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Home , Alessio Figalli, Artur Avila, Chern Medal, Nevanlinna Prize, Simons Foundation

Opening Ceremony August 1, 2018

Opening address , Chairman of the ICM 2018

Dear Minister of Education, Rossieli Soares da Silva, Dear Colleagues, Ladies and Gentlemen, When the Federal Polytechnical School of Zurich opened its gates to the participants of the first International Congress of , on the 9th of August 1897, was nowhere to be seen. Back then this was a rural country with just a handful of scientific institutions, none in mathematics. The first schools of engineering were being created in a few major cities. The first universities, with undergraduate courses in mathematics, were still a few decades into the future. The nation’s mathematical awakening took place in the 1950’s, when the In- stitute for Pure and Applied Mathemat- ics was created, Brazil joined Group 1 of the International Mathematical Union, and the biennial series of Brazilian Math- ematical Colloquia was launched. By 1962, Leopoldo Nachbin became the first Brazilian invited speaker at an ICM, in Stockholm. Seven years later, in 1969, the Brazilian Mathematical Society was created and became our adhering or- ganization to the IMU. The country you are visiting today is totally different, and so is its mathemat- ics. Early this year the IMU members voted Brazil’s promotion to Group 5. That news was greeted as a matter of national pride, by our media and our authorities, as much as by the mathematical community. As it should be. Four years ago, was distinguished with the , the first winner born, raised and educated in a developing country. And today, we are honored to welcome the delegates to the first International Congress of Mathematicians ever held in the southern hemisphere. Such progress, in such a short time, was the doing of generations of mathematicians, some of whom are unfortunately no longer with us. But it is fair to say that no one symbolizes the achievements of Brazilian mathematics better than Jacob Palis, without whose dreams and whose vision we would probably not be here today. 10

That is why I ask you to join me in electing Jacob the honorary president of the Congress, through a round of applause. However, for all the merits of the past, bringing ICM 2018 to Rio de Janeiro has always been about the future. Most of our efforts over the last six years went into taking advantage of this historical opportunity to promote mathematics amongst our youth. The Biennium of Mathematics, proclaimed by the Brazilian national parliament. The Math Festival we organized last year, with over 18,000 visitors of all ages. The Interna- tional Mathematical Olympiad, also last year and also a huge success. And all the other initiatives we organized or triggered during these two years, under the umbrella of the Biennium.

As a token of all that, we have here today a group of very special guests: the 576 gold medalists of the Brazilian Mathematical Olympiad for Public Schools. This is the largest and, I dare say, most impressive school competition in the world. It is held by IMPA every year, with the participation of over 18 million students from all over Brazil. This year, we have invited the gold medalists to attend the ICM opening. In a few minutes they will participate with us in the award ceremony for the Fields medal and the . And they will receive their own medals in this same room, tomorrow afternoon. I don’t think this is an experience they will ever forget. I would like to add a more personal note. Four years ago, in Seoul, I pledged that we would put the best of Brazilian ingenuity to the service of this Congress. Of course, I did not expect that we would be put to the test the way we were, just two days before this ceremony. That we passed the test is the definite proof that we have a tremendous team, enor- mously competent and fully dedicated to this dream of having an ICM in Brazil. I want them to know how grateful I am for what they achieved here in the last 48 hours. 11

I want to finish by thanking you for your presence here today, and by welcoming you to Brazil and to the Marvelous City. Enjoy Rio, enjoy ICM 2018, and, most of all, enjoy mathematics. Thank you!

