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#1#Springer#2#Advertisement#3#2018#4#A56389_ICM_ad_2018#5#HRM_A56389_ICM_ad_2018 1 4-6-2018 17:12:52 Contents

Welcome Message from the President of the IMU...... 4 Courtesy of IMPA Courtesy

Welcome Message from the Chairman of the ICM 2018 Scientific Editors Organizing Committee...... 6 José Maria Espinar Garcia IMPA, Rio de Janeiro, Brazil E-mail: [email protected] Welcome Message from the President of the Brazilian Mathematical Society...... 7 Pablo Guarino UFF, Rio de Janeiro, Brazil The Brazilian Math Olympiad of Public Schools, 10 Years E-mail: [email protected] Promoting Social Justice Through Academic Merit...... 8

Executive Editors Brazil’s Peculiar Journey to the Elite of Math...... 13 Raphael Gomide Corcovado Comunicação Estratégica, Rio de Janeiro, Brazil The Development of the Brazilian School of E-mail: [email protected] Dynamical Systems...... 15 Robinson Nelson dos Santos Springer, São Paulo, Brazil A Gender Perspective on Mathematics in Brazil...... 21 E-mail: [email protected]

Writers and copy editors: Karine Rodrigues and Sergio Algebraic Geometry in the Interface of Pure and Torres Applied Mathematics...... 24 Design: Miro (OMERCADOR), São Paulo, Brazil

Printing: Pigma, São Caetano do Sul, Brazil IMO-Style Problems: Are You Ready for a Gold Medal?...... 32

Biennium of Mathematics: Approaching Brazilians and Mathematics...... 34

Kleinian Groups in Several Complex Variables...... 39

Nonlinear Diffusion Processes: Geometric Ideas and Beyond...... 42

Olympiads Arouse Enthusiasm for Mathematics...... 49

The Development of Differential Geometry in Brazil...... 53

The Early Years of Applied Mathematics in Brazil: A Brief Account on How It Developed...... 56

Front cover: View of Morro da Urca and Guanabara A Glimpse of the Current Research Activities in Applied Bay from the Sugar Loaf Cable Car. Photo courtesy of Bondinho Pão de Açúcar. Mathematics in Brazil...... 63

Academic Events Mobilize Institutions in the School Break..... 66 The Rio Intelligencer is a gift from Springer on the occasion of the International Conference of Mathematicians 2018 in Rio de Janeiro, Brazil. Its contents and recommendations solely reflect the experiences and views of its authors. Solutions to “IMO-Style Problems: Are You Ready for a Gold Medal?”...... 70 Mathematics Editorial Springer Tourist Guide...... 74 Shigefumi Mori President, IMU Professor Emeritus, Kyoto University, Japan E-mail: [email protected]

Welcome to the ICM 2018 at Rio de Janeiro, Brazil, the first ICM ever held in the Southern Hemisphere The ICM is a quadrennial meeting, the single worldwide event covering the whole of mathematics, which is organized by the host country under the auspices of the International Mathematical Union (IMU).

If you are overwhelmed or intimidated by this mammoth event, you will be pleasantly surprised to learn that the ICM is orchestrated by fellow mathematicians who are nonexperts in organizing meetings. To familiarize you with ICM 2018, let us browse through some of its highlights and how it has been prepared and organized during the last four years or so.

Just before the previous ICM in 2014, the General Assembly of IMU elected the current Executive Committee (EC), which was the start for most of us. The members of the EC attended and experienced ICM 2014 especially in preparation for ICM 2018, and started their work in 2015. As my first task as IMU President, I asked Professor János Kollár to be Chair of the Program Committee (PC) and he accepted his role during ICM 2014. The whole PC, consisting of about 10 members, was selected and became complete shortly after the first meeting of the EC in 2015.

The PC decided the structure of 19 Panels of the ICM, and furthermore all of the Panel committees, consisting of about 150 members, were formed by Spring 2016. Altogether they selected about 20 plenary speakers and about 180 invited speakers of the highest scientific level by Spring 2017, which form practically the whole academic content of ICM 2018. We were happy to find out that the speakers were from about 40 countries, which provides great evidence that the latest frontiers of research in mathematics are explored in diverse countries worldwide.

When ICM 2018 opens, your attention will be drawn to prize winners. The Fields Medal and the Nevanlinna Prize are awarded to young mathematicians who have attained the highest results, which makes them unique among the most prestigious prizes worldwide. The Gauss Prize is awarded for mathematical research which has its foremost impact outside mathematics, and the Chern Medal is awarded for prominent lifetime achievement in mathematics. The latter prizes have no age restrictions. Nowadays the popularization of mathematics has attracted increasing public attention, and the Leelavati Prize awards the most outstanding contributions in this direction, again with no age restriction. The winners of these five prizes will be introduced at the opening and closing ceremonies of ICM 2018, and furthermore short videos will be presented, which are created under the patronage of the Simons Foundation. Besides these prizes, a woman with a top lifetime contribution to mathematics is invited to deliver the ICM Noether Lecture. The EC appointed six committees, which finally selected these prize winners and the lecturer by Spring 2018.

The Organizing Committee (OC) led by Chair Marcelo Viana, though mentioned last, started their work in 2012 or even earlier and wisely took advantage of two events. The venue of ICM 2018, the Riocentro Convention Center in Barra da Tijuca, chosen by the OC, was one of the hubs for the 2016 Summer Olympic Games, and thus the area as well as it connections with city centers and the Airport has been very quickly developed and managed. The OC could declare that the Biennium of Mathematics 2017–2018 has successfully absorbed the public attention of the country since they succeeded to host the 2017 International Mathematical Olympiad at Rio as well as ICM 2018.

The OC has launched the Open Arms program to offer travel grants to 550 mathematicians from developing countries to attend ICM 2018 under the support of the IMPA, the Brazillian Mathematical Society, and the IMU. The OC’s work over several years has culminated in ICM 2018 attracting thousands of participants. I should also mention that ICM 2018 is managed by many supporters and volunteers who have helped to bring the first ICM in the Southern Hemisphere to a great success.

We very much hope that you will enjoy the ICM, its academic and social programs, and furthermore the beautiful touristic places Brazil can offer, including the Sugarloaf Mountain and Christ the Redeemer. Marcelo Viana Chairman of the Rio ICM 2018 Organizing Committee

Managing Director, IMPA, Brazil E-mail: [email protected]

Welcome Message from the Chairman of ICM 2018 Organizing Committee

Dear Colleagues,

On behalf of the ICM 2018 Organizing Committee, the Instituto de Matemática Pura e Aplicada (IMPA) and the Brazilian Mathematical Society, it is my great privilege to welcome you all to Rio de Janeiro, and to the first International Congress of Mathematicians ever held in the Southern Hemisphere.

Hosting ICM 2018 is a great honor for the Brazilian mathematics community. We want to believe that it also pays tribute to the progress attained in so little time by a developing country where mathematics research is little more than six decades old.

When in 1893 Felix Klein called on the mathematicians of the world to Unite!, Brazil hardly existed at all in scientific or even academic terms. And here we are now, adding our own contribution to the venerable and ever lively tradition of the International Congresses of Mathematicians.

Back at the closing ceremony of ICM 2014, in Seoul, I pledged that we would put the best of Brazilian energy and ingenuity into delivering the best possible Congress. I am confident that we kept that promise. But you alone can judge how successful our efforts have been.

For us who endeavored to bring the Congress – as well as the International Mathematical Olympiad 2017 – to Brazil, the greatest challenge has always been how to turn the glamour of the occasion into a permanent legacy for our younger generations.

The Biennium of Mathematics Brazil 2017–2018, that we launched across the whole country with the support of the National Parliament and government, is our main response to that challenge. It is most gratifying that the response has been so positive.

I wish you all a very fruitful Congress and a memorable stay in the Wonderful City.

And do come back, as often as you like, for math and more! Paolo Piccione President, Brazilian Mathematical Society Full Professor, Institute of Mathematics and Statistics, University of São Paulo, Brazil E-mail: [email protected]

Welcome Message from the President of the Brazilian Mathematical Society

It is a great honor to welcome all participants of the 2018 International Congress of Mathematicians, in the wonderful city of Rio de Janeiro. For the first time in history, the world’s most important mathematical congress will be held in the Southern Hemisphere, and we are particularly proud of hosting it in our country. Aiming to offer a memorable event to all participants, the Brazilian Mathematical Society, together with numerous members of our community, have provided decisive contributions to the realization of this conference. For all of us, the ICM 2018 opening ceremony will be the culmination of several years of tireless preparation efforts.

Although it will only last nine days, our core goal has always been that of organizing a landmark event with long-lasting results. To quote the Brazilian poet Vinicius de Moraes, “let it not be immortal, since it is flame, but let it be infinite while it lasts’’.

Since we started preparing our bid to host the ICM, it was clear in our minds that Brazil could offer a magnificent environment for a major mathematical event. First of all, the last decades have witnessed an impressive expansion of our country’s mathematical significance, supported by sound and continuous investments through our research agencies, and by the enthusiastic leadership of several of our most renowned mathematicians. The striking consequences of this growth are undeniable and irreversible, as the Brazilian mathematical production has now achieved truly outstanding qualitative and quantitative levels.

Second, and equally important, the Brazilian academic community, as well as the entire Brazilian society, are genuinely multicultural and extraordinarily diverse, rendering them uniquely qualified to receive all scholars attending the ICM– from students and early-career mathematicians to world-leading researchers. We knew, by direct experience, that Brazil would be able to offer an unparalleled supportive and friendly hospitality to participants from all corners of the planet, and that everyone would feel comfortable in our home. This was also the spirit of our Open Arms Grant Program, through which a considerable number of travel grants were offered to mathematicians from developing countries.

For all these reasons, we have taken up the monumental challenge of organizing the ICM, and we have vigorously dedicated ourselves to this task. It is now time for all of us to enjoy together so many exciting mathematical lectures and exchanges of ideas, in the exquisite ambience provided by the city of Rio de Janeiro. Let us celebrate together one of the most diverse mathematical communities in the world – which is thrilled to receive your visit.

Welcome to ICM 2018 in Rio de Janeiro!

Claudio Landim Coordinator of OBMEP Deputy Director, IMPA, Brazil E-mail: [email protected]

The Brazilian Math Olympiad of Public Schools: 10 Years Promoting Social Justice through Academic Merit

Brazil’s primary and secondary schools can varied from state to state. Scores in southern be divided into three major groups according or southeastern states of Brazil such as Paraná to their administrative dependence: municipal (406) or Espírito Santo (405) significantly and state schools, federal schools and private surpassed those of students from northeastern schools. By 2015, approximately 85% of states, such as Bahia (343) or Maranhão (343). 15-year-old students were enrolled in a state or It is within this context that the Brazilian municipal school, 13.5% in private schools, and Mathematical Olympiad of Public Schools 1.5% in federal schools. (OBMEP) was launched in 2005. It had three According to the Programme for major goals at its creation: to detect students International Student Assessment (PISA) of with a talent for mathematics, to train them the Organisation for Economic Cooperation and to stimulate them to pursue their studies and Development (OECD), Brazilian students at the university and to improve the quality of perform much below the average of OECD math teaching in Brazil. countries. In 2015, the scores of Brazilian In order to identify talented students in this students in mathematics was 377, which is unequal and adverse context, a committee of significantly lower than that of OECD students university professors was formed to formulate (490). The same difference is observed in the math problems whose resolution did not require, sciences (401 and 493) and in reading (407 and as far as possible, knowledge in mathematics, 493). The performance in mathematics places but only reasoning, logic and creativity. Brazil behind most of the Latin American countries taking part in the PISA (Chile 423, The following problems for 8th and 9th Uruguay 418, Mexico 408, Costa Rica 400, graders (13 and 14 years old youngsters), taken Colombia 390, Peru 387), and just ahead of the from previous Olympiad exams, illustrate the Dominican Republic (327). style of the questions. All the past exams are available in Portuguese at the webpage http:// A more detailed analysis reveals a regional www.obmep.org.br/provas.htm. disparity, as well as a huge gap between municipal and federal institutions. In 2015, the Problem 1. An ant is at point A of the grid PISA scores in mathematics of students from shown in Figure 1 and wants to reach point B municipal schools was 311, while it was 369 passing through point R. It walks on the sides for state schools, 463 for private schools, and of the squares and only to the right or down. In 488 for federal schools. Scores varied also how many ways can the ant make this journey? Problem 2. What is the tens digit of the following sum? 9

Problem 3. Elisa places six dice on a Each die has faces numbered from 1 to 6 table, as in Figure 2, and then writes down and the sum of the numbers of two opposite the sum of the numbers of all the faces faces is always 7. What is the largest sum she can see as she goes around the table. Elisa can get?

Figure 1. An ant’s walk.

Figure 2. A stack of dice.

A student capable of solving problems such (15 to 17 year-olds). All students enrolled in a as these qualifies for the second phase of the public school take part in the first stage of the Olympiad. This means that one who does not Olympiad. The exam is applied on the same have a special inclination for mathematics but day throughout the country. There are 20 who prepares for the test by solving problems multiple choice questions which are graded by of past exams can qualify for the second the school teachers using a template sent by phase. With a little extra effort a bronze medal the OBMEP’s offices. The students with the 5% can be earned. This is an important feature of best marks of each school in each level qualify the project, and it distinguishes the OBMEP to the second stage, whose exam consists of 6 from other math olympiads. The problems problems divided into several items. are conceived so that any student can get an Almost 18 million students from 99.57% of award with some application. the municipalities of Brazil engage each year in the Math Olympiad. About 1 million of them An 18 million-student Olympiad take part in the second stage, applied in 9,000 Since the first event, the competition has centers around the country, supervised by been divided into 3 levels and 2 stages. The 46,000 inspectors. The tests are first graded first level comprises students from the sixth by 70 regional committees. At each level, the and seventh grades (11 and 12 years old), the 10,000 tests with the best marks are sent to second level pupils from the eight and ninth Rio de Janeiro and reinspected by a national grades (13 and 14 years old) and the third level committee. The best marks of this second students from the tenth to the twelfth grades grading are rewarded with 500 gold, 1,500 silver and 4,500 bronze medals. Around 46 Almost 18 million students from thousand honorable mentions are awarded, 99.57% of the municipalities of and the rules of the competition stipulate Brazil engage each year in the that at least 60 bronze medals are awarded Math Olympiad; about 1 million to each state to encourage students from all of them take part in the second parts of the country. stage, applied in 9,000 centers The OBMEP has been able to detect around the country, supervised talented youngsters. Gérson Tavares is a by 46,000 inspectors. typical example. The son of a glassmaker and a maid, Gérson worked during the day and studied at night. During a whole school year, impact has been observed in schools which he only took math classes on four Fridays. On engage in the project. When he first heard other occasions, the teachers did not appear. of the Olympiad, professor Geraldo Amintas He had never liked and had never excelled in from Dores do Turvo, a municipality of 5,000 mathematics. He could not believe it when inhabitants in Minas Gerais, thought that he learned that he had been admitted to the his little school would never stand out. The second stage of the Olympiad; even more so winners would come from colleges in urban when he won a gold medal. Gérson had no centers. But when he discovered that a boy idea he was good at mathematics, nor did his from a neighboring town had been awarded a teachers! After the medal, he started to train bronze medal he began to believe that perhaps on his own and won three more gold medals his pupils had a chance. The following year, in a row. He was the first four times champion Geraldo invited the students who qualified for of the OBMEP. After his Olympiad success, the second phase to participate in a special Gérson graduated in Electrical Engineering at group. the University of São Paulo, considered to be Studying math on Saturdays has never the best in Brazil. attracted many teenagers, and initially Every year, the Olympiad singles out membership was not very high. Many lived students like Gérson, with a gift for mathematics, in the countryside and did not have easy and trains them. The National Council for access to school. Even so in 2007 two of these Scientific and Technological Development students were awarded the first bronze medals (CNPq) offers a one-year grant worth R$ of Dores do Turvo. The director, Ângela Maria 100 per month (approximately US$ 30) to Campos, promoted a local awards ceremony. all medalists to attend classes in universities, The small celebration encouraged recalcitrants where they learn topics not seen in school, to engage in Geraldo’s project. The following meet young people with the same interests year, Thiago Moreira would win the school’s – and envision a university education. André first gold medal. Many others have since Alves, from Rio Quente in the state of Goiás, succeeded. Terezinha de Jesus school is today sums up the dedication of some students. one of the biggest winners of the OBMEP, By 2012, the program loaned participants the with 94 medals, of which 17 are gold. In 2012, translation of a Russian mathematics book, Dávila Meireles, a student of Geraldo’s, had the which was to be returned. Fearing not being second highest grade in Brazil at level 1. able to solve all problems in time and with no Dores do Turvo’s involvement in the Olym- money to pay for photocopies, André copied piad had a strong impact on the achievement by hand, on sheets of paper, the whole book. of all students in the national exams. From The books ended up being donated. 2005 to 2015, students’ performance in the The OBMEP has been able to identify and Basic Education Development Index (IDEB) train committed students, but its greatest jumped from 4.2 to 7.7 in the initial years of Elementary Education and from 4.3 to 5.5 in focusing on science (an approximate increase 11 the final years. In 2015, 51% of students in the of 10 points in grades – from 390 points in 9th grade year had an adequate learning in 2006 to 401 points in 2015 – does not represent mathematics, while in Minas Gerais the avera- a statistically significant change). ge is 20% – and in Brazil, 14%, according to Brazil’s average in reading has also remained the National Institute of Research and Studies stable since the year 2000. Although there was in Education Anísio Teixeira (INEP). With the a rise in the score from 396 points in 2000 to support of the school, parents and alumni, 407 points in 2015, this difference does not teacher Geraldo Amintas is changing the des- represent a statistically significant change. In tiny of the youngsters of Dores do Turvo. The the area of mathematics, there was a significant university has become a natural goal, and the increase of 21 points in the average of students invitation to join the special class in prepara- between 2003 and 2015. At the same time, tion for the Olympiad, a privilege. there was a decline of 11 points if we compare This phenomenon is not restricted to the the average of 2012 with the average of 2015.” small town of Minas Gerais. In many other These expressive examples of success schools, committed teachers have achieved have led the Institute of Pure and Applied similar results. A study by professor Francisco Mathematics (IMPA), which coordinates the Soares, former president of the INEP, points Olympiad, to create the program OBMEP at out that schools engaged in the OBMEP School, with the objective of universalizing offer elementary students the equivalent of initiatives such as Geraldo’s. another year and a half of math education. A national exam, taking advantage of the “The results found show a very significant existing logistics of the second stage of the impact of a school’s involvement with the Olympiad, selects teachers from all over the OBMEP on its students’ grade in mathematics. country. The qualified ones receive, over 7 This impact is greater as the time of school months, a scholarship of R$ 765 (US$ 250) involvement with the Olympiad increases, per month to attend a monthly meeting where indicating the importance of the school’s they are trained to prepare their students for ongoing involvement with this initiative.” the Olympiad. At the same time, they form in Actually, according to PISA’s country note their own schools classes with 20 pupils, which for Brazil (http://www.oecd.org/pisa/pisa-2015- meet for 2 hours every week. The students Brazil-PRT.pdf) “Brazil’s average in science has are trained to solve problems similar to those remained stable since 2006, the last PISA cycle of the Olympiad. Despite its short duration – Courtesy of IMPA Courtesy of IMPA Courtesy

Figure 3. Indians from the State of Mato Grosso attend Figure 4. The then President of Brazil, Dilma Roussef the second stage of the Olympiad at a school for (center), in 2013, at the OBMEP award ceremony with indigenous people. gold medalists from São Paulo State. from June to September – OBMEP at School Every year, books with original problems si- achieved exceptional results in 2016. The milar to those of the Olympiads are produced students’ average grade over the 900 teachers and distributed to all public schools in the cou- which took part in the project was double the ntry. More recently the OBMEP has launched national average, in all three levels. the project of a collection of open and colla- Furthermore, to improve the quality of borative textbooks in mathematics (https:// math teaching in Brazil the OBMEP invested www.umlivroaberto.com/wp/). The collection in the production of didactic material. A portal is available on the internet to all teachers, with (https://matematica.obmep.org.br/) was crea- the usual creative commons rights. They can ted with videos, workbooks, exercises and use the books in the classroom and modify or quizzes covering the entire school curriculum participate in the elaboration of the contents. from the sixth grade to the final year. Created in 2014, the number of enrolled students exceeds Over the years the OBMEP has become an the 200,000 mark, the number of views, 7 mil- educational project engaged in improving the lion, and the average viewing time, to my sur- quality of math education in Brazil, promoting prise, 14 minutes. social mobility by academic merit.

