hopefully this Congress will be a critical point in the development of modern math in China. The great Confucius guided China spiritually for over 2000 years. The main doctrine is "?"?pronounced "ren"?, meaning two people, i.e., human relationship. Modern science has been highly competitive. I think an injection of the human element will make our subject more healthy and enjoyable. Let us wish that this Congress will open a new era in the future development of math.

Today, mathematicians from all over the world are gathering here for the first International Congress of Mathematicians in the new millennium. On behalf of President Jiang Zemin and the Chinese government, I have the pleasure to extend to you our warmest welcome.

No one could have imagined the extraordinary evolution of science and technology over the past century. Space exploration, nuclear energy, computers and information technology, not to mention biological engineering, are all milestones that mark a new era of knowledge for humankind. Our social progress depends on scientific innovation, and mathematics is fundamental to science. Mathematics expressed the theory of relativity and the quantum mechanics in the early 20th century; since then mathematicians has played a vital role in inventing computers, designing space and energy programs, and investigating the structure of DNA molecules. Mathematics is the language of the universe.

Mathematical methods are used extensively in economics, medicine, agriculture, architecture, arts and all other fields of modern knowledge. As Roger Bacon pointed out, mathematics is the key to all branches of science. Today mathematics is the keystone of high technology, and, in a sense, the symbol of modern civilization. In this light, the Chinese government is especially delighted to see this congress being held in Beijing. As President Jiang Zemin clearly expressed when he met with Professor Chern Shing-shen, IMU President Palis and other mathematicians in October 2000, "the Chinese government fully supports hosting the 2002 International Congress of Mathematicians in Beijing. China wishes to take this opportunity to promote math research and education in the country, in an effort to bring them up to the world advanced level in the early 21st century and lay a solid foundation for the future progress of science and technology in China."

As a developing country, China is marching on the road toward modernization. It has been a century-long pursuit for the Chinese people to revitalize their country through development of science and education. This historical process has been even further accelerated in the last two decades by reform and opening up policies, as both young talents and accomplished experts emerge in great numbers on the international scientific scene. The Chinese government has fully supported all endeavors to pursue this development, including a series of programs launched nationwide to promote basic scientific research, especially in mathematics. For example, in the past four years, the National Science Foundation of China has doubled its funding for mathematics, and the government has allocated thousands of millions of yuan to support the Pilot Knowledge Innovation Program in the Chinese Academy of Sciences.We are aware that China still has a long way to go before reaching the advanced world levels in science and technology. Science knows no boundaries. The advancement of science requires peace, stability and cooperation. In this regard, I believe that the International Congresses of Mathematicians, with over a hundred years of tradition, sets the example. Hosting the 24th Congress in Beijing is a good opportunity for Chinese scientists to learn from and to cooperate with their colleagues abroad. I hope that this congress will mark a new starting point for the development of mathematics and science in China. As the first congress ever held in a developing country, I also hope that this congress will inspire a new era of international cooperation for global scientific community.

In about 10 minutes' time, the new Fields medallists and the winner of the Nevanlinna Prize will be announced and awarded. I would like to take this opportunity to offer them my sincere congratulations. Their achievements not only represent their distinguished contributions to mathematics, but to world cooperation and the well-being of all humankind.

In conclusion, I wish this congress a great success, and all our guests a memorable stay in China.

Good afternoon! Today, I feel very honored to be present at International Congress of Mathematicians 2002. Here, on behalf of Beijing Municipal Government and the thirteen million people of the city, I would like to extend my sincere congratulations to the opening of this congress and express my warm welcome to scientists and guests participating in the conference.

ICM is committed to the research in one of the most basic disciplines of human knowledge. The intellectual fruits achieved in the field by mathematicians exert far-reaching influences on the progress of science and technology of human society and on the development of social culture and people's way of life. The fact that this conference is the first of its kind in the new century and the first session ever held in a developing country has given special significance to this meeting. The Municipal Government and myself are very pleased to be able to provide support and service to the meeting and we wish to present our highest compliments to mathematician and their exploration of reason.

The mathematic tradition in Beijing can be traced back to ancient times. Since the end of the nineteenth century, Beijing has played an important role in promoting the scientific and cultural exchanges between the east and the west. The city has nurtured numerous brilliant mathematicians, from Zhu Shijie in the thirteenth century to professor Chen Xingshen who is present here today. Now, Beijing continues to maintain its position as China's major center of mathematic education and research. Some two thousand mathematicians from the mathematic departments of tens of universities and research institutions such as Chinese Academy of Sciences are engaged in the education and high-level research of the field in an all-round way. At the same time, they keep extensive and close contacts and cooperate with their colleagues from countries and regions around the world.