Welcoming Address , President of the International Mathematical Union

Minister of Education, distinguished guests, Prize Winners and families, every atten- dant of this event: Watching the scenes from past ICMs, I recalled myself the ICM in Seoul four years ago. Perhaps you also did Hyderabad in 2010, Madrid in 2006, Beijing in 2002, or even further. I even recall ICM 1990 Kyoto, when I received the Fields Medal. During that congress I was so tense and could not enjoy the congress and celebrate the oc- casion. That was a regret of mine if any. So I wish you especially the win- ners, plenary and invited speakers enjoy the congress as well as learn through it. Back to this ICM 2018 at Rio de Janeiro, we are familiar with the beautiful Brazil and its history and culture. Dur- ing the Congress, you will learn about the recent developments in mathematics and celebrate IMU Prize Winners. You will surely meet old friends, make new friends, since this is a unique opportunity to do so once every four years; thousands of mathematicians came here to participate in this ICM from all over the world. I also hope that you will find time to explore this magnificent place, its history, cul- ture, and beauty, and food, since this is the first ICM ever held in the Southern hemi- sphere. But above all, I think you will find that all this is made possible by the very smooth organization and devoted and sustained work of the Local Organizing Committee. I have had many occasions already, in the preparations to the ICM, to witness their dedi- cation and efforts, and we will continue to do so during the whole ICM. I would like to thank all of them on your behalf, for this work. To end this welcome address to the ICM, I want to express again my hope that you will enjoy the remainder of this Opening Ceremony and of the whole Conference. Thank you. 12

Award Ceremony Shigefumi Mori

The Gauß Prize has been awarded at the ICM for the first time in 2006 and is now awarded at every ICM. The Prize was established by the IMU and the German Mathe- matical Society. It honors a scientist whose mathematical research has had an impact outside mathe- matics either in technology, in business, or simply in people’s everyday lives. The 2018 Gauß Prize is awarded to David Donoho for his fundamental contributions to the mathematical, statistical and computational analysis of important problems in signal processing For the glory of the Opening Ceremony to be shared by later events of ICM, the award ceremony of Gauß Prize 2018 will take place at 7:30pm on Aug 4 (Sat) before the Social Dinner. The Award was awarded for the first time at the 2010 ICM and is now awarded at every ICM. It was established by the IMU and the Chern Medal Foundation in cooperation with the Simons Foundation. It is awarded to an individual whose accomplishments warrant the highest level of recognition for outstanding achievements in the fields of mathematics. The 2018 Chern Medal Award is awarded to Masaki Kashiwara for his outstanding and foundational contributions to algebraic analysis and representation theory sustained over a period of almost 50 years. For the glory of the Opening Ceremony to be shared by later events of ICM, the award ceremony of Chern Medal Award 2018 will take place at 7:30pm on Aug 4 (Sat) before the Social Dinner. The Leelavati Prize was awarded for the first time during the Closing Ceremony of ICM 2010 in Hyderabad. The prize was established by the IMU and by the government of India; it is presently funded by Infosys, as a permanent IMU prize to be awarded at every ICM. The Leelavati Prize accords high recognition for outstanding contributions to increas- ing public awareness of mathematics as an intellectual discipline and the crucial role it plays in diverse human endeavors. The 2018 Leelavati Prize is awarded to Ali Nesin for his outstanding contributions towards increasing public awareness of mathematics in Turkey, in particular for his tireless work in creating the Mathematical Village as an exceptional, peaceful place for education, research and the exploration of mathematics for anyone. Following the precedent set at its first awarding at ICM 2010, it was decided that the Leelavati Prize 2018 winner be announced during the Opening Ceremony, but that the award ceremony itself is part of the Closing Ceremony of each ICM.

Award Ceremonies for the Fields Medals We now start the ceremony for the Fields Medals. The Fields Medals have been awarded by the IMU since 1936. They recognize out- standing mathematical achievement for existing work and for the promise of future 13 achievement. From the start, they were meant for young mathematicians. The rule is now that to be eligible to receive a Fields Medal, mathematicians must have their 40th birthday after January 1st of the year in which the ICM is held. The Rolf Nevanlinna Prize has been awarded at every ICM since 1982. It was es- tablished by the IMU together with the Finnish Academy of Sciences. It recognizes outstanding contributions in mathematical aspects of information sciences. It is subject to the same age limit as the Fields Medal: to be eligible for the Nevanlinna Prize, the 40th birthday of the winner must be after January 1st of the ICM year.