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Image: Alexaldo/Peshkova/iStock [m] 014486c Brazil’s Peculiar Journey 13 to the Elite of Math

Sixty-four years after joining the IMU’s Group 1, Brazil lands in the team of the most developed research nations

When he was queried other countries – Germany, research federal agencies in about the possibility of moving Canada, China the USA, Fran- 1951 – “Conselho Nacional de to Brazil to lecture, Russian-A- ce, Israel, Italy, Japan the Uni- Desenvolvimento e Tecnoló- merican mathematician Oscar ted Kingdom and Russia – as gico” (CNPq) and “Coorde- Zariski (1899–1986) thought one of the top countries of nação de Aperfeiçoamento about what he would do in such origin of experts like Weil and de Pessoal de Nível Superior” a remote place. However, when Grothendieck. (CAPES) – brought impetus. Zariski heard that the Univer- A year later, the IMPA was set “In the 1950s, there was sity of São Paulo (USP) had up as the first research divi- practically no math research in also hired the French mathe- sion under the CNPq. Brazil. Now, sixty years later, matician André Weil (1906– we are among the champions Some time later, Brazil link.springer.com 1998), he asked himself: “why in the field”, highlighted Mar- pleaded to become a mem- not?” The two of them arrived celo Viana, managing director ber of the IMU. In 1954, it in the country in January 1945 of the Instituto de Matemá- was only possible to join for a season in which they sha- tica Pura e Aplicada (IMPA) under the status of begin- red long hours of conversation whilst observing the peculiar ner countries: Group 1. The about their favorite theme: al- aspects of Brazil’s trajectory initiatives taken from there gebraic geometry. upon the announcement of its demonstrate an interest to At the time, math research ascension to Group 5. “I don’t move ahead. In 1957, the first in the country was at its em- recall another country that Brazilian Math Colloquium bryonic stage. Even a decade has experienced such a tra- was organized at the IMPA, a later, when more foreigners jectory”, Viana declared in an biannual event that remains like Alexander Grothendieck interview in which he explai- to this day, representing an (1928–2014) were admitted ned that other elite nations, important milestone: math as visiting professors, related with a stronger tradition in the in Brazil began to grow arou- scientific production in Brazil field, had been admitted to the nd it. In the 1969 Colloquium, Knowledge Matters. Choose SpringerLink. Visit remained incipient. upper level groups by the IMU. the Brazilian Society of Math (SBM) was founded. Jumping in time to 2018, today Research agencies only • The World’s Leading Scholarship the scenery revealed is qui- From its joining of the IMU in the 1950s te different. Brazil, since Fe- until moving to Group 2 in • In the Most Complete Online Collection of STM Content bruary 2018 a member of the In a country where public 1978, the country began its select Group 5 of the Inter- health and agriculture were exposure overseas. Leopoldo • Delivered on the Fastest, Most Intelligent Research Platform We’ve Ever Developed national Mathematical Union prioritized, math rose slow- Nachbin (1922–1993), an IMPA • All from Springer – A Global Leader in Scienti c Publishing (IMU), now ranks, next to ten ly. The creation of two main researcher, became the first

Image: Alexaldo/Peshkova/iStock [m] 014486c Paulo Piccione (SBM), Marcelo Viana (IMPA), Maria Helena Guimarães (executive secretary, Ministry of Education) and Elton Santa Fé Zacarias (executive secretary, Ministry of Science, Technology, Innovations and Telecommunications), during the press conference for the official announcement: recognition after a history of relevant contributions to the field Courtesy of IMPA Courtesy

Brazilian guest speaker at the A jump in scientific ration between Brazilian mathe- 1962 International Congress of production maticians and their worldwide Mathematicians in Stockholm. counterparts equally reached Excellence in Brazilian re- In 1974, it was Maurício Peixo- an ascending rhythm. And search was evidenced by va- they have outstanding records, to’s turn as one of the institu- rious factors, such as consi- having obtained some of the te’s founders and speaker in derable growth in scientific most important international the Vancouver meeting. publication and notable pro- prizes. Artur Avila was awar- gress in qualitative and quan- As a result of stronger post- ded the Fields Medal in 2014 titative terms. In 2006, right graduate courses in the 1970s, and Marcelo Viana received the after its promotion to Group the education of leading re- prestigious Grand Prix Scienti- 4, it represented 1.53% of the searchers was consolidated. fique Louis D., delivered by the world math production (1,043 In 1981, Brazil moved to Group Institut de France, among other papers). A decade later, the recent achievements. 3 and reached the 21st cen- national production jumped to tury, already positioning itself 2.35% (2,076 papers). ICM 2018 and other interna- among the more outstanding tional, large scale events in the The math postgraduate pro- countries when it moved to field are an effect of the trust gram also progressed expressi- group 4 in 2005. granted by the world math vely and quickly reached inter- community to the developed Since then until last year, national standards. The IMPA Brazilian math. Brazil has alrea- when the IMPA and SBM put is one of the most respected dy demonstrated its capacity to research institutions in Brazilian forward their candidacy to join organize the International Ma- Science and one of the most the group of The most deve- thematical Olympiad (IMO), recognized centers of math re- loped nations in math resear- in July 2017 in Rio. Now it has search in South America. ch, Brazil has made significant the opportunity to prove, once contributions to world math, It is no coincidence that re- more, that its hard won reputa- as recognized by the IMU. gional and international collabo- tion was well deserved. Artur Avila 15 Fields medalist, 2014 CNRS-IMJ-PRG, France and IMPA, Brazil [email protected]

Paulo Brandão IMPA, Brazil E-mail: [email protected]

Vilton Pinheiro UFBA, Brazil E-mail: [email protected]

The Development of the Brazilian School of Dynamical Systems

The history of research in mathematics in laureated Brazilian mathematicians, Palis ob- Brazil is not a long one, chronologically speak- tained his PhD at Berkeley, having Steve Smale ing. There was certainly some mathematical as advisor and, after a short period at the Uni- activity before the 1950’s (Oscar Zariski and versidade do Brasil (currently known as Federal André Weil had visited the University of São University of Rio de Janeiro, UFRJ), he became Paulo in the 1940’s for instance), but in these a professor at IMPA. Palis has been the PhD ad- pre-historical times, even such great mathema- visor of more than 40 students, most of them ticians could not have a significant impact, as from all around South America. Taking into ac- the local community was not yet ready. Things count further generations there is a large num- would start to change significantly with the ber of dynamicists who are “descendants” from founding of IMPA by Lélio Gama, Leopoldo Na- Palis. Many of them remained in Brazil or South chbin and Maurício Peixoto in 1952. America, and make up for the bulk of the dy- Peixoto was the pioneer of dynamics in Brazil, namical research therein, while the others who and since, as of 2018, he is still an active mathe- spread through the other continents often con- matician who can be seen at times at IMPA, he tinue to have a strong connection with Brazil. is also a living display of how short is the history we are about to present. He established one of Lucky events the earliest results about structural stability [19], an influential work that attracted the attention Success is often said to be the consequence of Stephen Smale at a moment in which he was of good luck and hard work (to position your- starting to switch his focus from topology to dy- self so to be able to better benefit from the luck namics. Peixoto was introduced to Smale by the events). It is certainly the case here. One could topologist Elon Lima in 1957, while Peixoto was imagine that even in the best circumstances, a visiting Solomon Lefschetz in Princeton. The con- young PhD student in a small new institute in a nection with Lima and Peixoto would lead Smale country with no mathematical tradition would to visit Rio, where he developed both his proof of not have access to exceptional students right the Poincaré conjecture, and the concept of the away. As it turns out, the two very first students “horseshoe”, the paradigmatic example display- of Palis, Welington de Melo and Ricardo Mañé, ing a coexistence of chaos and structural stability. were exceptional, and the circumstances were ripe for them to develop. Despite Peixoto’s pioneering role, the Brazi- lian/South American school of Dynamical Sys- In 1969, just after concluding his PhD and re- tems can be considered to have been founded turning to Brazil, Palis started a seminar at IMPA, by Jacob Palis. One of the most influential and mostly about new results on Dynamics, with talks being given even on Saturday mornings. In Structural stability one of these occasions a young student asked Mañé left Montevideo to study at IMPA when Palis to allow him to participate in the seminar. he was 23 years old. Two years later, he conclud- Palis was surprised, especially because this you- ed his PhD and became a researcher of the same ng student had just been accepted to the Mas- institute. It was through him that Ergodic Theory ter’s Program at IMPA, but given the determina- arrived in Brazil. He was also a brilliant exposi- tion of the student, he agreed. This student was tor, with the capacity to talk about sophisticated de Melo and by 1972 he had already defended topics to students and researchers that were just his thesis, which would be published in “Inven- beginning in the subject. As a teacher, his lessons, tiones Mathematicae” [8]. even about basic topics became legendary. Own- Also in 1969, Palis managed to raise a gen- er of a very complex personality, brilliant, gay, a erous funding from Finep (a Brazilian govern- compulsive reader and very learned, having an mental agency) to organize an international intense nightlife, waking up only after noon, Mañé meeting in his research field.¹ was perceived as a super-hero by students. His death, as a consequence of AIDS, at the age of The Salvador Symposium on Dynamical 47 was a deep loss for Brazil and for the whole Systems in 1971, organized by Palis, Peixoto dynamics community. and Lima in Bahia, was the first big internatio- nal mathematical event in the country. It had He had a broad understanding of dynamical the presence of many leading mathematicians systems, and a remarkable technical range, al- such as E. C. Zeeman, G. Reeb, S. Smale and lowing him to be at ease both with the flexible R. Thom. This event would leave a deep mark C1-world and with the rigid holomorphic one. in the dynamical community and generated Let us describe quickly one of his most remar- strong international ties for IMPA. The Sympo- kable contributions, belonging to the former. sium also gave rise to a quite successful Pro- Inspired by Peixoto’s work about structural ceedings volume, edited by Peixoto [20]. stability of flows in two dimensional manifolds, The Symposium would also mark the first Smale began to speculate about the structur- time Mañé came to Brazil. In the year before, he al stability for dynamical systems in general. A had written a legendary letter² to Palis, who de- diffeomorphism f on a compact manifold M is scribes it thus: “I was astonished that a young structurally stable if it is topologically conjugat- Uruguayan was able to write such a letter, possi- ed to all diffeomorphisms in a neighborhood of bly full of incomplete or even wrong statements f (strictly speaking this notion depends on the and demonstrations, but with an immense taste precise regularity class of the diffeomorphisms about what the central problems of the area under consideration, but we will not pay too were at that time, and also the daring to pro- much attention to this matter here). In 1970, he pose demonstration schemes.” On the basis of and Palis showed that Morse-Smale diffeomor- this letter, Palis asked Mañé to come to Salvador phisms are structural stable in any dimension. to participate in the Symposium. A diffeomorphism f is called Morse-Smale if its nonwandering set³ Ω(f) consists of a finite One final example of chance influencing the number of periodic orbits, with all of them hy- events. Due to Smale and his students, the ini- perbolic, and the stable and unstable manifolds tial development of Dynamical Systems in Bra- of the periodic orbits meet only transversally. zil was very much influenced by the American The Morse-Smale diffeomorphisms were not school. But it would not take long for strong the only class of diffeomorphisms known at ties to develop with a different country, France. that time as being structurally stable. Indeed, At the time, military service was compulsory in in 1967 Anosov had already introduced a struc- France, but one could fulfill one’s obligations turally stable class of diffeomorph sm (the through civil service in a foreign country, sev- Anosov diffeomorphisms), each of them having eral young Frenchmen took advantage of this a dense set of periodic points. Because of that, possibility to go to at IMPA. Thus, mathemati- Smale introduced the concept of Axiom A dif- cians such as Étienne Ghys, Cristian Bonatti and feomorphism: f is Axiom A if Ω(f) is hyperbolic Jean-Christophe Yoccoz initiated their contact and the set of periodic points Per(f) is dense with Brazil in large measure due to military pol- in Ω(f). The Axiom A diffeomorphism contains icy, and when the policy changed, the connec- the Morse-Smale ones and, in 1970, Palis and tions were already firmly established and were Smale [17] announced the Structural Stability developing on their own. Conjecture: a diffeomorphism is structurally stable if and made through computer assisted proofs, start- 17 only if it is Axiom A and satisfies the transver- ing with the celebrated work of Lanford. Such sality condition, techniques were particularly suitable to es- tablishing the existence of hyperbolic periodic meaning that the stable and unstable ma- points of renormalization. nifolds of all points of nonwandering set meet one another transversally. In the beginning of the 1980s renormal- ization theory would start to be developed in The Structural Stability Conjecture influ- the complex setting by Douady and Hubbard enced a lot the research in dynamics in the de- in Paris, to understand certain features of the cades of 1970 and 1980. In particular, since the Mandelbrot set. Conditions were ripe for Sul- lost letter mentioned before, this conjecture livan (then traveling constantly between Paris was always in Mañé’s mind and finally, it was and New York), who had recently sensational- proved by him in 1988 [6], eighteen years after ly settled a fundamental conjecture about the 1 his letter, in the case of low regularity (C -dif- dynamics of rational maps (now known as the feomorphisms). Later S. Hayashi (1997) com- No Wandering Domains Theorem) by using 1 pleted the picture for C -flows [4]. The higher the tools of Teichmüller theory: he proposed regularity case remains wide open. a daring generalization of the Renormalization Conjectures as well as a program to establish it One-dimensional dynamics within the Douady-Hubbard framework. Welington de Melo was uncompromising in At the time, Sullivan had long developed ties his search for excellence. A tough teacher, fea- with Rio, having in particular proved with Mañé red by most students who were not close to and Sad (another student of Palis...) the densi- him, he held strong opinions about what were ty of Structural Stability for rational maps [7], the relevant questions in mathematics. On the in a work that introduced the fundamental no- other hand, for the few mathematicians that tion of a “holomorphic motion” (independently cleared a pretty high bar, he had a seemingly discovered by Lyubich). De Melo sensed that unlimited admiration. His mathematical Pan- something big was going on and dedicated theon was populated by people like Smale, Sul- several years to help Sullivan achieve his plan livan, Thurston and Gromov, with the first two (he would constantly travel to meet Sullivan being personally close to him as well. His first wherever he was, forcing his students later to works regarded the Structural Stability Conjec- be endlessly reminded about the excellent food ture, but after some years he would move on to and wine that both had enjoyed at those op- focus on one dimensional dynamics, which we portunities). He had earlier initiated a fruitful now discuss. collaboration with Sebastian van Strien (and later on, Marco Martens) and had established In the mid-1970s, while studying interval himself through fundamental results about real maps, Feigenbaum as well as Coullet and Tress- maps (regarding, in particular, the absence of er, discovered the existence of “universal scal- wandering domains under appropriate regular- ing law” in the transition from regular to chaotic ity assumptions), but he was more than happy dynamics through cascades of doubling bifur- to play a supporting role in this case. The re- cations. To explain this remarkable phenomena, sult was a technically formidable proof of part they independently introduced renormalization of the generalized renormalization conjectures theory in dynamics (it should be noted that all using a Teichmüller theory of certain lamina- three had a background in physics). At the cen- tions (solenoids) especially developed for this ter of the picture lies a “meta-dynamical sys- purpose. tem”, taking place on an infinite dimensional space of maps. The dynamics of this renormal- When time came to write it up, de Melo de- ization operator is reflected on the behavior of cided to just produce a book with van Strien, called simply “One Dimensional Dynamics” [11] families of the interval maps. Particularly, they which would become the basic reference in the showed that the observed universality would field for years, and whose final chapter was follow from what became known as the Feigen- dedicated to Sullivan’s program. baum-Coullet-Tresser Renormalization Conjec- ture, which immediately caught the attention Other mathematicians also took inspiration of both the mathematics and physics com- from Sullivan, but followed different routes. munities. Progress towards the understanding The theory developed intensively in the 1990s of the renormalization operator was initially though the work of McMullen and Lyubich, and an even more general version of the renormal- Many years after de Melo and Mañé ap- ization conjectures would become a fundamen- peared, another student would do his PhD with tal tool in establishing the Regular or Stochas- Palis and go on to have a marked influence on tic dichotomy, which described very precisely dynamics. Marcelo Viana was born in Rio, but the dynamics of typical quadratic maps. Mean- spent his childhood in Portugal, only return- while, in collaboration with Sullivan’s student ing to Brazil for graduate school. Influenced Edson de Faria, de Melo went on to develop the by Mañé, whose talks he admittedly appreci- renormalization theory in the setting of critical ated, he displays remarkable clarity even when circle maps [9, 10]. In addition to obtaining this presenting hard subjects, and became a natu- and other important results, he became the ral attractor for students in dynamics in Brazil. mentor of a younger generation of Brazilians Contributing to many different areas within dy- that he attracted to renormalization theory. namics, we can still describe a large part of his work as dealing with weakenings of the notion Welington passed away in 2016, a few weeks of uniform hyperbolicity. This includes the ones after a beautiful conference in honor of his 70th described above, but also another one, nonuni- birthday held at IMPA, which was attended by form hyperbolicity to which we now turn. many of his friends. Lennart Carleson had become convinced, Generalized hyperbolicity probably in the 1970s, that problems in dy- namical systems often involved an interesting During the 1960s and 1970s, the research on analytical component.4 Naturally fearless, he uniformly hyperbolic (Axiom A) systems went joined forces with Michael Benedicks in order far beyond matters of structural stability. Parti- to tackle the Hénon attractor, a central exam- cularly important was the work of Sinai, Ruelle ple in the theory, but whose understanding and Bowen establishing that Lebesgue almost seemed totally out of reach. Hénon maps are every orbit equidistributes according to one of diffeomorphisms of the plane of the form x,( y) finitely many probability distributions called → (1 – ax² + y, bx), and for certain values of the “physical measures”, which is the beginning of parameters (a, b) computer experiments indi- a vast theory establishing good statistical pro- cated the existence of a chaotic attractor. perties for such systems. However, at the same They first considered the limiting caseb = time, it became increasingly clear that uniform 0, which degenerates into a (one-dimensional) hyperbolicity was relatively special, in the sen- quadratic map. Jakobson had earlier shown that se that many systems of interest, were far from for many quadratic maps (i.e., for a set of pa- being uniformly hyperbolic. rameters a in a set of positive Lebesgue mea- A flexibilization of the notion of uniform hy- sure), the dynamics is chaotic. Orbits would perbolicity is the notion of partial hyperbolici- come near the critical point zero, which is in- ty. In uniform hyperbolicity, the tangent bundle finitely contracting, but then would spend some splits continuously into two invariant (under the time elsewhere, gaining expansion. In average, tangent map) subbundles TM = Eu⊕Es along expansion tended to prevail, giving rise to ex- which one sees uniform expanding and con- ponential growth of the derivative along typical tracting behaviors. In partial hyperbolicity, one orbits. But this “nonuniform expansion” process allows for a third subbundle, with “in between” is very delicate, and small changes in the param- behavior. Finally, there is an even weaker notion, eters lead to a very different dynamical behav- domination, which involves a splitting into n ≥ ior, with almost all orbits being now attracted to a single periodic orbit. Benedicks and Carleson 2 subbundles, TM = E1⊕...⊕En along which vec- tors are not required to have absolute (expand- first obtained a new proof of Jakobson’s Theo- ing or contracting) behavior, but there is control rem, focusing on controlling how fast the critical of the relative behavior: any vector in E must orbit could interact with itself, and then allowed i the parameter b to be non-zero but very small. dilate more than any vector in E . The under- i +1 The role of critical points is then taken over by standing of domination played a role already in tangencies between unstable and stable mani- the proof of the Structural Stability Conjecture folds, with the mild technical issue that a single by Mañé, and continued intensively in the 1990s problematic point is replaced by a Cantor set. by the generalized Brazilian school (so much so that domination was sometimes referred to as It is hard to overstate how difficult it was to “Brazilian hyperbolicity”), who connected it with comprehend the resulting work of Benedicks notions such as robust transitivity. and Carleson which established the existence of a positive measure set of parameters exhibi- two dimensions) the case of a tangency bet- 19 ting a chaotic attractor. One can get an idea by ween unstable and stable manifolds and sho- reading the last paragraph of the introduction wed that they could produce a nightmare sce- of [1]: Carleson,5 the solver of the Lusin Con- nario (the Newhouse phenomenon [15]): many jecture, comments about the difficulties they (topologically generic in an open set U) near- faced writing up the result. by systems would display infinite many sinks. What goes on in the “bad” open set U (called At IMPA, Viana took to himself to unders- tand this work, and having achieved this, he the Newhouse domain), all maps display tan- went on to generalize it. Together with another gencies between unstable and stable manifolds student of Palis, Leonardo Mora, he showed of (possibly non-periodic) orbits in a horse- [12] that attractors such as those constructed shoe. In a sense, what is going astray is that the by Benedicks and Carleson arise “naturally” in unfolding of any single tangency always gene- the process of unfolding homoclinic bifurca- rates many others. tions (see the next section), and hence they are Palis, in the 1980s, became increasingly inter- present for a large class of dynamical systems. ested in the unfolding of homoclinic tangencies The “Hénon-like” attractors which were pro- (homoclinic bifurcations), which he understood duced by this method are not uniformly hyper- as one of the main mechanisms through which bolic but were later shown to be also amenable complex dynamical behavior arises (such as the to a good statistical description. Particularly, typ- Hénon-like attractors of Mora-Viana). He dared ical orbits near the attractor equidistribute with to conjecture that the only way to be safe from respect to a physical measure _, which is non- appropriately generalized6 homoclinic bifurca- uniformly hyperbolic in the sense that there is a tions is to be in the presence of uniform hyper- measurable splitting TM = Eu⊕Es over μ-almost bolicity. This conjecture has been the subject of every point, along which one sees nonuniform intense research, particularly by his students En- expanding and contracting behaviors. rique Pujals and Martín Sambarino [22] (in two dimensions) and by Pujals and Crovisier in high- Palis the conjecturer er dimensions [3]. In a different direction, the work of Palis with If Palis has a mathematical hero, it is certainly collaborators like Takens, Viana and Yoccoz, Poincaré. During his talks, he often describes the showed that what tends to happen just after a famous mistake in his work about celestial me- chanics that had been recognized with the King generic bifurcation (in two dimensions) depends Oscar prize. Due to this mistake, Poincaré had to crucially on some fractal dimensions associated spend his prize money buying back a whole edi- to a horseshoe, in the sense that if the sum of tion of Acta Mathematica, but he got something dimensions is less than one then uniform hyper- much more important in return: the discovery bolicity prevails, and if it is larger than one then of the importance of homoclinic orbits. Those the set of parameters corresponding to tan- are trajectories that are asymptotic, in the future gencies is too large in measure to neglect. But and in the past, to a common periodic orbit. If Palis expected even more, that such tangencies the periodic orbit is hyperbolic this corresponds would be already inside Newhouse domains. His to the intersection of the unstable and stable student Carlos Gustavo Moreira would later join manifolds, and if the intersection is transverse forces with Yoccoz [14] to establish this, by first one quickly gets a complicated picture, about solving a related conjecture of Palis about arith- which Poincaré would later comment: “On sera metic sums of Cantor sets [13]. frappé de la complexité de cette figure, que je Inspired on many of the results mentioned ne cherche même pas à tracer. Rien n’est plus above, in 1995 Palis conjectured [16] the exis- propre à nous donner une idée de la complica- tence of a dense subset D of dynamical sys- tion du problème des trois corps et en général tems such that each map in D admits a good de tous les problèmes de Dynamique...” [21]. statistical description, in the sense that Lebes- Smale would later recognize that the picture gue almost every orbit is equidistributed with that scared Poincaré was so complicated be- respect to one of finitely many physical measu- cause it contained his famous horseshoe. res supported on attractors.7 This conjecture, But homoclinic phenomena gets even more known as the Global conjecture, caught the at- complicated. In the early 1970s Newhouse con- tention of the Dynamicist community, but des- sidered instead (for area contracting maps in pite intense research it remains open. Final remarks Brazilian goes on to describe a different (sim- pler) setting where a spectral gap does hold The three authors of this article did their PhD typically, to which the Israeli objects: “But can at IMPA, one of us as a student of de Melo, and you give a single example?” The Brazilian mum- the others as students of Palis. Having been in bles something about stability. At this point, touch since then with various different interna- the American starts to laugh. tional environments, we would like to mention a distinct characteristic of the Brazilian school 1 According to Palis, the funding that supported that congress of dynamical systems. From the beginning, stu- was offered by the Brazilian government by a very unexpect- ed meeting with the then current Finep director Pelúcio Fer- dents get exposed to the grand goal of achieving reira when Palis was returning from one of those Saturday a global view of dynamical systems. They should morning seminars mentioned above. Ferreira recognized try to explore the whole parameter space, and Palis and asked him why he was walking carrying so many book in a Saturday. Palis told Ferreira that IMPA was starting to describe the behavior of generic (both in the a new doctoral program and he was training the students. sense of topology and of measure) systems. It is He also commented that he was thinking that it would be of course an adaptation of the optimistic point useful for students to gain a broader view of mathematics if they organized an international conference. In the end of of view Smale had as he entered dynamics, but the talk, Veloso promised Palis US$ 150,000.00 to organize the influence of Poincaré also looms large. the meeting. This was an impressive amount for that kind of venue at that time in Brazil. We conclude with the following anecdote 2 Fittingly, this letter was lost when IMPA changed its physi- which illustrates our point. A Brazilian, an Is- cal location from downtown to Jardim Botânico. raeli and an American are walking along the 3 The complement of the set of points with wandering n sidewalk of the beach of Ipanema. The Israeli neighborhoods W, i.e., such that the f (W), n ∈ ℤ, do not intersect each other. talks about some recent work, the precise set- 4 The work of Michael Hermann in the 1970s settling the up of which is not relevant, but involving the Arnold Conjecture (on the linearization of circle diffeomor- choice of a few matrices, which need to dis- phisms) probably played an important role here. play a “spectral gap”. The Brazilian comments 5 According to Benedicks. that unfortunately it is difficult to get hold of 6 One must also consider bifurcations of so-called heterocli- nic type. a spectral gap. The Israeli disagrees with the 7 The conjecture has some refinements, including statements assessment, since one can provide many such about (stochastic) stability of physical measures, and some matrices, by considering the case of algebraic weak deterministic stability along generic parametrized fa- entries. The Brazilian replies that this is a tiny milies. Also, for one dimensional dynamics it is stated in a much stronger form: maps with a good statistical descrip- set, and that nothing is currently known about tion are not only dense, but typical (this formulation was the typical case. Then the tables turned: the inspired by the Regular or Stochastic result of Lyubich [5]).