Isn't it a pleasure to have friends from afar! The ancient and modern city continues its three thousand years history of civilization and composes its ode to the 2008 Olympics. We sincerely welcome you to tour around the city during your spare time. The city's historical monuments and sites will demonstrate you the charm of Chinese culture. The rapid development will bring your thoughts to the future of an international metropolis. I hope that all the guests will have a pleasant and efficient stay in Beijing and a beautiful memory in your heart.

May the conference a complete success!

After four years of preparation, the 24th International Congress of Mathematicians is now opening. It is my great honor on behalf of the Local Organizing Committee and the Chinese Mathematical Society to welcome you all to the ICM-2002 in Beijing.

Four years is long for expecting, but short for preparing. Since the 13th General Assembly of the International Mathematical Union in Dresden in 1998, at which Beijing was chosen as the site of ICM-2002, Chinese mathematical community has been racing against time to work for today's ICM-2002. The first step was the setting up of the Local Organizing Committee in September of 1998, right after the Berlin Congress. The Committee, consisting of representatives from Taiwan, Hong Kong and overseas Chinese mathematicians, has been cooperating closely with the Executive Committee of IMU to ensure a smooth and effective preparation of this Congress. The preparation of the Congress is a symphony of international cooperation. I would like to take this opportunity to thank colleagues world-wide who have rendered all kinds of help and assistance. I am indebted in particular to IMU President Jacob Palis, Past President David Mumford, and Secretary Phillip Griffiths for their all-out support. Special thanks goes also to my German predecessor Professor Martin Groetschel, whose experiences of organizing the Berlin Congress are really helpful to us. The preparation of the Congress has won wide social and governmental support in China. The support from the government is evidenced by the presence of President Jiang Zemin and other Chinese leaders at this opening ceremony. The financial support from the Chinese government was even more than expected. The Organizing Committee of ICM-2002 is grateful to the Chinese ministries and agencies that were listed on the slide shown left, the total of their funding is 10 million Chinese yuan, which amounts to about 1.2 million US dollars.

The spirit for the ICM-2002 has been high among the Chinese public. Many Chinese scholars, teachers, industrialists, and even students were eager to contribute not only to help to prepare a successful ICM, but also to make the Congress a new start point for development of mathematics in China. Regarding the donations only, the Organizing Committee has received contributions of 3 million Chinese yuan from universities, industries and individuals. This amount is significant in view of that China is still a developing country. Please watch the slides at left, which show the major donors, and I, in the name of the Organizing Committee of ICM-2002, would like to extend to them our sincere thanks.

While the financial support is important, the scientific program is always the core of the Congress. Thanks to the Program Committee headed by Professor Y. Manin and the 19 international panels, the selected 20 plenary lectures and 174 invited lectures will, I believe, represent the latest advancement and frontier achievements in our science. The lectures given by the newly awarded Fields medallists and winner of the Nevanlinna Prize will of course highlight the scientific program of the Congress. On the other hand, more than 1200 short communications and poster presentations arranged by the local scientific committee will reflect the widespread active participation in the development of mathematics in recent years.

Up to now, the ICM-2002 has 4,270 registered participants from 101 countries and regions, among whom 1 percent are from Australia, 3 percent from Africa, 56 percent from Asia, 16 percent from America, 24 percent from Europe. As the ICM held for the first time in a developing country, we see from above statistics that the percentage of the participation of mathematicians from developing countries is above 52 percent. The success of the financial program enabled us to make good our promise by various means to support financially about 400 scholars from developing countries and Eastern Europe (here I should thank the IMU for covering international traveling expenses for approximate 200 participants who are young mathematicians from developing countries and mathematicians from Eastern Europe, Africa and Latin America). In addition, the Organizing Committee has supported a number of mathematicians from western part of China as well.

Keeping in mind that it is the first ICM of the 21st century, the Organizing Committee has paid due attention to the programs for the general public, and considered it to be important for a new information era to attract the public to modern mathematics. Public talks on a range of topics and special activities related to the Congress were arranged for that purpose. Part of them are shown on the slide, among which I would like to mention here two examples: the Juvenile Mathematics Forum and the ICM-2002 Mathematics Summer Campus, both were organized to raise the enthusiasm of young generation to mathematics that may have impact on the future of mathematics.