The four 2018 Fields Medalists are, in the alphabetical order of last names of the laureates: is awarded a Fields Medal – for the proof of the boundedness of Fano varieties and for contributions to the minimal model program. 14

Alessio Figalli is awarded a Fields Medal – for contributions to the theory of opti- mal transport and its applications in partial differential equations, metric geometry and probability.

Peter Scholze is awarded a Fields Medal – for his having transformed arithmetic algebraic geometry over p-adic fields. 15

Akshay Venkatesh is awarded a Fields Medal – for his synthesis of analytic num- ber theory, homogeneous dynamics, topology, and representation theory, which has resolved long-standing problems in areas such as the equidistribution of arithmetic ob- jects.

Award Ceremony for the Nevanlinna Prize We now proceed to the Nevanlinna Prize. 16

Constantinos Daskalakis is awarded the Nevanlinna Prize for transforming our under- standing of the computational complexity of fundamental problems in markets, auctions, equilibria, and other economic structures. His work provides both efficient algorithms and limits on what can be performed efficiently in these domains.

Announcements Shigefumi Mori

I would like to make two remarks. In this ICM 2018 Rio de Janeiro, each winner of the IMU Prizes, Gauß Prize, Chern Medal, and Leelavati Prize will be introduced to you by a short movie in addition to the Fields Medal and Nevanlinna Prize. These videos are the result of a collaboration of the IMU and the Simons Foundation; the IMU is grateful to the Simons Foundation for having accepted to fund and produce these movies. Brazil is a great country. When I visited Rio last year to examine the ICM prepara- tion, it was in the year of the International Mathematical Olympiad and I saw people enjoying mathematics together as a team. Brazilians know how to foster through enjoy- ment talents of the next generation following Artur Avila the Fields Medalist 2014. Brazil is famous and strong in mathematics with IMPA as one of its big centers. Just in case you are not familiar, ICM is held by the host country under the auspices of International Mathematical Union (IMU), which is why I am here at all. I served the executive committee (EC) of the IMU 2 terms (1995–2002) before I became President in 2015, Jacob Palis also served IMU as Secretary and President. Naturally I learned 17 a lot from him. He is a great figure representing Brazil in my mind. We knew that he wanted to hold ICM in Brazil. So thank you, Jacob, and also Congratulations on fostering great talents who have made your dream come true.

IMU Status Report Helge Holden, Secretary General of the International Mathematical Union

It is my pleasure to report to you about the activities of the International Mathematical Union (IMU) during the last four years since we assembled in Seoul, Republic of Korea, for the ICM 2014. In addition to an Executive Committee (EC), the IMU operates five Committees and Commissions, namely the International Commission on Mathematical Instruction (ICMI), the Commission for Developing Countries (CDC), the International Commis- sion on the History of Mathematics (ICHM), the Committee on Electronic Informa- tion and Communication (CEIC), as well as the Committee for Women in Mathematics (CWM). A major part of the operative work of the IMU is executed by our Committees and Commissions. For reasons of brevity, I refrain from reporting on their activities here, and I recommend that you consult the respective websites to get a better under- standing of their work for the benefit of mathematics and mathematicians worldwide. I would like to take this opportunity to thank all our colleagues who have spent endless hours working for the IMU dur- ing the last term. Your efforts are very well appreciated — even if rarely ex- pressed in public — by the global mathe- matical community. In addition, I would like to express our deep gratitude for the generous donations and grants that the IMU has received from individuals and organizations worldwide. We very much depend on outside support to achieve our goals. The IMU is also issuing a free elec- tronic newsletter, the IMU-Net. Did you stop receiving it? Due to the GDPR (Gen- eral Data Protection Regulation) direc- tive introduced by the European Union in order to protect personal privacy, we had to cancel all current subscriptions. You will have to re-subscribe to the IMU-Net in order to continue receiving it. Please go to www.mathunion.org and resubscribe today! 18