References 1. Benedicks, M., Carleson, L.. The Dynamics of the Henon 13. Moreira, C.A.; Yoccoz, J.C.. Stable intersections of reg- Map. The Annals of Mathematics. 133, 73-169 (1991). ular Cantor sets with large Hausdorff dimensions. An- 2. Benedicks, M.; Viana, M.. Solution of the basin problem nals of Mathematics, v. 154, n.1, p. 45-96, 2001. for Hénon-like attractors. Inventiones Mathematicae, v. 14. Moreira, C.A.; Yoccoz, J.C.. Tangences homoclines 143, p. 375-434, 2001. stables pour des ensembles hyperboliques de grande 3. Crovisier, S. ; Pujals, E. R. . Essential hyperbolicity and dimension fractale. Annales Scientifiques de l’Ecole homoclinic bifurcations: a dichotomy phenomenon/ Normale Supérieure, v. 43, p. 1-68, 2010. mechanism for diffeomorphisms. Inventiones Mathe- 15. Newhouse, S.E.. Diffeomorphisms with infinitely many maticae, v. 201, p. 385-517, 2015. sinks. Topology. 13, 9-18 (1974). 4. Hayashi, S.. Invariant manifolds and the solution of the 16. Palis, J.. A Global View of Dynamics and a Conjecture C1-stability and ω-stability conjectures for flows. Ann. on the Denseness of Finitude of Attractors. Astérisque. of Math., 145, 81-137 (1997). 261, 335-347 (2000). 5. Lyubich, M.. Almost Every Real Quadratic Map Is Either 17. Palis J. and Smale S.. Structural Stability Theorems. In Regular or Stochastic. The Annals of Mathematics. 156, Global Analysis, Proc. Symp. Pure Math., volume 14, 1-78 (2002). 223-231, AMS (1970). 6. Mañé R. (1988) A proof of the C1 stability conjecture. Inst. Hautes Études Sci. Publ. Math. No. 66 (1988), 161-210. 18. Palis J.; Viana, M.. High Dimension Diffeomorphisms Displaying Infinitely Many Sinks. Annals of Mathemat- 7. Mañé, R. ; Sad, P. R. G. ; Sullivan, D.. On The Dynamics ics, v. 140, p. 207-250, 1994. Of Rational Functions.. Annales de I’Ecole Superieure - 1983, p. 0-0, 1983. 19. Peixoto M. On Structural Stability. Annals of Math., 69, 8. de Melo, W.. Structural Stability of Diffeomorphisms on 199-222. Two-Manifolds. Inventiones Mathematicae, v. 21, p. 233- 20. Peixoto, M. (editor). Dynamical Systems: Proc. Symp. 246, 1973. held at the University of Bahia, Salvador, Brasil, July 9. de Melo, W.; de Faria, E.. Rigidity of Critical Circle Map- 26-August 14, 1971. (Academic Press, 1973). pings I. J. Eur. Math. Soc., v. 1, n.4, p. 339-392, 1999. 21. Poincaré, H.. Les méthodes nouvelles de la mécanique cé- 10. de Melo, W.; de Faria, E.. Rigidity of Critical Circle Map- leste, Volume 3 H. Poincare. (Gauthier-Villars, Paris, 1899). pings II. Journal of the American Mathematical Society, 22. Pujals, E. R.; Sambarino, M.. Homoclinic tangencies and v. 13, n.2, p. 343-370, 2000. hyperbolicity for surface diffeomorphisms. Annals of 11. de Melo, W. C., Strien, S. V.. One Dimensional Dynamics, Mathematics, Princeton, v. 151, n.3, p. 961-1023, 2000. Springer-Verlag, 1993. 23. Viana, M. Dynamics: a probabilistic and geometric per- 12. Mora, L.; Viana, M.. Abundance of strange attractors. spective. Proc. Int. Congress of Mathematicians (Berlin, Acta Mathematica, v. 171, p. 1-71, 1993. Germany, 1998) vol 1, pp r. 21

Carolina Araújo Full Researcher, IMPA, Brazil E-mail: [email protected]

A Gender Perspective on Mathematics in Brazil

The under-representation of women in STEM recipients of national research grants is much fields, particularly in mathematics, is a global lower. The National Council for Scientific phenomenon. In Brazil, it is estimated that 49% and Technological Development (CNPq) is of researchers in all sciences are women, whilst the oldest and one of the most important in mathematics women account for less than scientific funding agencies in Brazil. It is the 25% [1]. In the last few years, gender matters Brazilian equivalent of the National Science in mathematics have been largely discussed. Foundation of the United States. Its “Research Debates about this theme have been organized Productivity” program provides individual independently by universities and in major grants to selected researchers in all areas of mathematical events nationwide. science, according to levels. Among the totality of CNPq Research Productivity grantees in Numbers from the Ministry of Education mathematics, probability and statistics, less show that in 2014 around 48% of the than 13% are women. In the highest level of the undergraduate degrees in mathematics in program, women account for less than 7%. Brazil were obtained by women, varying from 34% to 62% depending on the region of the Statistics from the most recent Mathematics country and type of degree. The percentage Colloquium in Brazil confirm this scenario. The of Ph.D degrees in mathematics awarded to Colloquium is the most important meeting of women in the same year was 24%. Women the Brazilian mathematical community and has account for approximately 40% of university been organized every two years since 1957. professors in Brazil. These numbers have not Among the 888 participants in the 31st Brazilian increased for the past 20 years. We refer to [2] Mathematics Colloquium (2017), 23.5% were for more complete statistics. women, whilst 16.8% of research talks were delivered by women. Among the 11 plenary In 1950, the first Ph.D in mathematics was lectures, only 1 was delivered by a woman. awarded to a woman in Brazil. Since then, the situation has clearly improved. It has been Gender disparity in mathematics in Brazil estimated that the percentage of women can be detected at an early age. The Brazilian researchers in mathematics in Brazil was 19% Mathematical Olympiad of Public Schools in the quinquennium 1996–2000, and close (OBMEP) provides expressive statistics. The to 25% in the quinquennium 2011–2015 [1]. 13th OBMEP took place in 2017, with over However, the percentage of women among 18 million participants from more than 53 Figure 1. Percentage of girls among all participants in the second stage of the Olympiad, and among all recipients of gold, silver and bronze medals at level 1 of the competition (11–12 years old), from 2007 to 2017.

thousand schools in Brazil, reaching 99.6% In the last decade, there have been a of Brazilian municipalities. The three graphs number of initiatives toward gender balance in above and below display the percentage of science as a whole in Brazil. In 2005 the CNPq, girls among all participants in the second stage together with the ministries of Education and of the Olympiad, and among recipients of gold, Science and Technology and other partners, silver and bronze medals in all three levels of launched the program “Women and Science”. the competition, from 2007 to 2017. These Its main goal lies in promoting the participation statistics were provided by the OBMEP. The of women in sciences and academia and fact that the performance of girls worsened as stimulating discussions on national gender they grow older, dropping dramatically to Level relations [4]. The program recovered and 3, is an eye-catcher and calls for special action. gave visibility to the history of women science Please refer to [3] for a detailed account of the pioneers in Brazil. Also, since 2013, women that OBMEP program. give birth or adopted a child while holding a

Figure 2. Percentage of girls among all participants in the second stage of the Olympiad, and among all recipients of gold, silver and bronze medals at level 2 of the competition (13-14 years old), from 2007 to 2017. 23

Figure 3. Percentage of girls among all participants in the second stage of the Olympiad, and among all recipients of gold, silver and bronze medals at the level 3 of the competition (15–17 years old), from 2007 to 2017. Performance worsens at they grow older.

CNPq Research Productivity grant have had around gender issues in the field. For the past their award extended for one year. two years, several groups have independently organized round tables, informal mentoring Despite these important recent initiatives, programs for female students and exhibitions national actions toward increasing the number portraying women mathematicians, which of women specifically in mathematics have have been widely displayed at universities, been scarce. There is no national association public open spaces and major science and of women in mathematics in Brazil, and until mathematical events. very recently the under-representation and challenges faced by women in the mathematical During the 2017–2018 official Biennium community were not often discussed in of Mathematics in Brazil, the national project forums. In the past few years, however, there “Matemática: substantivo feminino” has been has been a strong movement to bring forth promoting a series of debates in local universities, the debate on the gender gap. In various connecting women mathematicians, whilst parts of the country, women mathematicians focusing on regional specificities given the have gathered in groups, inviting their local continental dimensions of the country [5]. mathematical community to discuss the The inaugural table of this series took place current gender situation. The “Encontro at the 31st Brazilian Mathematics Colloquium. Paulista de Mulheres na Matemática” took Gender-related activities of the Biennium will place in March 2016, and was the first Brazilian culminate with the World Meeting for Women meeting devoted specifically to the work in Mathematics, taking place in Rio de Janeiro of women mathematicians and the debate as a satellite event of ICM 2018 [6].

References 1. Gender in the Global Research Landscape, report 4. CNPq - Programa Mulher e Ciência - http://cnpq.br/ available at https://www.elsevier.com/__data/assets/ apresentacao-mulher-e-ciencia pdf_file/0008/265661/ElsevierGenderReport_final_ 5. Matemática: substantivo feminino - https://matemati- for-web.pdf casf.wordpress.com 2. Brech, C. O “dilema Tostines” das mulheres na 6. (WM)2 - World Meeting for Women in Mathematics - Matemática, Revista Matemática Universitária - https:// https://www.worldwomeninmaths.org www.ime.usp.br/~brech/gender.html 3. Landim, C. The Brazilian Math Olympiad of Public Schools, 10 years promoting social justice through aca- demic merit, in this volume. Alicia Dickenstein Full Professor, University of Buenos Aires, Argentina Principal Researcher, CONICET, Argentina E-mail: [email protected]

Algebraic Geometry in the Interface of Pure and Applied Mathematics

1 Introduction abstract objects are enabled, researchers often find major new theorems and applications of The new era of applications of algebra these objects by experimentation. and geometry started in the 1980s, with the availability of personal computers and the The book Ideals, Varieties, and Algorithms [13], implementations of algorithms to compute written by D. Cox, J. Little, and D. O’Shea, first Gröbner bases, introduced by B. Buchberger in published in 1992, represented a timely vision his 1965 thesis written under the direction of W. that algebraic geometry and computational Gröbner. Two of the free and open-source CASs commutative algebra could be made for polynomial computations that emerged in accessible not just to mathematicians who that period and are now widely used and still were not experts in the area, but also to users in active development, are Macaulay2 [34] and of mathematics in engineering and computer Singular [20]. Macaulay2 was designed and science. It was followed by a graduate text by the written by D. Grayson and M. Stillman, with the same authors: Using Algebraic Geometry [14], aim of supporting computation in research in first published in 1998, in the same accessible algebraic geometry, commutative algebra, and style, which broadened the way we teach and related fields, including applications in biology, use (computational) algebraic geometry. physics, and statistics. Singular has been The maturity of the subject led to the developed under the direction of W. Decker, creation in 2016 of the SIAM Journal on Applied G.-M. Greuel, G. Pfister, and H. Schönemann, Algebra and Geometry (abbreviated: SIAGA). who head Singular’s core development team This journal offers a new home for exciting within the Department of Mathematics of emerging applications using tools from algebra, the University of Kaiserslautern. It has a geometry and topology. Its Editor in Chief is special emphasis on commutative and non- B. Sturmfels, the author of several fundamental commutative algebra, algebraic geometry, books (in particular, [42, 48, 49]) and a large and singularity theory. Further functionality number of papers introducing techniques of is obtained by combining Singular with third- algebraic geometry and combinatorics in many party software; this includes tools for convex different areas of mathematics and applications. geometry, tropical geometry, and visualization. Another pioneer CAS is CoCoA, developed in the In the following sections I will give a brief University of Genova [1]. One important aspect description of some of the application areas of these systems is that once computations of and the algebro-geometric concepts involved. 25 Figure 1. A kinematic mechanism design curve, computed with Bertini_real (https://dl.acm.org/citation. cfm?id=3056528). Each point on the curve represents a four-bar mechanism satisfying three control points and three poses, part of the family of Alt-Burmester problems.

Section 2 will be devoted to applications value of these invariant linear functions on the in biology and Section 3 will survey other concentrations). In many cases, these linear important applications. In turn, these invariants are easily predicted in terms of the applications lead to interesting and basic biochemistry. Limits of trajectories are steady theoretical questions; some of these questions states of the system, that is, solutions to the have been answered along the way and some algebraic system fk (x) = 0. Thus, questions about remain as directions of active research. Finally, steady states in biochemical reaction networks Section 4 contains other pointers to software under mass-action kinetics are fundamentally for polynomial system solving. questions about nonnegative real solutions to parametrized polynomial ideals. We refer 2 Biological applications the reader to the survey article [21] for basic Systems biology’s main goal is to understand definitions and further references and we the design principles of living systems. review here some advances developed after Algebraic geometry can be used to analyze the that article was published. standard models in the field. In particular, in the We introduced a general framework for realm of biochemical reaction networks, that biological systems called MESSI systems [43], is, chemical reaction networks in biochemistry, that describe Modifications of type Enzyme- the usual mass-action kinetics modeling of Substrate or Swap with Intermediates, and we the evolution of the concentrations of the proved general results based on the network different chemical species along time yields structure. Many post-translational modification an autonomous system of polynomial ordinary networks are MESSI systems. For example: the differential equations in the unknown motifs in [29], sequential distributive multisite vector of concentrations x of the n species as phosphorylation networks [44], sequential functions of time. This is indeed a family of processive multisite phosphorylation networks, polynomial differential systems associated phosphorylation cascades (as in Figure 2), to a labeled directed graph of reactions. The the bacterial EnvZ/OmpR network in [47], monomial terms come from the labels of the many two component systems, and all linear nodes by complexes in the given species, the networks. We showed that, under mass-action coefficients depend on the (positive) reaction kinetics, MESSI systems are conservative, and rate constants k that label the edges and the we simplified the study of steady states of these total production of each reaction (which is the systems by explicit elimination of intermediate difference of the labels of the target and source complexes (inspired by [30, 53]). nodes). The n real polynomials fk, i(x) carry a combinatorial structure inherited from G. A (bio)chemical reaction network is said to Linear dependencies among these polynomials exhibit multistationarity if there exist at least give linear conservation relations and the two positive steady states with the same total behavior of the system also depends on the amounts. Multistationarity provides a crucial values of the total amounts (the constant mechanism for switching between different response states in cell signaling systems and and (too) many parameters. The question enables multiple outcomes for cellular-decision is also difficult because in general, it is hard making. We identified an important subclass to find sparse real polynomial systems with of MESSI systems with toric steady states [44] many (positive) real roots (see for instance and we gave in this case an easy algorithm to Theorem 1.2 in [24]). determine the capacity for multistationarity. It Most of the different results to decide provides choices of rate constants for which the capacity for multistationarity of a given multistationarity takes place, based on the reaction network have been summarized in theory of oriented matroids. Theorem 1.4 of [40]. A consequence of these When a network has the capacity for results driven by applications is the first partial multistationarity, the next question is how to generalization of the classical Descartes’ rule predict (semialgebraic) regions in parameter to guarantee the existence of at most one space which give rise to multistationary positive root in the multivariate setting (which systems. The nice recent article [11] deals was hidden in [19]). The classical Descartes’ with this question based on degree theory, rule of signs was stated by Descartes in 1637 allowing for the determination of both in “La Géometrie”, an appendix to his “Discours multistationarity and monostationarity de la Méthode”. It gives a very simple bound for conditions depending on the rate constants the number of positive real roots of a univariate (while the conditions on the total amounts real polynomial in terms of the number of sign variations of its coefficients. We were also able to find an analog of Descartes’ rule of signs in There are many theoretical the multivariate case when the support of the and computational tools in polynomials in the system is a circuit [3] (with n real algebraic geometry, but + 2 monomials in n variables). This study shows we need precise answers for how difficult it is to even state a conjectural systems of biological interest, complete generalization in the multivariate usually with (too) many case, which is a widely open question. For other real questions, see [48]. variables and (too) many parameters In previous works, we developed in [15] the basic theory of toric dynamical systems (a.k.a. complex balanced systems) in the are not specified). A different approach using context of computational algebraic geometry results from real algebraic geometry has and showed that the associated moduli space been developed in [32], where we get only is also a toric variety. We proved this for open sufficient conditions, but jointly on rate detailed balancing systems whose invariant constants and total amounts. Other main polyhedron is two-dimensional and bounded, problems in the area are to develop tools to the simplest case of the main open conjecture find the maximal number of positive steady that the unique complex balancing steady states and to find regions in parameter space state is a global attractor of the trajectories. with the predicted number of positive steady In [37], we presented an efficient procedure states, or at least where lower/upper bounds for calculating steady state invariants that are apply. Another important question is the linear combinations of complexes and depend characterization of networks that allow for on selected variables. We showed how enzyme stable oscillations, as in the Lotka–Volterra bifunctionality can lead to different forms population model (which can be seen as of concentration control that are robust to arising from a directed graph under the changes in initial conditions or total amounts. mass-action kinetics modeling). There are Discrete dynamical systems have also been many theoretical and computational tools increasingly successful in modeling biological in real algebraic geometry, but we need networks and algebraic geometry provides precise answers for systems of biological powerful tools for their study. The literature is interest, usually with (too) many variables too vast to be cited in this survey, so we will 27