The 46 satellite conferences form a landscape of ICM-2002. The slides show the list of satellite conferences, which are distributed geographically over 26 cities in different parts of China as well as 6 cities in Japan, Russia, Singapore, South Korea and Viet Nam. Almost for each satellite conference there is a story of international cooperation, the participation in of a number of Fields medallists, winners of Wolf Prize and winners of Nobel Prize made the whole program even more inspiring. Though it has been a tradition of ICMs to have a series of satellite conferences, the ICM-2002 makes the satellite conference program broader in scale and more meaningful to a successful ICM. I would like therefore to express my thanks to all the local organizers of satellite conferences for their contribution.

Last but not the least, a few words about the logo of the ICM-2002. The design was based on a diagram drawn by the 3rd century Chinese mathematician Zhao Shuang to demonstrate Pythagoras theorem that appeared in ancient China first in Zhou Dynasty?11th century B.C-3rd century B.C?. Some inspirations were put in to transform it to our logo. Let me show quickly by the video how does it make sense. First, by opening the edge of the outer square and enlarging the square inside, it will symbolize that mind of mathematicians are open, and that China is open. Next, varying colors make the diagram more like a rotating pinwheel to symbolize the hospitality of Beijing people.(Pinwheel is a folk toy which you may see children in Beijing's hutong playing with and greeting you: "Welcome, welcome!) Welcome to ICM-2002, welcome to Beijing. Let us join hands to lift the veil of a new epoch of mathematics. I wish the congress a great success, and wish you all pleasant stay in Beijing.

Sixteen years ago I attended as an observer on behalf of the Chinese Mathematical Society the 10th General Assembly of the International Mathematical Union in Oakland, at which CMS became a member of the IMU. I am very happy to see that the cooperation between Chinese mathematicians and the international mathematical community has been developing rapidly and fruitfully since then, and the inspiring progress is demonstrated today by the opening of the 24th ICM in Beijing. It is a high privilege and an honor for me to extend to you my warmest welcome.

Our science-mathematics, is an age-old yet evergreen field of human knowledge. The vitality of mathematics is, it seems to me, from its dealing with the numerical relation and spatial form in the most general sense. Numbers and forms, in the final analysis, reflect the most essential characters of things in the actual world. It is therefore no strange that the abstract theories and methods investigated by mathematicians would pervade almost all fields of science and technology. "Each science", as pointed out by Karl Marx, "could be considered to be perfect only if it permits the successful application of mathematics".

Mathematics gives, directly or indirectly, impetus to the development of productive forces as well. I mention here only one example-the revolutions of the communication industry, which would not have been possible without the mathematical physics from Gauss to Maxwell, and more recently without Turing and von Neumann's ideas of computers. It is therefore not without reasons that Napoleon has once said "the advancement and perfection of mathematics are intimately connected with the prosperity of the State". I prefer to quote again non-mathematician's viewpoint on the value of mathematics to avoid arousing suspicion of mathematicians' boast.

We are at the beginning of a new century. The unique situation of mathematics, different from any previous century at the turn, appears to be caused by the impact of the computers. Computers provide new tools, raise new problems, and allow new applications of mathematics. All that, I believe by my own research experience, will make a genuine new century of mathematics. It might be more challenging and promising to Chinese mathematicians whose country is struggling for transition from a developing society to the information and knowledge-based society.

Modern mathematics has historical roots of diverse civilizations. Mathematical activities in ancient China can be traced back to early time. The major pursuit of the ancient Chinese mathematicians was to solve problems expressed in equation. Along this line they contributed the decimal place-value numeration, negative and irrational numbers, various techniques for solving equations… etc. It is believable that ancient Chinese mathematicians had active knowledge exchanges with middle Asia and even Europe through the Silk Road. Today we have railways, airlines and even information highway instead of the Silk Road, the spirit of Silk Road-knowledge exchanges and cultural mergence ought to be greatly carried forward. I hope that the International Congress of Mathematicians 2002, held for the first time in a developing country, will open a glorious new page in the universal cooperation of mankind and bring with a prosperous future of our mathematical sciences.

I wish the Congress a success, wish you all a nice stay in Beijing.