Membership Membership in the IMU is by countries. Regularly, I receive requests for individual membership, but the only way to adhere to the IMU is through an Adhering Organi- zation (AO), which can be, e.g., a national academy, a national mathematical society, or a university. The IMU may be considered as the “United Nations of Mathematics”, and the IMU itself is a member of one organization only, the International Science Council (ISC), which assembles international scientific unions and regional scientific organizations. The ISC was recently created as a result of the merger of the Interna- tional Council of Science (ICSU) and the International Social Science Council (ISSC). It is our expressed goal to have as many countries worldwide as members. By increasing membership, our work to support mathematicians and mathematics worldwide becomes more forceful. Membership in the IMU has been going through a positive development during the period 2015–2018. Even if some countries experience problems in terms of raising funds for settling the membership dues due to domestic problems, the IMU was able to increase its membership. In addition, four countries have been upgraded to a higher group of adherence. The following countries have applied for membership and been accepted as new members of the IMU (all in Group I): Malaysia, Kenya, Luxembourg, Morocco, Senegal, Thailand, Oman, and Kyrgyzstan. The following countries have been accepted as new Associate Members of the IMU for a four-year term: Paraguay, Bangladesh, and Uzbekistan. Madagascar, Gabon, and Papua New Guinea have been granted a second four-year term as an Associate Member of the IMU. Turkey has been upgraded to Group II, Chile and Portugal have both been upgraded to Group III, and Brazil to Group V. The Mathematical Council of the Americas has been admitted as a new Affiliate member of the IMU. (Membership is listed in chronological order.) Currently, the IMU has 88 regular and Associate Members and five Affiliate Mem- bers. While we are doing quite well compared to other scientific unions, we can and should do better. Note that 116 countries participated in the International Mathematical Olympiad in 2018, 193 countries are members of the UN, 206 countries participated in the Rio Olympic Games in 2016, 211 countries are recognized by FIFA, 215 countries are members of the IAAF! Thus there is ample room for improvement. I should add that each organization has its own interpretation of what constitutes a “country”. In Figure 1 we can see IMU membership displayed on a world map. It has always been a challenge for us to stay in contact with all IMU members. The means of communication have never been better, but yet it is not always easy to get feedback from all the AOs, probably because some countries have small and rather fragile national organizations. The majority of the members of the IMU are paying their membership dues properly, however, there are countries in arrears. Unpaid dues accumulate, and the IMU Statutes (Article 32) have clear regulations regarding IMU membership. Thus, the IMU EC tries to keep an active communication with the IMU members. In the coming term, the IMU will seek to increase membership while keeping its current members. 19

Figure 1: The world according to the IMU. Member countries are dark.

International Day of Mathematics — March 14 The IMU has taken the initiative to have the UNESCO proclaim March 14 (Pi Day) as the International Day of Mathematics (IDM). This date was chosen in consultation with the IMU Adhering Organizations, which enthusiastically support the idea to create an IDM. March 14, often written 3/14 in English, is already celebrated as Pi Day in many countries (and not only English speaking countries) around the world. Hence, the date was proposed by many countries when IMU consulted its members. March 14 is convenient for all parts of the world, including the Southern hemisphere. Every year, all countries in the world will be invited to celebrate the IDM in the schools and with the public, under a (non mandatory) chosen theme for the year. However, UNESCO still needs to be convinced that March 14 be proclaimed as the International Day of Mathematics. The ambition of the IMU is that the topic has gained necessary political support to be discussed at the General Conference of UNESCO in the fall of 2019, and that the official launch of the International Day of Mathematics will take place on March 14, 2020. The more countries support the proposal, the more likely it is to have UNESCO proclaim March 14 as the International Day of Mathematics.