Figure 2. A model of E enzymatic cascade

S0 S1

F1 P0 P1 P2

F2 F2

R0 R1 R2

F3 F3 just mention one sample recent article. Many from topological data analysis have been used problems in biomedicine and other areas of the to show how such codes reflect the structure life sciences can be characterized as control of the underlying stimulus space. In the case problems, with the goal of finding strategies of hippocampal place cell codes, which are to change a disease or otherwise undesirable responsible for tracking the animal’s position state of a biological system into another, in space, these methods have been used to through an intervention, such as a drug or show that correlations in neural activity reflect other therapeutic treatment. The paper [41] the underlying topological and geometric presents a method for the identification properties of the environment [33]. We refer of potential intervention targets in Boolean the reader to the survey [16]. molecular network models using algebraic Another area of interaction of algebraic geo- techniques. The proposed control methods metry and biology is the study of phylogenetic are useful and efficient for moderately invariants in evolution. Phylogenetic varieties large networks. contain the set of joint distributions at the lea- An important question in neuroscience ves of a tree evolving under a Markov model of is understanding the neural code and, in molecular evolution. These varieties are inte- particular, how the collective activities of resting from a biological point of view because neurons represent information about the they provide new tools of non-parametric infe- outside world. In many brain areas, the firing rence of phylogenetic trees, and also pose inte- patterns of neurons have been shown to encode resting algebro-geometric challenges. We refer information about an animal’s interaction with the reader to [9] and the references therein. its environment, including sensory inputs and the animal’s position in space. These 3 Other emerging applications experiments give clues about the intrinsic In recent years there has been a number of structure of neural codes, and how they encode advances about tensors and their different no- various stimuli. In particular, it has been shown tions of rank, which have applications in phylo- that the encoding of the maps connecting genetics, algebraic statistics, signal processing, external inputs to neural responses are often quantum information, convex algebraic geom- given by an arrangement of convex receptive etry, and combinatorial algebraic geometry. An fields. Methods from algebraic geometry and elementary introduction to tensors focusing on combinatorics are now being used to analyze some applications can be found in the article by the intrinsic structure of neural codes [17, 18]. P. Comon [10]; a standard textbook is [38] by The combinatorial data can be represented J. Landsberg; B. Sturmfels wrote a nice recent algebraically via the neural ideal, much as survey [52] and for the spectral theory there is simplicial complexes are algebraically encoded a recent book by L. Qi and Z. Luo [45]. There by Stanley–Reisner ideals. Moreover, ideas is a strong link between tensors and algebraic

1 geometry. For instance, decomposable tensors driving new ideas back into algebraic geometry have rank 1 and correspond to the Segre va- (see for instance [39]). There is a lot of recent riety, symmetric tensors correspond to homo- activity in multi-view geometry, the sub- geneous polynomials, symmetric tensors which discipline of computer vision that studies 3D are powers of linear forms have rank 1 and cor- scene reconstructions from images, and which respond to the Veronese variety, tensors of has deep foundations in projective geometry rank bounded by k form a dense subset of the and linear algebra. k-secant variety of the Segre variety, tensors of One of the earliest successes of applied border rank bounded by k comprise exactly the algebraic geometry has been to the area of k-secant variety of the Segre variety, etc. geometric modeling. In 1995, Sederberg and For general varieties, the concept of Euclidean Chen introduced a method for the study of distance degree – introduced in [26] – counts the implicitization problems in geometric modeling, number of critical points of the squared distance that they termed as moving curves and to a general point outside the variety, appealing to surfaces [46]. D. Cox realized that underlying classical intersection theory from the perspective their work was the algebraic notion of of computational algebraic geometry. For Segre syzygies [12], which opened up a fruitful area of varieties these numbers have been computed research. In 2002, L. Busé and J.-P. Jouanolou in [31]. The nearest point map of a real algebraic abstracted and generalized on a sound variety with respect to Euclidean distance is an basis the method of Sederberg–Chen [7] algebraic function (for instance, for varieties of via approximation complexes, a tool in low rank matrices, the Eckart–Young Theorem homological commutative algebra that had states that this map is given by the singular been developed by J. Herzog, A. Simis and value decomposition), which is of importance in W. Vasconcelos [35, 36]. They later produced control theory, geometric modeling, computer other interesting articles on the subject jointly vision, and low rank matrix completion. with M. Chardin. In [6] we unveiled in concrete terms the general machinery of the syzygy- Algebraic geometry and polynomial based algorithms for the implicitization of optimization techniques have long been used rational surfaces in terms of the monomials in to formulate and solve a number of problems the polynomials defining the parametrization. in computer vision [28]. Visibility computations The theoretical justification is not naive and with moving viewpoints lead to interesting and requires a good command of techniques of difficult problems in real algebraic geometry, (homological) commutative algebra. However, even for simple classes of objects (such as the algorithms do not require a heavy balls and polytopes) [52]. The emerging field of background and are easy to explain. algebraic vision attemps to introduce new ideas from moduli theory, representation theory, The relation between nonnegative real as well as numerical, real, and combinatorial forms and multivariate polynomials that can algebraic geometry into computer vision, also be written as a sum of squares is of great

Figure 3. A singular algebraic surface in the gallery of Surfer [49], a software designed to experience the relation between formulas and forms interest because of the active area of research the cohomologies needed. Stillman and his 29 on sum of squares optimization, which has co-authors found a method to compute the applications in many areas, notably control cohomologies of a large (but finite) class theory. The nice article [5] substantially of divisors, which often works by hand. The extends Hilbert’s celebrated characterization physical implications of the results of these of equality between nonnegative forms and computational algebraic geometric methods is sums of squares, giving geometric insight still a work in progress. to the different cases via the relation with projective varieties of minimal degree. In the Over the last decade, the need more recent work [4], they extend their study for frame-theoretic research from irreducible varieties to reduced schemes. has grown alongside the Their results have applications to the positive emergence of new methods in semidefinite matrix completion problem signal processing. Interestingly, and to the truncated moment problem on modern advances in frame projective varieties. theory involve techniques from Frame theory studies special vector algebraic geometry arrangements which arise in numerous signal processing applications. Over the last decade, the need for frame-theoretic research has grown Symbolic computation allows us in principle alongside the emergence of new methods to make computations with parametric systems in signal processing. Interestingly, modern of polynomials, that is, with families of algebraic advances in frame theory involve techniques from varieties. F. Rouillier and M. Safey El Din algebraic geometry, semidefinite programming, have implemented and constantly improved algebraic and geometric combinatorics, and certified computations with real polynomials representation theory (see for instance [8]). and real solutions, which were used in manifold applications, for example, in the study of the Finally, we refer to the forthcoming structural stability of n-dimensional systems in book [53], to be published by the American control theory, or in the discovery or preclusion Mathematical Society, for a good account of the of geometrical structures in certain 3D basics of the active area of algebraic statistics. manifolds [27]. This is an introductory graduate text book which includes a fair amount of background The inherent complexity of most nonlinear on probability, algebra and algebraic geometry, algebraic computations has led to the develop- statistics, and convex geometry. ment of software for polynomial system solv- ing based on *numerical algebraic geometry*. 4 Back to software The numerical solution of systems of polyno- mial equations is based on well-tuned algo- Applications require the solution to other rithms for homotopy continuation. A. Leykin problems in computational algebraic geometry. has been instrumental in adding numerical al- For example, in the recent and ongoing gebraic geometry methods to Macaulay2. Two research of A. Braun, C. Long, L. McAllister, M. other examples of useful free software for the Stillman, and B. Sung, it has been necessary numerical solution of polynomial systems are to find many (possibly singular) rigid divisors Bertini [2] and PHCPack [56]. Bertini is a gener- (defined in terms of sheaf cohomologies) on al-purpose solver that was created for research a Calabi–Yau 3-fold. In cosmological solutions on polynomial continuation, designed and written of string theory, if there are suitable rigid in C by D. Bates, J. Hauenstein, A. Sommese and divisors then one might see a bright signal in C. Wampler. There are many applications of Berti- cosmic microwave background experiments. ni to questions in robotics, see e.g. [25]. PHCpack D. Eisenbud, M. Mustata and M. Stillman had is an open source software created and main- found algorithms for computing the required tained by J. Verschelde. It was originally designed sheaf cohomolgies. These were implemented to implement polynomial homotopies, exploiting in Macaulay2 by G.G. Smith. Unfortunately, structure in order to better approximate all isolat- the implemented algorithms cannot compute ed solutions. The package also exports numerical irreducible decompositions, and can compute all also [22] for applications of multidimensional positive dimensional solution sets of a system. polynomial residues. There is currently a PHCpack Web Interface to solve polynomial systems. Acknowledgements I am grateful to Eduardo The basics of symbolic-numeric methods for Cattani, Giorgio Ottaviani and Bernd polynomial system solving, including resultants, Sturmfels for their generous input to improve discriminants, solving equations via algebras, this survey. residues and duality, primary decomposition, border bases, and numerical algebraic I also thank Magalí Giaroli and Danielle Brake geometry can be found in the book [23]. See for their help with the illustrations.

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015421 | Image: BrianAJackson/iStockphoto [m] Nicolau C. Saldanha Professor, PUC-Rio, Brazil E-mail: [email protected]

Carlos Gustavo (Gugu) Tamm de A. Moreira ICM Plenary lecturer Researcher, IMPA, Brazil E-mail: [email protected]

Emanuel Carneiro Chair of the Scientific Program Subcommittee of the ICM 2018 Researcher, IMPA, Brazil E-mail: [email protected]

IMO-Style Problems: Are You Ready for a Gold Medal? Take a break and test your problem-solving skills with a few challenges selected by three experts in Mathematical Olympiads

Here are some problems which are not so points P6 and P7 in the plane with P5P6 = a such technical and have various degrees of difficulty. that the closed polygonal line P1P2P3P4P5P6P7 Enjoy! is simple and is the boundary of a heptagon H such that it is possible to tile the plane with 1. Tamás and Emanuel are two mathematicians that enjoy games and puzzles. congruent copies of H. After discussing other games, Tamás introduces 3. Alice writes two distinct integers from the game "15 out of 3". In this game, there are 9 the set {0; 1; 2; …, N} on the back of two cards cards on the table, numbered from 1 to 9. Each and places them on the table so that the player in his turn takes a card. The player who numbers are not visible. Bob chooses a card is the first to obtain three distinct cards adding and examines the number written on that card. up to 15 is the winner. If all cards are taken and He then decides which card to choose. His aim no one wins, the game is declared a draw. is to choose the card with the largest number. (a) Does someone have a strategy to win Thus, if Bob always keeps the first card chosen, the game? his probability of victory is 1/2.

(b) Emanuel, who knows several popular (a) Prove that there exists a strategy which games and puzzles, realizes that this will give Bob a probability of victory game is closely related to something strictly greater than 1/2. he already knows. Can you figure out (b) What is the largest probability of victory his reasoning? If you want to hear this that Bob can guarantee? story from Emanuel himself, you can find it in the articleJogos e feijoada no 4. Rosencrantz and Guildenstern like to play São Paulo's, http://www.obm.org.br/ heads and tails. Rosencrantz always bets on content/uploads/2017/01/eureka_27.pdf heads and Guildenstern always bets on tails. They like to invent new ways of playing. 2. Consider a > 0 and a simple polygonal line Their latest way of playing uses a common P1P2P3P4P5. Assume that the polygonal line is contained between the line P1P5 and the parallel coin. Before starting, they agree on a positive line passing through P3. Prove that there exist integer N and they toss the coin many times counting the outcomes until there have been distance between the points P and Q). Prove 33 exactly N heads. Each head is worth one point that d(f(P), f(Q)) = d(P,Q) for all P and Q. for Rosencrantz and each tail is worth one point 6. Let G be a simple, finite and connected for Guildenstern. When they stop, whoever has graph. A hunter and an invisible rabbit play a fewer points must pay the winner one coin for game on the graph G. The rabbit is initially on a each extra point he had. vertex ω0 (not known to the hunter). The hunter

For instance, on Saturday morning they freely chooses a vertex υ0: if υ0 = ω0 the rabbit is agreed on N = 5 and obtained the following captured and the game is over. Otherwise, the

results (with H for heads and T for tails): rabbit moves invisibly along an edge from ω0 to ω (thus ω and ω are adjacent and in particular HTHTHHH 1 0 1 distinct). In general, in the k-th round the hunter

and the game thus finished with a score of chooses freely a vertex υk: if υk = ωk the rabbit is 5 to 2 and Guildenstern paid three coins to captured and the game is over. Otherwise, the

Rosencrantz. rabbit moves invisibly along an edge from ωk to ω . The hunter then chooses υ , and so on. (a) What is the probability of a tie? k+1 k+1 The hunter knows the rules above and (b) What is the most probable final score? knows the graph G. After the k-th round the (c) What is the probability that Rosencrantz hunter knows that ωk ≠ υk but he gains no wins? further information. For which graphs G does the hunter have a strategy that guarantees that (d) Is this a fair game (in the sense that he will capture the rabbit in finite time? the expected value of the payment is zero)? Recall that in a simple graph edges are not directed and every edge joins two distinct 5. Let ⨍ be a function from the plane to vertices. the plane. Assume that d(P,Q) = 1 implies d(f(P), f(Q)) = 1 (where d(P,Q) is the euclidean See solutions on pages 66 to 69

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015420x_210x138ma_4c.indd 1 18/11/2013 19:34:59 Biennium of Mathematics: Approaching Brazilians and Mathematics The Biennium 2017–2018 activities allow for a new focus on the subject in Brazil

Generations of Brazilians partnership with the Brazilian So far, success has been im- have promenaded on Copa- Mathematical Society (SBM) to pressive. In the course of four cabana’s shore without rea- organize in 2017 the first Fes- days (from 27th to 30th of April, lizing the mathematics in the tival of Mathematics in Rio de 2017), the festival took place in symmetrical black and white Janeiro which was modeled af- the following schools located waves drawn on the board- ter the United States National in three different geographical walk of Portuguese cobbles- Math Festival. areas of Rio de Janeiro: Escola tones. This kind of associative The free event intended for Eleva, in Botafogo (south), Es- thought is not so difficult, but audiences of all ages brought cola Sesc de Ensino Médio, in requires a more sensitive look together 18,000 people in Jacarepaguá (west); and Nave at the specific area of knowled- about 60 workshops with ac- do Conhecimento Cidade Olím- ge that forms part of daily life tivities based on mathematical pica, no Engenho de Dentro but is many times presented as concepts. There were also exhi- (north). Despite the rainy wea- something disconnected from bitions of art, 3D prototypes, ther and political protests that reality. games, videos and lectures by sparked a national outage and The IMPA believes in trans- renowned researchers, market affected public transportation, forming the way people obser- professionals and students from the festival attracted approxi- ve reality and is driven by the Brazil and abroad. The IMPA mately 18,000 visitors. mission to promote mathema- seeks to do everything within Participants acted both as tics’ multiple practical uses in its reach to bring mathematics visitors and protagonists. They research, teacher training, dis- to the population at large and interacted and enjoyed the ac- semination of knowledge and show that, in addition to being tivities available for hours or integration with other areas. part of daily life, mathematics days on end given that many Accordingly, it established a can be fun and accessible. chose to visit the event more A boy takes a surprisingly smooth ride in a bicycle with square wheels: events show how mathematics can be fun and accessible Courtesy of IMPA Courtesy than once in order to streng- couldn’t have been better in cations and of Education, and then their knowledge or create the Brazilian scenario. By being with the sponsorship of Banco pleasurable stories with mathe- chosen for the first time to host Nacional de Desenvolvimento matics. two of the most important Econômico e Social (BNDES), The Festival of Mathematics world events in mathematics the initiative includes national was widely covered by the mains- – the International Congress and international actions to tream media and there were live of Mathematicians (ICM) and stimulate, popularize and pro- broadcast transmissions. In inter- the International Mathema- mote improvements in the tea- views, Marcelo Viana, the Mana- tical Olympiad (IMO) – Brazil ching of mathematics country- ging Director of the IMPA, stres- has been witnessing rising ex- wide, highlighting issues rele- sed the importance of presenting pectations among researchers, vant to personal and economic the relevance and attractiveness teachers and students of ma- development. of mathematics, thereby elimina- thematics about the 2018 ca- Since the implementation of lendars. the Biennium and in addition to ting the wrong impression that In order for the debate on the preparations for the IMO it is hermetic and uninteresting. mathematics to be inclusive, and the ICM the IMPA has been “Mathematics is a fundamental the Brazilian National Congress devoted to the challenge of component in the formation of enacted a federal law in Novem- creating a schedule to open up human beings. Learning it is an ber 2016 setting up the Brazi- the topic for a broader discus- exercise of citizenship and indivi- lian Biennium of Mathematics sion and the help the country dual achievement. At the same 2017–2018, honoring Joaquim achieve new levels in teaching, time, as a discipline, it is a pre- Gomes de Souza (1829–1864), research and scientific dissemi- ponderant factor for a country’s a pioneer mathematician from nation of mathematics. productivity”, said Viana. Maranhão State. A strong communications Extensive Debate Through the support of the effort has been made, suppor- The moment to bring ma- ministries of Science, Technolo- ted by strategic action planning, thematics to the forefront gy, Innovation and Communi- including the objective to pre- or at times members of the pu- During the event, the most frequent blic with almost closed eyelids scenes revealed a public with watchful from bursts of laughter. There eyes that were curious, staring in were also plenty of generatio- surprise or at times members of the nal encounters. The engineer public with almost closed eyelids from Onofre Laerte Camargos, 75, bursts of laughter for example, marveled at the tetrahedral kite workshop, for which he signed up with Paulo sent these projects in traditional zens of renowned institutions Henrique, his 7 year-old grand- media and social networks to such as the UFMG’s Itinerant son. In a message to the festi- expand the Biennium’s reach. Museum (Universidade Federal val, Patricia Camargos, Onofre’s de Minas Gerais) the Itinerant Fun and creativity daughter, wrote that she was Museum of Life and Planeta- The Festival of Mathema- delighted with the event and rium at Fiocruz the “Fundão” tics was one of the first ideas that her father enjoyed himself Project of UFRJ (Federal Univer- to come from the paperboard. in magic activities with cards, sity of Rio de Janeiro), the Film Whilst living up to its name by the construction of Polyhedra Society of Mathematics of UFF bringing together a series of ac- with straws and Lawson’s Mi- (Federal University Fluminense) tivities in an event of great pro- nimum Surface. “Thank you and SESI Mathematics. The IM- portions, it relied on the inter- for your kindness in providing section of mathematics with the PA’s VISGRAF laboratory pre- us with such a wonderful and most diverse forms of expres- sented an orchestra of compu- memorable family get together. sion such as colorful Platonic ters and, in partnership with the We were deeply touched!”, she solids constructed with origami; Museum of Moving Image and summed up. constructing musical rhythms Sound (MIS) and @ambos.art, The Festival was surprising via mathematical combinations; the exhibition “+Copacabana: not only for those who frown on sculptures created from poly- the mathematics from ground mathematics and have no idea nomial equations; magic num- up to the museum” in which about how interesting and within bers; and a model transformed one can find what is behind the reach it can be. Even those who into an animation by means of virtual pattern of waves in the have long been touched by the computer programs. postcard of Rio’s worldwide fa- beautiful aspects of the discipline The activities of the festival mous beach promenade. were delighted by what they were selected from 274 original During the event, the most saw, as in the case of 23 year- suggestions sent to the IMPA frequent scenes revealed a pu- old Cesar Ilharco Magalhães, a from all over the country. The blic with watchful eyes that Google software engineer and chosen activities came from do- were curious, staring in surprise one of the event’s speakers: The French mathematician Étienne Ghys, from the École Normale Supérieure de Lyon, explores the geometry of the official soccer ball of the 2014 World Cup at the Math Festival in Rio Courtesy of IMPA Courtesy

“I was very happy to see children de Janeiro and the Senegale- easy to understand and eye- and youngsters playing with se Khadim War also presented catching. The same positi- mathematics, the twinkles in their stories of love for mathe- ve assessment was given by their eyes, the smiles. That matics, touching the public. John Bush, a professor from was fascinating", he said after Currently living in Italy, Kha- MIT who delivered a lectu- speaking on the theme of dim, doctor in the area of dyna- re on mathematics in sports; machine learning to a packed mical systems, summed up how Rogério Martins, professor at audience who could hear about mathematics has no borders the New University of Lisbon, his work experience in Zurich, and can make a difference any- who enchanted all with his Switzerland. where. “It doesn’t matter whe- area calculator bicycle; Caro- ther you’re in Rio, Senegal, Italy lina Araujo, a researcher from Inspirational or the United States. Mathe- IMPA, talked about mathe- Magalhães and other spea- matics is something in common matics found in nature; and kers also embody the inspiratio- for all”, he noted, stressing that Étienne Ghys, French director nal side of the festival. Twice the discipline can help solve of research at Lyon’s École gold medalist in the OBMEP everyday problems. Normale Supérieure and the (Mathematics Olympiad of Bra- The public that assessed IMPA’s special visiting profes- zilian public schools), he explai- the festival as “excellent” in sor, who spoke about science ned how math reveals multiple a qualitative query also had in fashion and the soccer ball. scenarios and can transform the chance to discover that The festival revealed that lives. Tabata Amaral from the even high-level research per- mathematics can be remar- city of São Paulo, Pietro Pepe formed by field luminaries kable and popular, well de- and Alessandra Yoko from Rio in Brazil and abroad may be serving of photos and selfies. Many of the pictures were ta- ken side by side with Aramat, “I was very happy to see children and the blue macaw mascot that youngsters playing with mathematics, loves math, designed by pu- the twinkles in their eyes, the smiles. blic school students, winners That was fascinating” of a contest. Shigefumi Mori making ugly faces. Brazil’s posts and information about from Japan and president of main news broadcast “Jornal Math and the events of 2017 the International Mathema- Nacional”, featured a spe- and 2018. tical Union (IMU) and Helge cial four-piece series on ma- In conclusion, one should Holden from Norway, Secre- thematics having the IMPA's not show disbelief if overhea- tary-General of the institution, director Marcelo Viana as a ring the following kind of ques- received numerous requests leading character, reaching tion in a walk by the beach pro- for autographs from students an audience of over 80 million menade in Copacabana: “Hey, as popular celebrities do. people, in November. do you realize that the pattern If the underlying idea was Mathematics has been the of the waves on the sidewalk that the festival and other ac- subject of over 3,000 news has much to do with mathe- tivities of the Biennium were stories since 2017. The IMPA matics?” However, one should designed to place the Brazilian created social media outlets not expect ordinary people to public in close contact with for itself, for the Biennium of discuss the symmetry of the mathematics, it is now possible Mathematics and the ICM, and to affirm that after 2017 ma- these have become the main design in terms of the Group thematics will be more often reference in Math for the me- Theory created by Évariste Ga- than not talked about. And, dia. The social media network lois. This kind of talk could pro- even better, the public will has reached over 15 million bably only be overheard among discuss mathematics without people with interesting and fun participants of ICM 2018. 39

Jose Seade Researcher, National Autonomous University of Mexico, Mexico E-mail: [email protected]

Kleinian Groups in Several Complex Variables

Kleinian groups were introduced by H. Much of the theory of Kleinian groups Poincaré in the 1880s as the monodromy has been generalized to discrete groups of groups of certain 2nd order differential isometries of . This is a rich and fascinating equations on the complex plane . For decades theory that has been studied for decades, and they have played, and continue to play, a will certainly continue to be studied for many major role in many branches of mathematics, as more decades to come. And what about the for example in Riemann surfaces and Teichmüller other side, automorphisms of ? That is the theory, automorphic forms, holomorphic subject we explore in this article. dynamics, conformal and hyperbolic geometry, The study of discrete subgroups of 3-manifolds theory, etc. Kleinian groups are PSL(n+1; ) puts together, under the same discrete groups of Möbius transformations, label, several important families, as for instance: , a, b, c, d ∈ , . • Subgroups of the affine group Affn ( ; ) = The group of Möbius transformations can : These act on preserving be regarded in several ways, three of these are: a projective (n — 1)-space.

As PSL(2, ), the group of automorphisms • Subgroups of PU(n; 1), the group of 1 of the complex projective line . holomorphic isometries of the complex

2 hyperbolic space .. These act on As Conf+( ), the orientation preserving conformal automorphisms of the sphere. preserving a ball that serves as model for . • More generally, subgroups of PU(p + 1, As Iso+( ), the orientation preserving isometries of the real hyperbolic 3-space. n — p), the projectivized Lorentz groups.

This brings great richness into the study • Groups of isometries of . Every such of Kleinian groups, and it is natural to search group has a natural embedding in PU(n, 1) and for generalizations to higher dimensions. It therefore in PSL(n + 1, ). There are many other is right here that a dichotomy springs: what embeddings of these groups in PSL(n + 1, ). does higher dimensions mean? This can mean The discrete subgroups of PSL(n + 1, ) conformal automorphisms of the n-sphere, that act on with a non-empty invariant which is the same as isometries of the real set where the action is properly discontinuous hyperbolic (n + 1)-space ; and this can were called in [14, 15, 16] complex Kleinian also mean holomorphic automorphisms of the groups. Their study has become an active field complex projective n-space. of research (see the bibliography). Besides the aforementioned remarkable . The sphere plays the role of a mirror, families of subgroups of PSL(n + 1; ), there splitting the sphere into two parts which are are many other ways of constructing complex interchanged by the involution. In [16] there is Kleinian groups (we refer to [7, 14] for more an analogous construction of Schottky groups on this topic). Two of these are especially in . Now the “mirrors” are boundaries of interesting, namely Schottky groups and tubular neighbourhoods of projective n-spaces groups constructed via twistor theory. Let us in . Each mirror splits into two say a few words about these. parts that can be interchanged by elements (involutions) in PSL(2n + 2, ). Given a finite Twistor theory springs from the pioneering number of such mirrors, pairwise disjoint, the work of Penrose, see [13]. The idea is that to corresponding involutions generate a complex every oriented Riemannian 2n-manifold M one Schottky group. In the particular case n = 1, the associates its twistor space , a fiber mirrors are products . bundle over M with fiber the set of all complex structures on the tangent space which are A key concept in this discussion is that of compatible with the metric and the orientation. the limit set. Classically, in real and complex Under certain conditions on the metric in M hyperbolic geometry, the limit set Λ is the set of one has that is canonically a complex accumulation points of the orbits. The set Λ is a manifold, and there are deep relations between closed invariant subset of the sphere at infinity. the conformal geometry in M and the complex Whenever we have a discrete subgroup Ґ of geometry in . Conf( ), the action splits into two invariant sets, Λ and its complement Ω . It is in Λ  where The paradigm of twistor theory is the the dynamics concentrates; the action on 4-sphere with its usual metric. Its twistor Ω is properly discontinuous and the quotient space is 3 which fibers over with Ω/Ґ is an orbifold with a rich geometric fiber SO(4)/U(2) = . In this case the group structure. The literature on this topic is vast

Conf+ has a canonical and enthralling. This is a paradigm in conformal embedding in PSL(4; ). It was proved in [15] dynamics. that Conf ( ) acts on the twistor fibers by + Now, what about the limit set for complex isometries with respect to the usual metric Kleinian groups? There is a significant difference on . This has important implications with with respect to the hyperbolic groups above, respect to the dynamics of the action. coming essentially from the fact that we do not Schottky groups play an important role in have in general what Misha Kapovich calls the conformal dynamics and in complex geometry. convergence property [10, p. 495]. The definition These can be constructed by considering of the limit set as the accumulation points of inversions on (n — 1)-spheres in the sphere the orbits (of points) is not good enough in

Figure 1. A wild knot as a limit set. Figure by A. Arroyo, see [1]. general. This is illustrated by the following complement is properly discontinuous, so its 41 example from [11], which can be easily adapted quotient by the group action is an orbifold, to projective spaces. Take the diffeomorphism in our case with a projective structure. For of given by . subgroups of Conf( ) it coincides with the This has as fixed points, and all orbits usual limit set. accumulate at these two points. Yet, the action The Kulkarni limit set is especially interesting is not properly discontinuous on all of \ in complex dimension two. The following . table compares properties of the limit set in Ravi Kulkarni in [11] proposes a refinement dimensions 1 and 2. of the definition of limit set: It consists of the There is a lot more already known, and much accumulation points of the orbits of points, call more to be known. We hope this article will this L1, together with the accumulation points of motivate the reader to explore this fascinating the orbits of compact sets in the complement topic, which is still in its childhood. We refer of L1. This is a closed invariant set with the to [7] and the bibliography here for more advantage of granting that the action on its details.

The Kulkarni limit set Λ Kul contains 1, 2, 3 or ∞-many lines, The limit set Λ contains 1, 2 or ∞-many points and it contains 1, 2, 3, 4 or ∞-many lines in general position

We know examples where the region of discontinuity ΩKul The region of discontinuity Ω has 0, 1, 2 or has 0, 1, 2, 3, 4 or ∞-many connected components. ∞-many connected components Not known if these are the only possibilities.

The group is elementary if Λ  has The group is elementary if Λ Kul has finite cardinality finitely many lines in general position

If the group is non-elementary: Λ  is If the group is non-elementary ΛKul is a union of lines and the closure of the set of fixed points of it is the closure of the set of invariant repulsive lines of loxodromic elements loxodromic elements

Its complement ΩKul is the largest set where Its complement ΩKul is the largest set where the action is the action is properly discontinuous properly discontinuous

Ω is also the region of equicontinuity ΩKul is also the region of equicontinuity If there are no projective subspaces with finite orbit, then The action on Λ is minimal the action on the space of lines in ΛKul is minimal

References [1] A. Arroyo, J.-M. Othoniel. Nudos salvajes. Edition [8] A. Cano, J. Parker, J. Seade. Action of -Fuchsian groups Othoniel, Paris, France, 2017. on . Asian J. Math. 20, No. 3, 449–474 (2016). [2] W. Barrera, A. Cano, J-P. Navarrete. The limit set of [9] A. Cano, J. Seade. On Discrete groups of automorphisms discrete subgroups of PSL(3, ). Math. Proc. Cambridge of . Geometria Dedicata 168 (2014), 9–60. Philos. Soc. 150 (2011), no. 1, 129–146. [10] M. Kapovich. Hyperbolic manifolds and discrete [3] W. Barrera, A. Cano, J-P. Navarrete. Subgroups of groups. Reprint of the 2001 hardback edition. Modern PSL(3, ) with four lines in general position in its limit Birkhäuser Classics. Birkhäuser (2009). set. Conform. Geom. Dyn. 15 (2011), 160–176. [11] R.S. Kulkarni. Groups with domains of discontinuity. [4] W. Barrera, A. Cano, J-P. Navarrete. On the number of Math. Ann. 237 (1978), 253–272. lines in the limit set for discrete subgroups of PSL(3, ). [12] J.-P. Navarrete. On the limit set of discrete subgroups of Pac. J. Math. 281, No. 1, 17–49 (2016). PU(2; 1). Geom. Dedicata 122, 1–13 (2006). [5] W. Barrera, A. Cano, J-P. Navarrete, J. Seade. Complex [13] R. Penrose. The twistor programme Rep. Math. Phys. 12 Kleinian groups, in “Geometry, Groups and Dynamics”, (1977), 65–76. edited by K. Gongopadhyay et al. A. M. S. Contemp. [14] J. Seade, A. Verjovsky. Actions of Discrete Groups on Math., vol no. 639. Complex Projective Spaces. Contemporary Math., 269 [6] A. Cano, L. Loeza, A. Ucan-Puc. Projective Cyclic (2001), 155–178. Groups in Higher Dimensions. Linear Algebra Appl. 531, [15] J. Seade, A. Verjovsky. Higher Dimensional Complex 169–209 (2017). Kleinian groups. Math. Ann., 322 (2002), 279–300. [7] A. Cano, J.-P. Navarrete, J. Seade. Complex Kleinian [16] J. Seade, A. Verjovsky. Complex Schottky groups. groups. Series “Progress in Mathematics” vol. 303, 2012, Geometric methods in dynamics. II. Asterisque 287 Birkhäuser. (2003), 251–272. Eduardo V. Teixeira Professor, University of Central Florida, USA E-mail: [email protected]

Nonlinear Diffusion Processes: Geometric Ideas and Beyond

1 Diffusion else their stationary versions, the so-called elliptic operators. Diffusion is a phenomenon accounting for average, spread, or balance of quantities in a The simplest way to appreciate the given process. These constitute innate trends connection between diffusion and second- in several fields of natural sciences, which in order elliptic operators is by the following naïve turn justify why diffusion is such a popular looking question: in a domain Ω of n, find a concept among scientists across disciplines. function ƒ : Ω → such that at each point y In the realm of mathematics, the study of ∈ Ω, ƒ( y) equals the its own average over any diffusion is often related to second-order ball centered at y. In slightly more precise differential operators of parabolic type — or mathematical terms, we seek for the relation

(1.1)

for all y ∈ Ω and all 0 < r such that Br( y) ⊂ Ω. function ƒ satisfies the averaging property (1.1) if, The answer may sound surprising at first sight: a and only if, it satisfies the so-called Laplace equation

(1.2)

The operator appearing in (1.2) is called the function that satisfies it. Understanding this Laplacian, and it is the prototypical example principle is paramount to many different areas of a second-order elliptic operator. Intriguing of pure and applied mathematics. mysteries surround the equivalence between That averages in (1.1) are taken over per- (1.1) and (1.2). For starters, while (1.1) requires fectly symmetric balls conveys the idea of just local integrability of ƒ to make perfect homogeneity of the medium, i.e. there is no mathematical sense, equation (1.2) involves preferred direction for diffusion. As for the second-order derivatives of ƒ, which in principle, partial differential equation (PDE) counterpart, have no reason to exist. This is a key point I want (1.2), homogeneity translates into an ideal, rota- to emphasize for now; somehow the averaging tional invariant, constant coefficient operator: property (1.1) bears a regularizing effect to a the Laplacian. 2 A million ways to say Laplacian The boundary condition 43 is a way to declare that one is looking for Analogy is one of the most powerful features membrane configurations attached to the of mathematics and the mathematical theory of given wire. It is easy to see that a minimizer diffusion is blessed with many such correlations, of the above functional will satisfy (first in a in which problems coming from rather different weak sense and later in the classical sense) the disciplines lead to a common, unified mathema- Laplace equation: . tical treatment. Here is a small pool of samples: Problem 2. What is the terminal temperature Problem 1. What is the equilibrium position of distribution in a room, prescribed a fixed-in- an elastic membrane attached to a given wire? time wall temperature?

Basic physical principles pertaining to the Let denote the temperature theory of elastic membranes predict that the distribution in the room Ω, with prescribed membrane will adjust itself so as to minimize wall temperature . The laws of the surface tension. Thus, a first-order appro- thermodynamics postulate that the heat ximation yields the following minimization flow, , streams from the regions with high problem for the membrane position: temperature to regions with low temperature. Thus, should be proportional to ̶∇푢. Let V ⊂ Ω be fixed. Since inV no heat is been added nor subtracted, one should have:

As V was taken arbitrarily, one finally given by , i.e. = 1 if deduces that the temperature distribution, ( ), 푢 x x ∈ D and = 0 if x ∉ D. Let x ∈ Ω be the in the room must satisfy: Δυ = : 0 position of the ant and δ > 0 the incremental Problem 3. What is the probability of an ant step of the ant towards four possible leaving a room through a door before hitting directions: upwards, downwards, left or right. the wall? If 푢(x) denotes the probability of the ant Let Ω denote the room, D ⊂ ∂Ω arriving at the door D before hitting the wall denote the door and be ∂Ω \ D, starting from x ∈ Ω , one can write

where e = (1, 0) and e = (0, 1). This is or is precisely . Dividing the above 1 2 x + δe2 because, being at x, the probability of the ant expression by δ2 and reorganizing the terms, to move to either x + δe1 or x ̶ δe1 or x ̶ δe2 one reaches:

Letting the incremental step size δ go to indeed admit a unified mathematical treatment zero, one finds out that the probability u is through the study of the Laplace equation. ruled by the following PDE: Δυ = 0: Even more importantly, such a consolidation yields a bridge between different disciplines, While I must disclose that rigorous allowing meaningful exchanges of insight, justifications of the above deductions are a bit which often promote decisive advances in a more laborious, I hope to convey that problems field that would hardly be even conjectured coming from very different backgrounds do otherwise. Figure 1. Problems coming from very different realms are linked up through a unified mathematical theory.

3 Diffusion in complex materials with respect to the Hessian argument. In this theory, the mathematical manifestation of the More realistic models require more involved diffusion attributes of the operator convert into differential operators, which may have a monotonicity condition on F with respect to divergence or non-divergence structures, the natural order in the space of symmetric depending on the nature of the model. matrices. Energy considerations, such as in optimization problems or in thermodynamics, often give The current literature on the general rise to differential operators in divergence theory of second-order elliptic and parabolic form, whereas probabilistic interpretations of differential operators is vast, dense, and diffusion lead to operators in non-divergence regarded as rather challenging, especially when form. From the mathematical perspective, it comes to understanding the regularizing leading (second-order) coefficients convey the effects of diffusion. Indeed, analytic approaches tangible properties of the medium in which to regularity theory mostly involve intricate phenomena take place, which, in turn, represent estimates which are, in general, hard to grasp. their physical complexities. For instance, in the As stated by Mingione (2006), "Regularity heat conduction Problem 2, if one takes into methods are sometimes not very intuitive, account the heterogeneity of the medium then and often overburdened by a lot of technical one ends up with a divergence form equation complications, eventually covering the main, with non-constant coefficients, say div(γ(x)Δ푢) basic ideas.” = 0, where 0 < γ (x) < ∞. In some models, not only is diffusion 4 A geometric idea of diffusion anisotropic, but it can also degenerate at In contrast to the overwhelming complexity some (a priori unknown) subregions of the of the usual mathematical treatment of domain. Such considerations lead to nonlinear nonlinear elliptic operators, if one goes back differential operators of degenerate or singular to the very essence of the idea of diffusion, type. For instance, in the membrane Problem 1, namely averaging, it becomes more intuitive if instead of minimizing 2 , one considers ∫ ∣ ∇v ∣ dx that a unified regularity theory could emerge high powers, say p , with p > 2, then a ∫ ∣ ∇v ∣ dx from genuine geometric insights. Thus, (very) minimizer will satisfy the so-called p-Laplace loosely speaking, an operator should be equation, div( p--2 ) = . One should note ∣∇푢∣ ∇푢 0 considered elliptic if that, not only is the p-Laplace operator non- linear, but actually its “coefficients”,∣∇푢∣ p--2, “it prescribes a balance on how much a degenerate along the set of critical points of solution bends towards each direction.” the solution, . :={∇푢(x) = 0} Of course, this is not intended to be a Generalizations of the ant Problem 3 yield a mathematical definition per se; nonetheless, it mathematical treatment of problems in control bears very powerful insights, which, remarkably and game theory. Emerging equations are of enough, yield the development of a robust non-divergence form and often involve fully regularity theory solely based upon such a very nonlinear structures, F(x, D2푢), that is, nonlinear weak, intuitive notion of an averaging process. The roots of such a radical approach While the solution to Hilbert’s 19th pro- 45 probably go back to Ennio De Giorgi and his blem was a major event, it turned out to magnicent solution to Hilbert’s 19th problem, be a mere manifestation of a much grea- [8]. In a commemorative article, [7], Enrico ter intellectual endeavor; the foundation of Bombieri (1997) mentions a chat he had with De De Giorgi’s theory of minimal surfaces. This Giorgi on how he got the idea to solve Hilbert’s constitutes a rather successful theory deve- 19th problem. De Giorgi replied as if it was all loped by De Giorgi and his collaborators in an indirect consequence of another problem, the 1960s, where a weak notion of perime- much more difficult, that he was studying at ter yields a geometric-measure treatment of that moment, namely the isoperimetric problem the classical Plateau problem, of minimizing in several dimensions. Bombieri records that in an area given a prescribed boundary. It is a his explanation, he kept moving his hands as parallel endeavor to the famous, and equally if he was touching an invisible surface, and successful, Federer–Fleming program, laun- showing how to perform his operations and ched in [10]. transformations, cutting and pasting invisible masses from one side to the other, leveling and 5 Flatness implies regularity filling the peaks and valleys of theses surfaces. One of the supporting pillars of De Giorgi’s “I then realized that De Giorgi looked at theory of minimal surfaces is the method these functions of several variables literally as of flatness improvement, [9], which states geometric objects in space. ... To me, it was an that if a minimal surface S is “flat enough”, unusual way of doing analysis, a field that often say in B1, with respect to a direction , requires the use of rather fine estimates, that then in B , S is even flatter, probably with ½ the normal mathematician grasps more easily respect to a slightly tilted direction . Heu- through formulas than through geometry”, ristically, the proof of such a result goes as comments Bombieri in [7]. He concludes by follows: suppose, seeking a contradiction, saying “... Perhaps the only other mathematician that the result is not true. That is, there exists

I met with a geometrical intuition similar to that a sequence of minimal surfaces Sj in B1 that of De Giorgi’s was Luis Caffarelli, of whom De are 1/j flat with respect to a direction j; Giorgi was a friend and had a deep esteem.” I however the prospective flatness improve- ment is not satisfied in B . By compactness, will come back to Caffarelli in a moment. ½

an appropriate scaling of Sj converges to the

graph of a function ƒ. By the minimality of Sj, ƒ turns out to be a harmonic function, i.e., ∆ƒ = 0. Being very smooth, the limiting function ƒ satisfies the flatness hypothesis. Hence, for

jo sufficiently large, one reaches a contradic- tion to the assumption that no flatness im-

provement were possible for Sjo. The flatness improvement result explicated above is instrumental to ultimately prove that flat enough minimal surfaces are smooth. Indeed, in [9], De Giorgi shows that if a minimal Figure 2. Geometrically surface S is “flat enough”, say inB , then in B it speaking, this function 1 ½ belongs to the class of is the graph of a C1,α-function. functions entitled to be a solution of an elliptic equation, as it presents The motto then becomes flatness implies a "fair” bending balance. regularity. Little did we know that such a slogan would propagate in many different branches of mathematical analysis, the theory of free boundary problems being one of them. Figure 3. Flatness improvement yields regularity: as enters it is trapped within a strip of width , for universal numbers and . Ultimately this yields a pathway that conducts S to pass through the origin in a smooth fashion.