Elections The governing body of the IMU is its General Assembly, which meets every four years, just before the ICM. We have just held the 18th General Assembly of the IMU in São Paulo, Brazil. One of the important decisions taken at the General Assembly was to elect the IMU leadership for the term 2019–2022. The Executive Committee for the term 2019–2022 comprises President Carlos Kenig (USA), Secretary General 20

Helge Holden (Norway), Vice Presidents Nalini Joshi (Australia) and Loyiso G. Nongxa (South Africa), and Members-at-Large are (), (Russia), Paolo Piccione (Brazil), R. T. Ramadas (India), Gang Tian (China), and Günter Ziegler (Germany). For the first time the EC has a representative from Africa, and also all five continents are represented in the EC. The majority of the members of the Commission for Developing Countries are elected by the General Assembly. For the term 2019–2022 the President will be Dipendra Prasad (India), Secretary for Policy Olga Gil-Medrano (Spain) and Secretary for Grants Alf Onshuus (Colombia). Members-at-Large are Mama Foupougnigni (Cameroon, rep- resenting Africa), José Maria P. Balmaceda (Philippines, representing Asia), and An- drea Solotar (Argentina, representing Latin America). In addition, the EC will appoint two members and ICMI will appoint one member to the CDC. For the International Commission on the History of Mathematics Isobel Falconer (UK) and Catherine Goldstein (France) were elected as Members-at-Large for the term 2019–2022. The International Commission on Mathematical Instruction elects its own Executive Committee at the ICMI General Assembly, which takes place in years congruent to 0 modulo 4, while members of the IMU Committees are appointed by the EC.

Members of Committees The Executive Committee of the IMU appointed some of the most important commit- tees for this congress. The scientific program of the ICM here in Rio was the responsi- bility of the Program Committee with members János Kollár (chair), Ngô Bảo Châu, Irit Dinur, Jianshu Li, Hiraku Nakajima, Sorin Popa, Enrique Pujals, Laure Saint-Raymond, , Eitan Tadmor, Ulrike Tillmann, and Aad Van der Vaart. The respon- sibility of the Program Committee includes the decision of the scientific structure of the program, that is, how to split contemporary mathematics into sections and their relative sizes, decide the chair and members of a panel for each section, which in turn recom- mends speakers in its section, and finally the Program Committee decides all speakers, both plenary and sectional. All of this work is done within a time-span of two years, and in complete secrecy. A truly gigantic task! We will enjoy the result of their work during the coming ten days. 21

The selection of the winners of the IMU awards is decided by prize committees, again working in complete confidentiality. The Fields Medal committee comprised Shigefumi Mori (chair), Hélène Esnault, Ed- uard Feireisl, , Nigel Hitchin, John Morgan, Hee Oh, Andrei Okounkov, M.S. Raghunathan, Kenneth A. Ribet, and . The Nevanlinna Prize winner was selected by the following committee: Tony F. Chan (chair), Manindra Agrawal, Emmanuel Candès, , Nick Higham, and . The Gauß Prize committee comprised Björn Engquist (chair), Maria J. Esteban, Mo- toko Kotani, Martin Nowak, Andrew Stuart, Nick Trefethen, and Bin Yu. The Chern Medal recipient was decided by the following committee: Caroline Series (chair), Jordan Ellenberg, Gerhard Huisken, Michio Jimbo, and Benoît Perthame. The Leelavati Prize committee comprised Gert-Martin Greuel (chair), Shihoko Ishii, Andrés Navas Flores, Sun Sook Noh, and Simon Singh. The ICM Emmy Noether Lecture committee consisted of Irene Fonseca (chair), Geor- gia Benkart, Kathryn Hess, Nalini Joshi, Izabella Laba, Dipendra Prasad, Fernando Ro- driguez Villegas.