6 Free boundaries of the notion of differentiable operations defined on a priori merely measurable sets, and De Giorgi’s core ideas and geometric hence De Giorigi’s geometric measure theory is insights were particularly important to the a perfect fit for such an endeavor. development of the variational theory of free boundary problems. Free boundaries Following the pioneering works of H. Lewy, are mathematical manifestations of sharp G. Stampacchia, J.L. Lions, D. Kinderlehrer, changes in the parameters that describe the among other eminent mathematicians, Luis problem. Typically, different physical laws are Caffarelli was the leading figure in the pursuit to be prescribed in distinct, a priori unknown of a systematic geometric approach to the subregions of a domain. This is the case, for investigation of free boundary problems. instance, for problems involving interfaces Caffarelli’s 1977 article, [3], on free boundary between materials, different states of matter, etc. regularity for the obstacle problem is a Free boundaries also arise in physical reactions landmark in the theory. The problem asks for where interfaces retain some portion of the the equilibrium position of an elastic membrane, system’s energy, viz., latent heat, membranes, u, restricted to lay above a given obstacle ψ(x). dead cores, flux balances, and so forth. Thus, That is, the obstacle problem is the membrane mathematical models of free boundary Problem 1 with the extra condition of u laying problems typically require weak formulations above the obstacle:

While the existence and uniqueness of indeed true and is the content of an important a solution to the obstacle problem can be Theorem first proven by J. Frehse (1972). established exactly as in Problem 1, its regularity The next, and rather more involved issue is theory is rather different. Indeed, solutions to to understand the smoothness of the interface Problem 1 are differentiable infinitely many times, whereas the optimal regularity for the between the contact set membrane restricted to lay above an obstacle and the non-contact set , the drops to C1-1; that is, the best one can hope for so-called free boundary of the problem, Ґ. Luis is boundedness of second derivatives. This is Caffarelli proved in [3] that if the contact set

Figure 4. The membrane problem in the presence of an obstacle. is thick enough in a neighborhood of a free regularity theory for a very general class of two- 47 1 boundary point x0, then Ґ is a C surface around phase free boundary problems. Here, however, x0. In slightly more precise terms, Caffarelli Lipschitz estimates for the free boundary showed the existence of a critical density ϱ(r) present themselves as an intermediary step. such that if x0 ∈ Ґ satisfies for some0 < r ≪ 1, Caffarelli’s free boundary regularity slogan r,α Width(C ∩ Cr(x0)) > ϱ(r), then Ґ is a C surface then becomes: flatness implies Lipschitz, and ∞ (and thus C , by a result from [12]) around x0. Lipchitz implies differentiability. Four years later, in 1981, Caffarelli, partnering with H. Alt, published what would 7 Back to diffusion become a magnum opus of variational free Recently, a radical new geometric approach boundary theory. In [1], Alt and Caffarelli study to the analysis of diffusive PDEs has been regularity properties of non-negative local launched, in which degenerate points of minimizers to the discontinuous functional, ellipticity are seen as part of what has been min: Here, termed “non-physical free boundaries”. stands for the characteristic function Heuristically speaking, in the realm of of the set . Since this is a discontinuous function, one should expect the Laplacian of a mathematics it is often the case that the minimizer to behave as a Dirac mass along the complexity of a given problem is encoded set of discontinuity, namely within some special entities pertaining to the free boundary of the problem. This is it: singularities, bifurcations, degeneracies, indeed the case; Alt and Caffarelli show that blow-ups, discontinuity, sharp changes, etc. a local minimizer, u, behaves linearly along Significant advances on the problem depend the free boundary, in particular it is Lipschitz upon a critical understanding of such distinct continuous (the optimal regularity of the elements and how they affect the order of problem). They also show that, in some very the model. When it comes to the analysis of weak sense, the normal derivative of u along diffusive PDEs, geometric insights from free the free boundary is constant. The most boundary theory provide a rather powerful delicate part of the program is to show the toolbox to investigate those special points; the differentiability of the free boundary, an so-termed non-physical free boundaries. -negligible set. This is accomplished by a rather By a way of example, the 1930s Schauder elaborate implementation of methods along estimates bear the premise that the smoothness the lines of flatness implies regularity, involving of the gradient (or the Hessian, depending on non-homogeneous blow-ups. whether the problem is in divergence or non- The investigation of sign changing divergence form) of a solution to a second- minimizers of discontinuous functionals of order linear elliptic equation could never exceed Alt–Caffarelli type, as above, is motived by the continuity of the medium. That is, if the problems in the theory of jet flows, phase coefficient, , of a divergent form operator transmission, among others. From the div is α-Hölder continuous, for mathematical perspective though, the analysis some 0 < α < 1, then the gradient of a solution to of sign changing minimizers is rather more the homogeneous equation div involved than that of its one-phase counterpart. is also α-Hölder continuous, for the same In particular, establishing Lipschitz estimates exponent α. This is a celebrated result, which for such minimizers required a powerful new is far from being elementary, or even intuitive. tool, namely a monotonicity formula, in the Even less intuitive is the fact that the continuity spirit of geometric measure theory. This is the of the gradient can be superior to the continuity content of the celebrated work of Alt, Caffarelli of the coefficient, but only along its critical and Friedman (1984), [2]. set , [14, 15].

Within the theory of free boundary regularity, Such an improved regularity estimate the motto flatness implies regularity attains its becomes even more appealing in the context of apogee in Caffarelli’s trilogy [4, 6, 5], where degenerate elliptic equations, as in the theory he develops a rather complete existence and of the p-Laplacian. This is because the critical Figure 5. Improved regularity along critical points: at a generic point, the gradient (resp. hessian) of solutions may develop sharp cusps; at a critical point, however, it can only develop much smoother corner-like singularities.

set is precisely the region in geometric, and were largely influenced by the which the diffusion attributes of the operator general free boundary framework mentioned collapse. Strikingly, even if the medium does above. Several applications and enhancements not have power oscillation decay, the gradient of these methods have been successfully set of a solution does, but only around points of . forth in the past few years, leading to a plethora The formal statements of such results are of other unanticipated results. This is currently a bit too technical to be stated here; however a rather active line of investigation and it is likely it is noteworthy that the core ideas behind that the analogy described herein will bear fruit the proofs of these theorems are genuinely in other branches of mathematical analysis

References [1] Alt, H. and Caffarelli, L. Existence and regularity for a [9] E. De Giorgi, Frontiere orientate di misura minima. minimum problem with free boundary. (1981) J. Reine (Italian) Seminario di Matematica della Scuola Normale Angew. Math. 325, 105–144. Superiore di Pisa, 1960-61. Editrice Tecnico Scientica, [2] Alt, H.; Caffarelli, L.; Friedman, A. Variational problems Pisa 1961 57 pp. with two phases and their free boundaries. Trans. Amer. [10] Federer, Herbert; Fleming, Wendell H. Normal and Math. Soc. 282 (1984), no. 2, 431–461. integral currents. Ann. of Math. (2) 72, 1960, 458–520. [3] Caffarelli, L. The regularity of free boundaries in higher dimensions. Acta Math. 139 (1977), no. 3-4, 155–184. [11] Frehse, Jens On the regularity of the solution of a [4] Caffarelli, L. A Harnack inequality approach to the second order variational inequality. Boll. Un. Mat. Ital. regularity of free boundaries. I. Lipschitz free boundaries (4) 6 (1972), 312–315. are C1,α; . Rev. Mat. Iberoamericana 3 (1987), no. 2, [12] Kinderlehrer, D.; Nirenberg, L. Regularity in free 139–162. boundary problems. Ann. Scuola Norm. Sup. Pisa Cl. [5] Caffarelli, L. A Harnack inequality approach to the Sci. (4) 4 (1977), no. 2, 373–391. regularity of free boundaries. III. Existence theory, [13] Mingione, Giuseppe Regularity of minima: an invitation compactness, and dependence on X. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15 (1988), no. 4, 583–602 (1989). to the dark side of the calculus of variations. Appl. Math. – [6] Caffarelli, L. A Harnack inequality approach to the 51 (2006), no.4, 355 426. regularity of free boundaries. II. Flat free boundaries are [14] Teixeira, E. Regularity for quasilinear equations on Lipschitz. Comm. Pure Appl. Math. 42 (1989), no. 1, 55-78. degenerate singular sets, Math. Ann. 358 (2014), 241– [7] E. Bombieri Ennio De Giorgi, Rend. Suppl. Acc. Lincei - 256. Serie IX - vol. VIII, 105–114 (1997). [15] Teixeira, E. Hessian continuity at degenerate points in [8] E. De Giorgi, Sulla differenziabilita e l’analiticità delle nonvariational elliptic problems, Int. Math. Res. Not. estremali degli integrali multipli regolari, Mem. Acc. Sci. IMRN 2015, no. 16, 6893–6906. Torino Cl. Sci. Fis. Mat. Nat. (3), 3 (1957), 25–43. Olympiads Arouse Enthusiasm for Mathematics

Annual competitions inspire learning and On with ICM 2018! the resurgence of new talent whilst having an impact on teaching and research in Brazil

Just imagine the whole po- tended for public school stu- jobs market, thus showing the

pulation of Chile or Holland dents from the sixth grade of OBMEP’s great impact on the

simultaneously focused on a elementary school to their se- daily lives of students, teachers

math test. The scene is farfet- nior year in high school and, and teaching centers.

ched, but gives an idea of the since 2017, has also included And what about the popu-

dimension of something very students from private institu- lation of an entire city? Cocal

tangible, the Brazilian Mathe- tions. dos Alves, a town of 5,572

matical Olympiad of Public In addition to the math inhabitants in the northeast sta-

Schools (OBMEP). Created by tests, the Olympiads encom- te of Piauí, is on the tail end of

the IMPA and the Brazilian So- pass a number of training ini- the Human Development Index

ciety of Mathematics in 2005 tiatives: the Scientific Initiation (HDI) ranking – one of the worst and held annually in order to program (PIC), the Scientific 30 in Brazil – but there have encourage study in the field Initiation Program and Masters been so many prizes awarded and to find new talent, the (PICME), the Olympic Intensive by the OBMEP to its inhabitants OBMEP had the participation of Training Centers (POTI) and the that the city became known as 18.2 million students last year. OBMEP in Schools. the “national capital of mathe- Superlative in the total Participants who are awar- matics”. number of registrants, with a ded gold, silver or bronze me- range that includes almost all dals are ensured entry into the Motivational stories (99.6%) 5,570 Brazilian mu- scientific initiation program and Sandoel de Brito Vieira, 24 nicipalities, the OBMEP stands an annual scholarship. It is of- years old, knows Cocal dos Al- out as the largest school com- ten the beginning of a success- ves very well. Born and raised petition in the world. It is in- ful career in academia or in the there, the OBMEP’s five-time As well as reports of stu- dent life, there is no shortage of exciting stories among tho- se who can teach – the OBMEP rewards students, teachers, schools and municipal secre- tariats of education. Reports on relevant changes in the workplace and expertise from involvement with the compe- Courtesy of IMPA Courtesy tition show that the Olympiad The OBMEP's 500 gold medalists gather every year in Rio to receive their prize for being the best among 18 million students from all over Brazil is also a professional training tool, thereby exercising an im- medalist achieved his dream talent countrywide, including portant role in a country where through a placement at the PIC students and teachers. criticism over the performance and by being the first in his fa- From the inexhaustible of math teachers relates to the mily to enter university. Further source of inspirational stories deficiencies in their training. “The OBMEP was my most to the completion of his Bache- through the Olympiads, peo- important program for conti- lor’s and Master’s degrees at the ple like Vieira aim for careers nuing education as a teacher”, Federal University of Piauí, Vieira in research. Others dream of admitted Antonio Cardoso do has flown even higher. At pre- attending university, as in the Amaral, who has taught Vieira sent, Vieira is a PhD student at case of Lucy Degli Esposti, 31, and dozens of other medalists the IMPA. He is one of several from small Bom Jesus do Ita- from the school Augustinho examples of the OBMEP’s initia- poana (RJ) who, after leaving Brandão, in Cocal dos Alves. tive to find new talent from any school and entering the novitia- Improvements in social stratum or region of the te, reversed direction and won teaching country and to promote social gold in her first OBMEP partici- In addition to exciting life inclusion through the dissemi- pation. Lucy has taken off and stories, independent studies that nation of knowledge. now aspires to a degree in Ci- analyze the positive effects of the The OBMEP math exam vil Engineering. Some people Olympiads help us understand emphasizes logical reasoning choose to follow a successful why it has become a prominent instead of acquired knowledge corporate career as in the case national public policy and has and formulas, thereby facilita- of Alessandra Yoko, 25, control contributed to the improvements ting the discovery of unknown and automation engineer. in the quality of education. In a 2014 assessment, then 51 teacher at the Federal Univer- sity of Minas Gerais and Presi- dent of the National Institute of Educational Studies and Re- search (INEP), Francisco Soares, proved that schools with active involvement in the Olympiad present an average of 26 points in improvement in the national Courtesy of IMPA Courtesy educational test (“Prova Brasil”) Claudio Landim and Marcelo Viana at the OBMEP ceremony with the Pernambuco – which with regard to school State delegation of gold winners attendance represents an addi- to select and train students to for the first time, the country tional 1.5 years. More recently, represent Brazil in international hosted the oldest and most pres- Diana Moreira, of the University competitions. tigious scientific Olympiad for of California (USA), concluded In 2017, for the first time high school students, the Inter- in her doctorate from Harvard – during the Biennium of Ma- University that the benefits of national Mathematical Olympiad thematics – the two Olympiads being awarded or receiving an (IMO), created in 1959. On its were integrated into a single th honorable mention from the 58 year, the competition orga- school competition, counting OBMEP are not limited to the nized by the IMPA was attended on the participation of stu- winners themselves but also ex- by students from 111 countries dents of the sixth grade up to tend to their classmates. who gathered in Rio de Janeiro their Senior year of high school. Before the creation of the to solve challenging mathemati- OBMEP, the IMPA already held, The two phases of the OBMEP cal problems. in conjunction with the SBM, an qualified students for the OBM, Among the 615 competi- annual competition for students which brought together com- tors, there were six Brazilians, from public and private schools: petitors – including 1,223 stu- national and international me- the Brazilian Mathematical Olym- dents in a category apart. dalists selected by IMPA and piad (OBM). Since then, the com- At the IMO, students SBM: André Yuji Hisatsuga and petition has undergone changes from five continents Pedro Henrique Sacramento in format whilst always maintai- Relying on its experience de Oliveira, São Paulo; Bruno ning its objectives to stimulate from two large scale annual na- Brazil Meinhart and George Lu- the study of mathematics, to tional competitions, Brazil prepa- cas Alencar, Ceará; João César identify talented candidates and red for a new challenge in 2017: Campos Vargas, Minas Gerais; Brazil was placed 37th in the ranking – falling from the pre- vious year, when it placed 15th. According to the international jury, the math test applied in Rio was one of the most diffi- cult in IMO’s history. As pointed out by Brazilian Artur Avila, an IMPA resear- cher, Fields medal winner of Courtesy of IMPA Courtesy 2014 and gold medalist in the The Fields medalist Artur Avila with two OBMEP medalists and the OBMEP coordinator Severino Cirino from the Pernambuco State 1995 IMO, the main goal of the Olympiads should not just Chandra and Davi Cavalcanti of a special prize to reward the be to attain a place among the Sena of Pernambuco. five young women who contri- first but to bring educational At a hotel in Barra da Tiju- buted the most to the perfor- benefits “with consequences in ca where the event was held mance of their countries. The research”. “It is important to in Rio, there was excitement, suggestion was approved. From show that mathematics can be enchantment and concentra- 2017 onward, this initiative will fun, to create an environment tion surrounding mathematics. be replicated in subsequent of discipline and focus, to sour- The IMO's Babel went beyond years. ce talented people by careful the mix of languages, gathered The IMPA promoted a round observation in the best institu- different dress modes, pets and table discussion on “Women in tions in the country.” distinct ages. Despite the diffe- Science”. Among the panelists The IMO has been broade- rences, the Rio gathering was a were Carolina Araujo, researcher ning the debate on mathemati- place of respect and harmony. at the IMPA; OBMEP medalist Between the excited groups Tabata Amaral, founder of the cs in the country and has been hailing from five continents, fe- movement “Mapa Educação”; helpful in consolidating Brazil’s male participation was noticea- and Larissa Lima, IMO medalist reputation on the world stage. bly lower, about 10% of con- in 2002. Also, the IMO shows the coun- testants, reflecting a historical Brazil started to participa- try’s ability to hold large scale, picture of the competitions. To te in the IMO in 1979 and has complex international events bring attention to the importan- won 25 medals and 32 honora- which bring together hundreds ce of diversity and to encourage ble mentions. In the 2017 IMO, of teenagers from very different women to participate in the IMO South Korea won first place fol- cultures. the IMPA proposed the creation lowed by China and Vietnam, On with ICM 2018! Manfredo P. do Carmo (1908–2018)

The Development of Differential Geometry in Brazil

I will divide the development of Differential D be a compact domain of a minimal surface, Geometry in Brazil into three periods: and g : D → S2 be its Gauss map. Assume that g is one-to-one. Then if g(D) is contained in a (1) Pre-history – From 1800 to 1957 hemisphere, D is stable; and if g(D) contains a (2) Beginning of History – 1957 to 1970 hemisphere, D is unstable. Chern asked how (3) Consolidation – 1970 to the present the hemisphere entered the result. Was it the day and will quote the periods by their numbers. area? Yes, it is the area, the estimate is sharp, and g is not necessarily one-to-one. In (1), I only found 2 papers, one written by Otto de Alencar and the other by Lelio Gama [2]. The question of [2] has been around (details on these works can be found in [3], pp. for a long time (see [3], p. 142). At the time we 370, 371) . They were published in local journals. published our result, D. Fisher-Colbrie and Rick Schoen published a more general result. In 1957, the First Brazilian Colloquium of Mathematics was organized. This was to be [4]. This was an example of a long-awaited repeated each odd year and had a strong surface. The equation of the surface appeared influence on all areas of Mathematics that were in Costa’s thesis but a small detail was left being cultivated in Brazil. unproved. Bill Meeks and David Hoffman proved this detail and produced a large number In (2), at least ten papers on Differential of similar examples. This created a discussion Geometry were published in journals of of priority that was soon solved and the surface international circulation (for details see [3], pp. became known as “the Costa surface”. 372–374]). [5]. The following question is attributed to In 1970 the IMPA started a Ph.D. program that E. Callabi: “Is there a complete minimal surface included Differential Geometry as a Thesis topic that lies in a semi-space?” The authors of this and I was put in charge of the Seminar on the paper proved more, namely, they proved that subject. It did not take long for most Departments there exists a complete minimal surface in the of Mathematics in the country to have people space between two parallel planes. with Doctor’s degree in Differential Geometry. [6]. This is an extraordinary work. The new ideas Within the Period of Consolidation, Brazilian needed for the proof of the Willmore conjecture mathematicians solved some problems in are varied and their number is surprisingly high. Differential Geometry that were challenging The authors point out that they are using the our colleagues throughout the world. See [1], minimax theory of minimal surface to prove the [2], [4], [5], [6], [7]. full statement of the Willmore conjecture. The [1]. This is the answer to a question by Chern. paper by Marques and Neves was written when A paper by Schwarz has the following result. Let Marques was a Professor at the IMPA. [7]. This was a breakthrough on the What we have now is a stable community problem of finding the minimal number of in Brazil contributing to Differential Geometry. points the Gauss map could omit. Previously We would like to mention that an American there had been a result by Osserman that the mathematician, H. Rosenberg, who had a Gauss map could not omit a set of positive career in France (quite a feat in itself), was so capacity. There are various examples that impressed by the Geometry Seminar at the omit four points and none that omit five IMPA that he has decided to join us. points. It turned out that the final answer was that it cannot omit five points.

REFERENCES 1. Barbosa J.L. and do Carmo M.P. On the size of a stable 5. Jorge L.P.M. and Xavier F. A complete minimal sur- minimal surface, American J. of Mathematics 98(1976) face in R3 between two planes. Annals of Mathematics 515–528. 112(1980) no. 1. 203–206. 2. do Carmo M.P. and Peng C.K. Stable minimal surfaces 6. Marques F.C. and Neves A. Min-max theory and the in R3 are planes, Bull. of A.M.S. (New Series) 111(1980) Wilmore conjecture. Ann. of Math. 179(2014), no. 2, 477–490. 683–782. 3. do Carmo Manfredo Perdigão. Selected Papers, 497 7. Xavier F. The Gauss map of a complete minimal surface pp. Research in Differential Geometry in Brazil 367– cannot omit 7 points. 394, Springer Verlag, 2012. 8. Annals of Mathematics 113(1981), no. 1, 211–214. 4. Costa C.J. Example of a complete minimal immersion 9. Xavier F. The Gauss map of a complete minimal surface in R3 of genus one and three embedded ends, Bol. Soc. cannot omit 7 points. Annals of Mathematics 113(1981), Brasil. Mat. 15(1984) no. 1-2, 47–54. no. 1, 211–214.

Manfredo Perdigão do Carmo: geometer admired in a variety of languages Emeritus IMPA researcher, founder of his own field of research in Brazil, died in April, 2018 at the age of 89

As featured on the cover of than others, but everybody when he came to intern for the the seminal work “Differencial can learn some mathematics. institution from the beginning Geometry of Curves and These people will have learned of 1959 until mid-1960. Once Surfaces”, Manfredo Perdigão to think independently, which a member of the IMPA, he fol- do Carmo, who died last will be useful for the rest of lowed the foundational scenes April at the age of 89, left an their lives.” in the history of mathematical admirable legacy in a variety research in Brazil, as led by Whatever Carmo learned, of languages. Mauricio Peixoto, among other he passed on to others. From pioneers, and he also became Beyond the foundational the unforgettable lessons by a protagonist. books on his specific field of professor Benedito de Mo- study, the emeritus IMPA re- rais in Maceió, an experience In an interview granted to searcher, born in Maceió (AL) he shared with his child- his fellow countryman and in 1928, left his mark as a ded- hood friend Elon Lages Lima alumnus Hilário Alencar, for- icated and well-humored mas- (1929–2017), also an emeritus mer president of the Brazilian ter who opted for mathemat- researcher at IMPA – to the Mathematical Society and pro- ics instead of engineering. dialogues held at the First fessor of the Federal Universi- Brazilian Colloquium in Math- ty of Alagoas (UFAL), Carmo Initially, Carmo dedicated spoke about that period of ematics (CBM) in 1957 and in his time “to study foreign time: “In the beginning of 1960, subsequent years at the IMPA. works in solitude, very slowly Steve Smale, on an invitation and filled with many unsolvable Carmo arrived at the IMPA by Peixoto, came to visit the problems.” Years later, in a through a referral by Lages IMPA – that is when he found speech for his students, he Lima when they both met as his proof of the Poincaré Con- observed that: “some people adults. Carmo was already jecture for dimensions greater can learn more mathematics married and the father of a son or equal to five.” Carmo added: “Peixoto and of transforming Brazil into a Brazilian Academy of Science Lima frequently met to discuss place of excellence in mathe- (ABC) as an ICM conference questions in which they matical research. participant in 1978 and as were interested. When I was president of the Brazilian So- “Manfredo sets an example present on these occasions, ciety of Mathematics (SBM) for all of us. He gave up a very I just remained quiet and from 1971 to 1973. He retired promising career abroad to bedazzled, witnessing the in 1997 but kept himself active. return to Brazil and founded creation of a new mathematics. the Brazilian school of Carmo was a source of in- This was the crucial point in differential geometry, one of spiration for a new generation my experience when I decided the most active and successful of mathematicians like Fer- that, if possible, I would like to in Brazilian mathematics. His nando Codá who arrived at become a mathematician. I books inspired generations of the IMPA in 1999 and also got never regretted that decision.” students”, as pointed out in involved in differential geom- Carmo’s choice brought Carmo's obituary by Marcelo etry, remaining there until he benefits to a legion of people Viana, the IMPA's managing took the position of researcher passionate about mathematics. director at Revista Piauí. at Princeton University, United States. He took responsibility for the Upon his repatriation, creation and consolidation of Manfred worked as a professor In 2009, Carmo generously field research in differential at the then recently created told Codá about achieving his geometry in Brazil. He University of Brasilia (UnB), dream of developing research became a reference in an area where he participated in a excellence in Brazil and associated with the Chinese collective resignation request highlighted the importance mathematician Shiing-Shen due to a disagreement with of a future generation of Chern (1911–2004), who was his the military regime after the mathematicians: “the legacy doctoral mentor at Berkeley, 1964 coup. Two years later, he that I have to pass on to the University of California. In became an IMPA researcher. young Brazilian geometers, 1963, he presented his thesis people like you, has to Through the years, Man- “The Cohomoloy Ring of continue.” fredo combined his research Certain Kahlerian Manifolds”, activities at the IMPA with the One of Carmo’s books is published in the journal writing of books and experi- used by Codá in his classes at “Annals of Mathematics”. ences as a visiting professor Princeton. Such is the legacy Having established himself overseas, and as a manager. of a geometer from Alagoas abroad, Carmo could have He was granted the Guggen- (a Northeastern state of Brazil, chosen not to return to his heim Scholarship and became whose capital is Maceió) that homeland. However, he de- visiting professor at Berkeley, should be brought forward to cided to bet on the country’s University of California. He future generations in as many future and fell for the dream engaged as a member of the languages as possible. José Alberto Cuminato Associate Professor, Institute of Mathematics and Computer Sciences, University of São Paulo, Brazil E-mail: [email protected]

José Mario Martinez Professor, State University of Campinas, Brazil E-mail: [email protected]

The Early Years of Applied Mathematics in Brazil: A Brief Account on How It Developed

In this report we give a brief historical ac- As a result of the university reform of 1968, count on how Applied Mathematics has develo- the universities in Brazil underwent a profou- ped in Brazil in the last 60 years. The appearan- nd change. Prior to the reform, mathematics ce of the first applied math research groups in and statistics teaching were spread across the Rio and São Paulo states and their contribution various schools and colleges of the universi- to the birth of institutions such as the LNCC, ties. With the reform, lecturers in mathema- SBMAC, and many of the Applied Mathematics tics and statistics from distinct schools and Departments existing today, is described. colleges were transferred to the newly crea- ted Institutes/Departments of Mathematics. 1 Introduction The reform promoted the transition from a university structured as a “confederation” of The first research groups in numerical ma- schools to one structured in departments. thematics appeared within engineering schools As the groups in Applied Mathematics grew around 1950, to meet the need of teaching Nu- stronger within the math departments, ideas merical Calculus to engineers. This is sometimes for new Applied Math departments and labora- also true for mathematics as a whole, as some of tories dedicated to Applied Math and modeling the Math Institutes/Departments existing today began to circulate. From this movement, many started as part of Engineering schools/depart- of the most important institutions dedicated ments. The Applied Math research groups were, to Applied Mathematics, existing today, were in the beginning, part of general math depart- conceived. ments and only later on, with the increase in the number of people dedicated to numerical ma- The historical notes in Sections 2 and 3 are thematics, did departments of Applied/Compu- concentrated on the States of São Paulo and tational Mathematics start to appear. Undergra- Rio de Janeiro because in those states the mo- duate courses in Mathematics started earlier on, dern history of Applied Mathematics of Brazil around 1934, with the founding of the Faculty of begins. However, it must be warned that in the Philosophy Sciences and Letters of the Univer- present day there exist Applied Mathematics sity of São Paulo (FFCL-USP), which greatly be- activities in all the states of Brazil, including nefited from the immigration of scientists, mainly Southern states (Rio Grande do Sul, Santa Ca- italians, who fled from the fascist regimes. Also tarina, Parana), North-East states, Center sta- in 1943, a Bachelor’s degree in Mathematics was tes (notably, Goias), and also in the Northern started at the University of Rio Grande do Sul. states. 2 The Evolution of Applied lativity and motivates the interest of both Phy- 57 Mathematics in São Paulo State sics and Mathematics students who complete their graduate studies in this department. 2.1 Applied Mathematics in Campinas In 1978 the area of Biomathematics was restricted to a single member of staff who, a The Department of Applied Mathematics few years later, emigrated to the United States. of the University of Campinas (UNICAMP) However, the area was reconstituted by adding was founded in 1976. The master’s program researchers of Numerical Analysis and by hiring in Applied Mathematics started in 1977, and new researchers. Currently, the group is active in epidemiological models and the application the doctorate in 1983. With few exceptions, of fuzzy theory to Medicine. all the research groups established in 1978 still exist today. The Numerical Analysis group has worked on the numerical solution of partial differential The research in Optimization and Opera- equations and applications to the oil industry tions Research involves theoretical aspects of since 1978. More recently, starting from a group optimization, implementation of algorithms, of Inverse Problems, a new group was created and applications. A number of scientific pa- that works in modern Applied Mathematics topics such as Neural Networks, Image Proces- pers have been written in these subjects and sing and Machine Learning. Moreover, Discrete free software with many downloads and appli- Inverse problems are also addressed within this cations to Physics, Chemistry and Engineering context with applications to the determination have been developed. In 2007, a Ph.D. student of protein structures. on Optimization won the CAPES Grand Prize in Exact Sciences. This was the first, and, for 2.2 Applied Mathematics in São Carlos now, the only time, that this award has been and beyond given to a mathematician. From Campinas traveling northwest into the In the Computational Geophysics Group, State of São Paulo we find several institutions hosting Applied Mathematics groups: São Pau- Applied Mathematics tools are effectively lo State University (UNESP) at Rio Claro and applied to the resolution of oil prospection Rio Preto, São Paulo University (USP) at São problems, in most cases with financial support Carlos, and the Federal University of São Carlos from Petrobras. (UFSCar). The development of Applied Mathe- The leaders of this group have won interna- matics at these institutions was initiated by Prof. tional awards in recognition of their scientific Odelar Leite Linhares, under whose guidance and technological activity. many students began their Applied Mathematics The area of Mathematical Physics is prolific careers. The Institute of Mathematical Sciences in obtaining results in classical and quantum re- in São Carlos was created in 1971. In 1969, Ode-

Some books for the basic course of the First School on Applied Math. Source: LNCC Library In 1975, the first Ph.D. thesis in 3 Applied Mathematics in Rio de Numerical Analysis in Rio de Janeiro Janeiro was defended at the IMPA. By the end of the 1960s, some activities Following these developments, in basic Numerical Analysis started at Ponti- new groups working in fical University of Rio de Janeiro (at the time Mathematics applied to PDEs that commonly known by the acronym PUC-RJ involved some numerical content, and currently by PUC-Rio). As a result of the- appeared at the Mathematics se activities, by the beginning of the 1970s, Institute of UFRJ the first Masters dissertations on Numerical Analysis were defended in the Department of Informatics.

Later on in the mid-1970s a more structured lar had moved away from his duties at USP to organization of Applied Mathematics was esta- create the first Department of Computer Scien- blished, with the return to Brazil of newly gradua- ce and the first Brazilian undergraduate course ted doctors from abroad. A number of Applied in Computer Science in Campinas. In 1972 he Math research groups sprang up from this new returned to São Carlos and repeated his succes- reality: a team working in Numerical Analysis of sful experience at the Institute of Mathematical PDEs emerged at the Brazilian Center for Physi- Sciences. The successful examples of Campinas cs (CBPF) (an LNCC embryo group) and also at and São Carlos attracted the attention of lectu- the Department of Informatics of PUC-Rio; also, rers from other Brazilian universities in various Mathematical Programming and Optimization regions of Brazil, who sought to do their Ph.D. research groups arose at the Federal University in this area. After his retirement from USP, Odelar of Rio de Janeiro (UFRJ). In 1975, the first Ph.D. moved to the Department of Computer Science thesis in Numerical Analysis in Rio de Janeiro and Statistics at Rio Preto, helping to start and was defended at the IMPA by a lecturer from consolidate an undergraduate course in Compu- the Federal University of Ceará, under the gui- ter Science and to start a Masters course in Com- dance of the PUC team. Following these deve- putational Mathematics. The achievements of lopments, new groups working in Mathematics Prof. Odelar are now visible in the subject areas applied to PDEs that involved some numerical of Computation and Computational Mathematics content appeared at the Mathematics Institute at São Carlos, Rio Preto, and Campinas. of UFRJ. In 1978–1980 the LCC – the Laboratory Odelar's legacy fructified in a good number for Scientific Computing) (which later became of achievements in the area of influence of São the future LNCC, the aggregated N standing for Carlos. In the last decades of the past century se- National), was founded, based at the campus of veral researchers completed their Ph.D. in highly Praia Vermelha, UFRJ. In the early eighties, un- reputable centers of Applied and Numerical Ma- der the leadership of Paulo Jorge Paes Leme, an thematics around the world. By keeping their informal interdisciplinary graduate program on links with the institutions where they studied, Applied Mathematics was started at the Techni- they were able to create strong groups in Nume- cal-Scientific Center of PUC-Rio, focusing spe- rical Analysis, Fluid Mechanics and Operations cifically on Scientific Computing and involving Research. Numerical Analysis studies soon evol- a dozen faculty-researchers from several de- ved into the development of methods for solving partments of this university. This program suf- PDEs with applications to Fluid Mechanics. fered several obstacles, mostly due to the con- tradictory interests of the many departments Effective computer packages that display involved, and ended up not surviving. However, the solutions of turbulent Fluid Dynamics pro- Paes Leme had the opportunity to structure a blems were elaborated and disseminated. On similar program in the newly created Research the other hand, Operations Research flouri- Institute of the State of Rio de Janeiro based in shed with a concentration on Scheduling, Cut- Nova Friburgo. ting and Packing problems. A great number of Ph.Ds in this areas are now professors in several In parallel, a graduate course started at the universities of Brazil. Department of Informatics of PUC-Rio in Nu- merical Analysis of PDEs. The first doctoral the- Although the relationship 59 sis in this area was defended in 1985 by a gra- between mathematics and duate of the University of Paris 6. Without ne- computing is old, computer glecting formal mathematical aspects, the ac- graphics, with the availability tivities of both groups were strongly industry- oriented. Paes Leme directed most of his from the mid-1980s of more group’s activities to numerical simulations of powerful and easy-to-access flow in porous media. The focus was in particular, displays, introduced new and on oil reservoir simulations, in the framework important elements of research: of a research project sponsored by Petro- high-definition imaging and bras. The PUC group, in turn, exploited new interactivity finite-element techniques applied to the cal- culation of large deformations of hyperelastic equations and topological data structure. Star- bodies, and to the viscoelastic flow of polymer ting from the 1990s, the group began to apply melts. R & D projects in these fields were spon- the results of its research on topological data sored by multinationals needing to model pro- structure in projects with Petrobras, working in cesses involving these types of materials, such collaboration with its Petroleum Research Cen- as the Pirelli Power-Cable branch of Brazil. ter (CENPES).

The second half of the 1980s saw the LNCC’s Late in the 1980s, more consistent activi- consolidation and recognition as a leading scien- ties in Applied Mathematics/Scientific Com- tific computing center in Brazil. This institution puting started at IMPA, in Numerical Analysis, was transferred from Rio de Janeiro to Petropo- Optimization, and Mathematical Economics. lis in 1998. At the same time, in the Department In the Department of Informatics of PUC-Rio, of Mechanical Engineering at PUC-Rio a group of seven Ph.D. theses were defended from 1988, more than ten Ph.D.s in Mechanical Engineering, on Numerical Analysis of PDEs and/or Scien- conducting research on modeling in the Mecha- tific Computing, based on the Finite Element nical Sciences with a strong emphasis on Ma- Method. The last of these theses was comple- thematical Analysis, carried out sandwich-type ted in December 1995 by a CNPq scholar from internships in laboratories in the Paris region Nanjing, China, who is currently a researcher at through bilateral exchange projects, thus helping the LNCC. Other students of this program went to strengthen the Applied Mathematics research to work across the Guanabara Bay, in Niteroi at groups in Rio de Janeiro. At the same time in the the Fluminense Federal University (UFF) as fa- Department of Mathematics of PUC-Rio, a new culty and, in partnership with other doctors in research group named Matmidia started in 1985 the area trained at universities in Rio de Janeiro, which contributed significantly to the streng- started research groups and training activities thening of applied mathematics. Matmidia had in the Institutes of Computing and Mathemati- computer graphics as its main area of focus and cs at UFF. The graduate program in Computing concentrated on conducting research into the at UFF also engaged in Optimization and rela- relevant connection between math and compu- ted areas. The graduate program in Computing ting, establishing an interface between applied at UFF is also engaged in intense research acti- mathematics and industry and contributing to vity in Optimization and related areas. the teaching of basic differential calculus using computational resources and staff training. Al- 4 The Role of SBMAC though the relationship between mathematics On the occasion of the 25th anniversary of and computing is old, computer graphics, with the Brazilian Society of Applied and Computa- the availability from the mid 1980s of more po- tional Mathematics (SBMAC), Prof. Odelar Leite werful and easy-to-access displays, introduced Linhares wrote: new and important elements of research: high- definition imaging and interactivity. The Matmi- “SBMAC was born in Belo Horizonte, Minas dia Group sought to use these characteristics Gerais, on 1/11/1978, at the end of the First Na- to explore, at the beginning of its activities, two tional Symposium on Numerical Analysis, which areas of research: implicit ordinary differential was then held at UFMG (Federal University of Minas Gerais), under the auspices of its Depart- us to understand the importance of the LNCC, ment of Computer Science. (...) In this way, a USP, UFSCar, UNICAMP, UFRJ and UFMG to dream that was long cherished by the commu- applied mathematics at the time is the fact that nity of lecturers and researchers in Applied and most of the original board members of SBMAC Computational Mathematics (including Nume- were affiliated to these institutions. The CN- rical Analysis) and in several other related sec- MAC conference has been held continuously tors, all over Brazil, had been realized. The need every year (apart from a recent period of four to create such a Society had long been felt and years when it became biannual and the retur- widely debated in many of the scientific mee- ned to its annual frequency). It is important to tings attended by the community. The Applied emphasize that, after the foundation and struc- Math community had then become aware that turing of SBMAC, it started sponsoring acade- within other societies there was no place for mic meetings in various parts of Brazil, as well the leading role that they anxiously wished to as directing efforts for the publication of the play in formulating appropriate policies for tea- Bulletin of SBMAC – an informative publication, ching, research, and dissemination in Applied with space for the dissemination of activities and Computational Mathematics and, conse- in the area of Applied and Computational Ma- quently, for the scientific and technological de- thematics and for discussions promoted by its velopment of the country.” members, in the context of science and tech- nology and education in Brazil. It should be noted that some of the issues The Bulletin of SBMAC, as announced in discussed in Belo Horizonte had already been the editorial article of its first edition, was considered in the First School of Applied Ma- created with the purpose of dissemina- thematics, held in São Paulo in July of 1977. ting general information about SBMAC, and The first Board of Directors of SBMAC was giving information about meetings, symposia, elected the following year, in 1979, at the II Nu- congresses – both past and planned. Soon its merical Calculus Symposium held in São Carlos, scope was expanded and by 1980 it showcased SP. At that time, it was decided that the Sym- sections on: a) Proposed Problems; b) Solutions posium would be called, from then on, the Na- of Proposed Problems; c) Reviews of Books; tional Congress of Computational and Applied d) Articles of a didactic or historical nature and Mathematics (CNMAC), and its bylaws were e) Articles of current interest, such as literature voted on then. Further evidence that may help reviews, open issues and new lines of research.

Figure 1. Poster and participants of the First National Symposium on Numerical Analysis at the Federal University of Minas Gerais, 1978 Another milestone in the consolidation of From the start, the journal of 61 the Applied Math community in Brazil was the SBMAC set out high standards creation by SBMAC of the journal Computatio- for its published papers. nal and Applied Mathematics (MAC), the first Articles should have relevant in Brazil on this subject. This was accomplished in 1982, at the general assembly of SBMAC du- results, preferably exposing (or ring the third CNMAC meeting held in Maringá leading to) a computational State of Paraná. Currently, the journal is named approach. They should include Computational & Applied Mathematics (COAM) complete mathematical and is published by Springer. It is the main pu- demonstrations and not only blication of SBMAC. In its 36 years of continuous state results obtained existence, it has never missed a single edition. Mathematics in Brazil was a landmark for re- From the start, the journal of SBMAC set out search in this area. high standards for its published papers, as can be inferred from its Editorial policy: Applied SBMAC now has around 600 active mem- and Computational Mathematics – the Scienti- bers, and promotes the national conference fic Journal of SBMAC – publishes original works CNMAC every year and regional meetings (ER- in the different areas of Applied Mathematics. MAC) in different parts of the country. Publica- Articles should have relevant results, prefera- tions sponsored by SBMAC include: Annals of bly exposing (or leading to) a computational CNMAC (publishes selected papers presented approach. They should include complete ma- at CNMAC); SBMAC Bulletin (an electronic ne- thematical demonstrations and not only state wsletter); Computational & Applied Math (SB- results obtained. In general, reviews will only MAC´s main journal published by Springer); be published if previously commissioned by Notes in Applied Mathematics (a book series the Editorial Board. In order to facilitate grea- publishing the contents of mini-courses delive- ter exchange with the international community, red at CNMACs); Proceeding Series of SBMAC authors should preferably write in English. This (electronic publication of papers presented at will also help maintain the high quality standard ERMACs); SBMAC SpringerBriefs (a book se- desired for the Journal as it opens up for more ries on current hot topics); and TEMA (a journal choices in selecting reviewers. Writing in Por- dedicated to articles of national interest). tuguese, authors should seek to avoid foreign terms by striving to find the exact equivalents SBMAC has been a member of the ICIAM since in the vernacular and thus enriching our tech- its creation and from 2019 will join SeMA, SIMAI nical vocabulary. In fact, the creation of a publi- and SMAI in the sponsorship of the ICIAM's La- cation focused on Applied and Computational grange prize.

The first issue (1982) of the journal “Computational and Applied Mathematics”, published by SBMAC. This journal is currently published by Springer 5 Conclusions ject to poor working conditions. Moreover, the national scientific-mathematical environ- Applied Mathematics in Brazil did not start ment has expressly benefited from the vo- with mathematicians, but with engineers. This is cation, talent, and orientation of expressive not surprising since, in fact, professional mathe- leaders such as Odelar Leite Linhares, Marco matics in Brazil, as well as in many other coun- Antonio Raupp, and the main leaders of IMPA tries, also began in that way. Brazil has a strong and LNCC. tradition in Engineering, as a result of the pre- sence of state-owned and private companies Today, the Brazilian presence in the in- linked to the production of energy, civil cons- ternational pantheon of Mathematics and its truction, and transport. These enterprises are applications is incontestable and is expressed the result of industrial development policies put through the publications, organization of and in place by the national governments since 1930. participation in scientific meetings, internatio- As a consequence, high quality Applied Mathe- nal prizes and invited lectures. matics has been and continues to be developed At the same time that financial support in Departments of Electrical Engineering, Me- seems to be decreasing, the hopes for the con- chanical, Communications, and Transport Engi- tinuation of the implicit national project that led neering, among others. It is not by chance that to the current state of affairs in Brazilian scien- institutions for the promotion of scientific and ce with regard to Applied Mathematics rest technological research, such as the CNPq and with institutions dedicated to promoting the CAPES, have been created concomitantly with integration of the Mathematical Sciences with companies such as Petrobras and Eletrobras. Industry, such as the Center for Mathematical Fortunately, government support for re- Sciences Applied to Industry (CEPID-CEMeAI) search and development in the Mathematical in São Carlos and the Institute of Industrial Ma- Sciences has not lacked in its essential aspects, thematics (IMI) in Curitiba. being in general well-correlated with the de- Acknowledgements mand and the emergence of talent in the area. It is to be hoped that the recent trend of cuts The authors would like to acknowledge the in research budgets will not continue, although invaluable contributions to this text of Profs there is enormous concern about this. Vitoriano Ruas (CNPq researcher at PUC-Rio), In addition to state support, other factors Mauricio Kritz (LNCC), Geovan Tavares (PUC- contributed to the development of Brazilian -Rio), Eliana X. Linhares de Andrade (IBILCE), Mathematics and Applied Mathematics. One Jose A. Salvador (UFSCar) and the Ph.D. stu- of them has been the generous opening of dent Marta F. dos Anjos (UFRN). The text in the the country and its institutions to scientists section about SBMAC was mostly taken from and teachers from bordering countries, of- reference [8] with permission. We are very ten harassed by oppressive regimes or sub- thankful to the author. References [1] Boletim da SBMAC, Number 1, 1980. springer.com [2] LNCC, LNCC comemora três décadas de vida. http:// www.lncc.br/lncc/noticias.php?idt noticia=674, 2016. [3] O.L. Linhares, Fato Historico SBMAC, Sociedade Brasileira de Matematica Aplicada e Computacional, Rio de Janeiro, 1988. Guccia was the mathematician [4] O.L. Linhares, Uma breve história dos primórdios da behind the journal Rendiconti SBMAC. Available in: https://www.dcce.ibilce.unesp.br/ sbmac/eventos/xxvicnmac/25anos.htm, 2015. del Circolo Matematico di Palermo. [5] Matemática Aplicada e Computacional, SBMAC, Vol. 1, No. 1, 1982. [6] V.O. Santos, Uma história da Sociedade Brasileira Join Prof. Guillermo Curbera, de Matemática durante o período de 1969 a 1989: biographer of Guccia, and Criação e Desenvolvimento. Ph.D. Theses, UNESP, Rio Prof. Luigi Ambrosio, journal editor, Claro, 2014. [7] A. Taitelbaum and E. Brietzke, Um pouco da historia do for a special event about the Instituto de Matemática da UFRGS, no date available. journal and its founder. [8] M.F. Anjos, Um estudo preliminar sobre os movimentos de institucionalização da Matemática Aplicada no Bra- sil, 2016. Available in: http://www.15snhct.sbhc.org.br/ resources/anais/12/1474321500 ARQUIVO SNHCMar- Where: Springer booth, ICM Rio tadosAnjoscorrigido. pdf When: Monday 6 Aug. at 16h45 63