Permanent stable office of the IMU Arguably the most important decision made at the 18th General Assembly was the decision regarding the permanent office of the IMU Secretariat. Since January 2011, the Secretariat of the IMU has been based in the heart of the city of Berlin, Germany, generously supported by a grant from the Federal Republic of Germany and the State of Berlin and hosted by Weierstrass Institute for Applied Analy- sis and Stochastics (WIAS). The annual financial support exceeds the total membership dues received from the IMU members. The grant funds are spent on office rent, salaries for the staff, and to support various IMU activities. As of April 2018, the premises of the IMU Secretariat have been in Hausvogteiplatz 11A, near Gendarmenmarkt, see Figure 2. The venue consists of four office rooms, the IMU Archive, a lounge, a meeting room, as well as auxiliary rooms. The IMU staff comprises IMU Manager (100% position, responsible for overseeing IMU business and, in particular, the EC), CDC & ICMI Ad- ministrator (100% position, administration of CDC and ICMI), IMU Technician (50% position, in charge of email, internet, webpage, backup), IMU Accountant (100% posi- tion, membership dues, accounting), IMU Archivist (60% position, responsible for the IMU Archive). In addition, there is the Head of the Secretariat (unpaid), who is a senior scientist employed at WIAS and serves as a liaison between the IMU Secretariat, the IMU, and the WIAS. The IMU Secretariat staff members are all regular employees of WIAS, and all decisions regarding salaries, working conditions, pensions, sick leaves, vacations, etc., are handled by the WIAS, while the actual work that is carried out is the responsibility of the IMU. The IMU Secretariat provides full administrative support for the IMU Executive Committee as well as the Commission for Developing Countries (CDC) and the Inter- national Commission on Mathematical Instruction (ICMI). In addition, the IMU Secre- tariat has full responsibility for all financial transactions of the IMU. The IMU Archive, 22

Figure 2: The IMU Secretariat is located in the dark building on the 5th floor. containing the historical material from the early days of the IMU up to today, is physi- cally located in the IMU premises. The IMU Secretariat is responsible for maintaining the IMU website as well as the mail servers. In accordance with the decision taken at the General Assembly in 2010, the contract between WIAS and the IMU was made for a ten-year period. Thus it was on the agenda of the 2018 General Assembly to decide what would happen past this period. The matter was discussed and I am happy to report that the General Assembly unanimously decided to extend the contract with WIAS indefinitely. Thus the IMU Secretariat will remain in Berlin for the future. This gives the IMU stability and a reliable way to handle all financial issues of the complete slate of IMU activities. The fact that the IMU has a permanent Secretariat with a very able staff has resulted in a considerable increase in the total volume of activities of the IMU. Without the dedicated staff at the IMU Secretariat, and the generous financial support from Germany, the level of activity of the IMU would have to be drastically reduced. The tradition of handing over the IMU “Secretariat” to the next President’s and Sec- retary General’s location and the “IMU Archive” consisting of a number of cardboard boxes, a tradition that dates from the inception of the IMU in the 1920’s, has come to an end once and for all. A new IMU website After considerable work done by many individuals and a commercial company, the IMU as well as its Committees and Commissions have for the first time a uniform ap- pearance on internet. The new website allows for the prominent exposure of news as well as the current requests to the Adhering Organizations. Now the IMU is able to display in real time on its homepage the names, including photos, full citations, brief 23 videos, and popular write-ups, for the winners of all the IMU awards, that is, at the very same moment when the names are announced by the IMU President at the ICM Opening Ceremony. Please check www.mathunion.org and see for yourself. The ICM in 2022 It may not feel right to talk about the ICM in 2022 before we have even started the scientific program of the current ICM in 2018 here in Rio de Janeiro, Brazil. But as we can see by looking around, an ICM requires meticulous long-term planning, and it is not too early to start the careful preparations for the next ICM four years from now. By tradition, the Adhering Organizations — our members — were asked to submit bids for the ICM in 2022. We received two bids — one from France (with venues Paris for the ICM and Strasbourg for the IMU General Assembly) and one from Russia (with venue St. Petersburg both for the ICM and the IMU General Assembly). Both bids were of excellent quality and very professional. After site visits, the Site Selection Committee made its recommendation to the IMU General Assembly, which at its meeting in São Paulo decided to accept the bid from Russia. Thus the ICM 2022 will take place in St. Petersburg, Russia. The IMU wants to express its gratitude to our colleagues both in France and in Russia for preparing two excellent bids. But right now we are in Rio de Janeiro, in Brazil, and we are looking forward with great expectations to the ICM 2018!