Jinyun Yuan Professor, Federal University of Paraná, Brazil E-mail: [email protected]

A Glimpse of the Current Research Activities in Applied Mathematics in Brazil

Applied mathematics in Brazil can be and computational mathematics (2) and traced back to 1848. The first Mathematics applied and computational mathematics with Ph.D. thesis was written by Joaquim Gomes habilitation in different areas (12) [4]. There de Sousa at Escola Militar da Corte at Rio de are also many production-engineering courses, Janeiro. In his thesis “O Modo de Indagar Novos which are based on applied mathematics. With Astros sem o Auxílio de Observações Directas” regard to graduate programs in applied and (“Search modes for new stars without direct aid computational mathematics, there are 11 Ph.D. observations”) [3], Gomes used perturbation programs and 21 master programs in Applied theory in search of new stars. Gomes was and Computational Mathematics; 32 Master le- born in the state of Maranhão in 1829 and vel degree programs and 21 Ph.D. programs in moved to Rio de Janeiro in 1840. Inspired by Mathematics in which applied and computatio- his studies in Physics and Chemistry at the nal mathematics disciplines are offered [5]. School of Medicine of Rio de Janeiro, Gomes In research, there are 500 research groups concentrated on his studies in Mathematics at on applied and computational mathematics [6] Escola Militar da Corte. His thesis was approved in the whole country. Below are some of the in October 14th, 1848. noteworthy groups:

In Brazil, the study of Applied Mathematics 1. Continuous and Discrete Optimization evolved with the support of two entities crea- groups led by Professor Alfredo N. Iusem ted in 1951: Conselho Nacional de Desenvolvi- (IMPA), José Mario Martinez Peres (University mento Científico e Tecnológico (“CNPq”) and of Campinas – Unicamp), Clovis C. Gonzaga Coordenação de Aperfeiçoamento de Pessoal (Federal University of Santa Catarina – UFSC), de Nível Superior (“CAPES”). Additionally, the Carlos Humes Jr. (University of São Paulo – Brazilian Society of Mathematics and the Bra- USP), Nelson Maculan Filho (Federal University zilian Society of Applied and Computational of Rio de Janeiro – UFRJ) and Celso Ribeiro Mathematics have been fostering the develo- (Federal University of Fluminense – UFF). This pment of applied mathematics in Brazil. Today, group has made valuable contributions to con- there are 26 Bachelor degree courses divided tinuous and discrete optimization, for example, as follows: industrial mathematics (4), compu- metric space and numerical optimization me- tational mathematics (4), business mathemati- thods and operational research. Additionally, cs (1), applied mathematics to business (1), ma- they have been training several generations of thematics (2) and scientific computing, applied young researchers and established some new research centers at the Federal University of The study of Industrial Goiás (UFG), the Federal University of Piauí Mathematics is relatively new (UFPI) and the Federal University of Paraná in Brazil. Mathematicians (UFPR). Their work has made strides on Bana- began to officially study ch space and Hilbert space optimization, mani- industrial mathematics in 2000; fold optimization, complexity, numerical algori- thms for combinatorial optimization, non-linear before that, there were only and non-smooth optimization, stochastic opti- a few individual consultants mization and their applications. in industrial mathematics, especially in production 2. Geophysics groups such as the one led by engineering Martin Tygel (just retired recently) and Joerg Schleicher (Unicamp), Milton J. Porsani (Fede- ral University of Bahia – UFBA), Jesse Carva- 3. Numerical Analysis and Simulation groups lho Costa (Federal University of Pará – UFPA), led by Abimael Loula and Frederic Valentin Walter E. Medeiros (Federal University of Rio (National Laboratory of Scientific Computing Grande do Norte – UFRN), Marcelo Assumpção – LNCC), Alexandre Roma and Saulo R.M. (University of São Paulo – USP), George Sand Barros (University of São Paulo – USP), Alvaro França (Seismic Observatory – OS), André Rei- Coutinho (Federal University of Rio de Janeiro – UFRJ), José A. Cuminato, Murilo Tomé and naldo Rodriguez Papa (National Observatory Gustavo Carlos Buscaglia (University of São – ON), and Luiz Andau and Alberto Luiz Coim- Paulo at São Carlos – ICMC-USP), Dan Marchesin bra (Graduate and Research in Engineering of and André Nachbin of Instituto de Matemática Federal University of Rio de Janeiro – COPPE- Pura e Aplicada – IMPA) and Fermín Bazán and -UFRJ). The development of geophysics in Bra- Antonio Carlos G. Leitão (Federal University of zil is owed to a graduate program sponsored Santa Catarina – UFSC). Several new centers by Petrobras, the Brazilian state – owned oil were established such as the ones at State company. In the 1960s, two specialists came University of São Paulo in Presidente Prudente annually from North America and Europe to (Unesp-PP) and São José do Rio Preto, UFPR, the Federal University of Bahia to lecture and Unicamp, USP, etc. supervise the work of Ph.D. theses supported by Petrobras for its graduate program. At pre- 4. The study of Industrial Mathematics is re- sent, almost all geophysics researchers have a latively new in Brazil. Mathematicians began to degree from that program. officially study industrial mathematics in 2000.

Joaquim Gomes de Sousa (left) and his work: the first math Ph.D. thesis developed in Brazil Brazilian legislation requires that energy compa- of cooperation by the Federal University of 65 nies invest 1% of their gross income on resear- Paraná and the Industrial Federation of Paraná ch, development and innovation, thus fostering State (FIEP). In 2011, the University of São Paulo the development of industrial mathematics. For at São Carlos created the Center of Mathematical example, the Federal University of Paraná’s op- Science Applied in Industry (CeMEAI). The timization group started to use mathematical project CEPID supported by São Paulo State modeling and numerical optimization to solve Foundation (FAPESP) put together researchers restoration and reconfiguration problems for the of industrial mathematics in order to organize distribution network of the Paranaense Energy field related activities in the State of São Paulo. Company (COPEL). The group of geophysics The groups of industrial mathematics tried at Unicamp works on mathematical problems to set up a national network, following the related to oil exploration by Petrobras. Prior to model of The Mathematics of Information 2000, there were a few individual consultants in Technology and Complex Systems (MITACS) industrial mathematics, especially in the field of [8] in Canada and the Industry Doctoral production engineering. Training Centre (IDTC) of the Australia Due to sectorial demand, a first undergra- Technology Network of Universities (ATN) duate course in industrial mathematics was es- [7] in Australia. Unfortunately, this initiative tablished by the Federal University of Paraná failed as a result of poor current economic and in 2000. More than 120 people have graduated political conditions in Brazil. from this course and approximately 50% of its The National Confederation of Industry graduated students work in areas directly re- (CNI) [2] and the Brazilian National Agency lated to the industry. Some of these students of Industrial Development (ABD I) [1] pay established their own highly profitable com- much attention to and focus on activities of panies. Subsequently, several Brazilian univer- industrial mathematics because of the recent sities such as the Federal University of Goiás implementation of Industry 4.0. (UFG) at Catalão, the Applied Mathematics School of Foundation of Getúlio Vargas (FGV), We apologize for not mentioning the names the Federal University of Rio Grande (FURG), of all researchers in the aforementioned groups the Federal University of Ceará (UFC), State and for excluding many excellent groups in University of Londrina (UEL) and the Federal other areas. University of Espírito Santo (UFES) have also inaugurated courses in the area. These cour- Acknowledgments ses have gradually attained popularity. We would like to thank Ricardo Biloxi, Cassio The Institute of Industrial Mathematics (IMI) Oishi, Mario Martinez and Nelson Maculan for was established in December 6, 2010 as a result their support.

References [1] www.abdi.com.br/ [6] http://dgp.cnpq.br/dgp/faces/consulta/ [2] www.portaldaindustria.com.br/cni/ [7] www.atn.edu.au/IDTC/ [3] Joaquim Gomes de Sousa, O modo de indagar novos [8] www.mitacs.ca astros, Editora da UFPR (Press of the Federal University of Paraná), Curitiba, Brazil, 1992. [4] http://emec.mec.gov.br/ [5] Report on Evaluation 2010–2012 at https://docs. google.com/viewer?a=v&pid=sites&srcid=Y2FwZX- MuZ 292LmJyfHRyaWVuYWwtMjAxM3xneDo0N- 2Q2NDU4YTIxMmMyM2U Academic Events Mobilize Institutions in the School Break Scientific programming gains pace faster with the biennium of mathematics

The title of math’s world cluding lectures, courses, exhi- institution delivered lectures in

capital is adequate for ICM’s bitions and workshops, to show several Brazilian states. that “Math is everywhere“. host cities, but Rio de Janeiro Innovations in has accelerated the pace long The vast program offered at Neuromath

prior to the ten days of the the National Science and Tech- Among the scientific events that took place, in June the Uni- world’s biggest event in the nology Week (SNCT) covered versity of São Paulo (USP) held, area. Since the beginning of at least one thousand Brazilian a Conference to discuss a rela- the Biennium of Mathematics municipalities, in an action for tively new area of study: “neu- 2017–2018, mathematics has the popularization and disse- romath”, a focus of the insti- become a frequent subject in mination of science coordina- tution's Centerfor Research, the Brazilian press and social ted by the Ministry of Science, Innovation and Dissemination networks in Rio, Cidade Ma- Technology, Innovation and in Mathematics, which brings ravilhosa, as well as other re- Communications (MCTIC). together mathematicians, neu- gions of Brazil. The movement The IMPA has been one of roscientists, physicians and has been intense, even during the institutions that opened its computer scientists. school holidays. doors to show to the teachers Also in June, the Fede- In 2017, for example, in ad- and students of basic educa- ral University of Ceará (UFC) dition to the series of academic tion what their researchers brought together speakers events, teaching and research already know: it is possible, from 12 Brazilian institutions, institutions across the country yes, to have fun with math. In plus researchers from Finland have promoted about 90 thou- addition to workshops, exhibi- and Spain, to its 10th Wor- sand free activities for the dis- tion of 3D objects and video kshop on Geometric Analysis. semination of the discipline, in- sessions, researchers of the In July, the Institute of Ma- 67 Courtesy of IMPA Courtesy

Participants of the 31st Brazilian Mathematical Colloquium in 2017: 40 programming activities for 5 days

thematics and Computer Scien- chers of international renown. of São Paulo, took office. He ces (ICMC) in São Carlos hosted It is also the gateway for those will manage the entity until the international meeting of wishing to follow an academic 2019. commutative algebra and re- career. Among its thematic events, lated areas. Given the limited In 2017, on the 60th anniver- the IMPA held a program on number of researchers in Bra- sary of the Colloquium, thou- Parameter Identification in Ma- zil, the event was held to foster sands of participants accom- thematical Models. Created the formation of study groups panied the 40 programming with the goal of bringing toge- on the subject and to promote activities for five days. It was a ther leading specialists in the scientific cooperation with spe- scientific marathon: eleven ple- area of numerical analysis and cialists from Latin America. nary sessions, eleven thematic mechanical engineering, these The IMPA welcomed par- sessions, two lectures, a rou- specialists spent two months ticipants of the most tradi- nd table, six advanced courses, in the institution to discuss tional and academic event in three introductory courses and the main current trends in re- the country: the Brazilian Ma- three poster sessions. search and promote scientific thematical Colloquium. Held every two years since 1957, During the event, the new collaboration among Brazilian this event transforms the ins- President of the Brazilian So- and European communities of titution’s corridors into a noisy ciety for Mathematics (SBM), researchers. and colourful Babel, formed Paolo Piccione, a professor at The IMPA's activities in by undergraduate and post- the Institute of Mathematics connection with the thematic graduate students and resear- and Statistics of the University program included two scien- At the IMPA, a total 400 students hailing Militar in Rio de Janeiro for a from the country’s 20 States and others promising exchange of expe- from 13 countries enrolled in the summer riences. program; lecturers were brought from Students from 20 Princeton University, Oxford University, States of the country

MIT, Universidad de Chile and the Math started out 2018

University of Birmingham, among others with renewed energy. Many

Brazilian institutions, among tific meetings: (1) new trends finance and quantitative risk them the federal universities of in identification parameters analysis. Ceará and Rio de Janeiro, the for mathematical models and “We also discussed the University of Brasília and the (2) Research in 2017 – in whi- role played by models to stu- ICMC, took advantage of the ch mathematical applications dy volatility in its many forms, school holidays to offer popu- in the financial market were especially with regard to the lar summer courses designed discussed. Held at the IMPA new financial instruments that to promote the interaction be- since 2005, this International arose to protect against ex- tween students and research Conference has brought to Rio cessive volatility and swings,” groups in Brazil and abroad. prestigious world-renowned said the researcher, adding At the IMPA, a total 400 names such as Bruno Dupire, that the meeting had also ad- students hailing from the Director of quantitative resear- dressed topics about mode- country’s 20 States and others ch of Bloomberg in the United ling, to better understand the from 13 countries enrolled in States; Marco Avellaneda from dynamics of commodity prices the summer program. In or- the Courant Institute of Ma- and interest rates. der to add to the buzz of the thematical Sciences (USA); and In November the IMPA and Extremal and Structural Com- Emmanuel Gobet from the SBM held the third national binatorics workshop, lecturers Ecole Polytechnique (France). symposium of Teacher Edu- were brought from Princeton IMPA researcher and orga- cation in Mathematics (PRO- University, Oxford Universi- nizer of Research in Options FMAT). For three days, re- ty, Massachusetts Institute of Jorge Zubelli considered the searchers, teachers, undergra- Technology (MIT), Universidad 2017 event one of the most duate and graduate students, de Chile and the University of productive ever, based on the students who work in the field Birmingham, among others. debates around data analysis, and graduates of PROFMAT The carioca Luize D’Urso, algorithmic negotiations and (master in Mathematics in Na- student of PUC-Rio, was one machine learning, themes of tional Network) from all over of the summer school partici- great relevance in the area of Brazil gathered at Colégio pants. Seven times gold med- Courtesy of IMPA Courtesy

Artur Avila speaks to an attentive audience at the 31st Brazilian Mathematical Colloquium in 2017

alist of the Brazilian Mathemat- After the Carnival, the pace Dynamics and Geometry (Flu- ics Olympiad of public schools remained just as intense in the minense Federal University, Ni-

(OBMEP), she exchanged expe- teaching and research units in terói); Group Theory (Cabo Frio); riences with people interested mathematics. The IMPA offered XII Brazilian Workshop on Con- in mathematics and was able seminars on Graphs and Ram- tinuous Optimization and 2nd to take lessons with the institu- domness; and, at the Federal Conference BRICS on Mathe- tion’s prestigious teachers. University of Paraíba, João Pes- matics (Foz do Iguaçu, Paraná);

In January the IMPA held soa, there was the 20th School Combinatorics: Extremal, Prob- an enrichment program of ac- of Differential Geometry. abilistic and Additive (Maresias, tivities for middle school math On the eve of ICM 2018, São Paulo), Nonlinear Dispersive teachers (PAPMEM), with a series of satellite events was Equations (Florianópolis, San- live lessons broadcast to 70 held in Brazil: TeX Users Group ta Catarina) and (WM)² World

University centers across the (IMPA); Weakkam Rio 2018 Meeting for Women in Mathe- country. (PUC-Rio); Complex Foliations, matics, in Riocentro. Solutions to “IMO-Style Problems: Are You Ready for a Gold Medal?”

Solution to Problem 1 (with uniform distribution). The game is equivalent to tic-tac-toe. He then chooses a card and examines it. If Indeed, write the integers from 1 to 9 as a the number on the card is smaller than X, he magic square: changes card; otherwise, he keeps the card. We 2 7 6 may assume that Alice knows Bob’s strategy but does not have any power to predict the 9 5 1 result of the random selection.

4 3 8 Assume Alice selects numbers i < j to write Notice that three distinct integers in this set on the cards. If X < i or X > j, the probability add up to 15 if and only if they are on a straight of Bob winning equals ; if i < X or X > j, Bob line (horizontal, vertical or diagonal). The is guaranteed to win. Thus, his probability of article mentioned in the statement includes a winning (before making the random selection) few related remarks and many anecdotes. is . Alice should therefore choose a pair of the form j = i + 1 and Solution to Problem 2 in this case Bob’s probability of winning equals 1 1 Consider the parallel line to P P through P . + , solving item (a). 3 2 1 2 2͈ The point P will belong to this line and we will 7 As to item (b), we claim that under have . The point P will belong to 6 reasonable hypotheses, this is the best the parallel line to P P through P and we will 3 4 5 strategy for Bob. We assume here have ; more precisely, we will have that his strategy consists of a function . and a Rotating the heptagon H = P P P P P P P 1 2 3 4 5 6 7 random variable Y with uniform distribution by 180 around the middle point of P P we º 1 2 in [0,1]. Alice is assumed to know F but is not get the heptagon H´ = P´ P´ P´ P´ P´ P´ P´ 1 2 3 4 5 6 7 capable of predicting Y , which is randomly with Translating P´1 = P2, P´2 = P1 , P´3 = P7, P´7 = P3. selected after she plays. Bob then examines the heptagons H and H´ infinitely many times the card, which shows a number k: if F (k, Y) = by , we cover a toothed 1 he changes card, otherwise he keeps it. The strip, which, translated infinitely many times previous example is equivalent to by , allows us to tile the plane. In this way, we tile the plane with the heptagons and , with disjoint Notice that more elaborate random interiors, all congruent to H. processes can be converted into this model. Solution to Problem 3 In this model, let ai = Prob [F (k,Y) = 0]. It Bob adopts the following strategy. After is not difficult to check that, given any such Alice has played but before choosing a card, function F, if Alice plays the pair (i,j) (where he randomly selects an element X from the set i < j), Bob’s probability of winning is equal to Since , there Going back to the game proposed in 71 exists i such that 1 , proving that no the statement, if we are only interested in ͕i −͕i ≤ +1 ͈ determining if Rosencrantz wins, we may function F is better for Bob than , the first interrupt the game as soon as the first player example. reaches N points. This reduces the game to Furthermore, any different choice of the the variation above. The desired probability numbers ak is strictly worse. therefore equals .

(d) We may consider this game as a repetition Solution to Problem 4 of N subgames consisting in tossing the coin until

(a) A score of N to k corresponds to a the first head. The payments can be made after sequence of N H and k T where the last letter each such subgame, without affecting the final is an H . Each sequence has probability 2(-N-k). amount. In each such subgame the expected Thus, we must choose a subset of k elements time T of appearance of the first head can be calculated from the following equality, which among the firstN + k -1 positions (to be the T). considers what happens after the first toss: The probability of a score of N to k is therefore N = (1/2) • 1 + (1/2) • (T +1) , which gives T = 2. So the expected value of the payment from Guildenstern to Rosencrantz after such a subgame is 1 – (2 – 1) As a special case, the probability of a draw is = 0 , which implies that the game is fair. Solution to Problem 5 First of all, notice that the images of the vertices of an equilateral triangle of side 1 also (b) Using the expression above and form an equilateral triangle of side 1. Therefore, simplifying, we have given two equilateral triangles of side 1 with a common side, the opposite vertices to the common side can have either the same image or different images at distance . In other words, if A and A´ are points such that d (A, A´) = and therefore , then d(f (A), f (A´)) ∈ {0, ,}. We will show that, indeed, d(f (A), f (A´)) = . If f (A) = f (A´), then taking B with d (A, B) = 1 and d (A´, B)= , we would have d((f(A), f(B)) = 1 ⇔ d(f(A´), f(B)) = 1 which is a contradiction. Thus d(A, A´) = similar equivalences hold for > and for ≤. Thus, d((f(A), f(A´)) = ⇒ d((f(A), f(A´)) = . for fixedN and changing k, PN,k grows up to This implies that any triangular lattice formed k = N – 2, we have PN,N-2 = PN,N-1 and then PN,k by vertices of equilateral triangles of side 1 with decreases. Thus, for fixedN , the two most disjoint interiors and tiling the plane is preserved probable scores are (P – 2) to N and (P – 1) to N; by f, in the following sense: the image of this these two have equal probability. lattice will be another lattice of the same type.

(c) Consider the following variation. In particular, points at distance n are taken to ). Rosencrantz and Guildenstern play heads and points also at distance n (for any n ∈ tails until the first player reaches N points; this This last fact implies that triangles of sides 1, first player is then declared the winner. In this and (as in Figure 1) variation there are no draws and by symmetry are preserved by the function f since their the probability of each player winning equals . vertices belong to a triangular lattice of side 1. Figure 1.

Using a procedure analogous to the d(f(X), f(P)) + d(X, P) – d(X, Y) | ≤ d(f(Y), f(P)) + previous one, we will now consider the image < a contradiction. d(P, Y) 4ɛn , of the vertices of two triangles of this type Solution to Problem 6 with a common side of length , each triangle being a reflection of the other A tree is thin if it contains a path C with along the common side. If X and Y are the vertices P1, P2,..., Pn such that any other vertex vertices of these triangles opposed to the Q is at a distance at most 2 from some Pi. common side, we have (as before) that d(X, Y) Equivalently, a tree is thick (not thin) if it contains = ⇒ d f(X), f(Y) ∈ {0, } where is twice the tree in Figure 3; that is, if it has a vertex ɛn ( ) ɛn ɛn P the height of the above triangles in relation and three other vertices Q1, Q2 and Q3 belonging to the common side. Since the area of the to three different branches in relation toP and triangle ABC equals , we have such that d(Qj, P) = 3 for j = 1, 2, 3. In other words, given a tree G look for two vertices at maximal

distance and consider the simple path P1, P2,..., Pn joining them: call this path C. Consider a vertex We will show that the images of points at Q at maximal distance d from C: if d ≤ 2 then G is distance are indeed distinct. Let be such ɛn kn thin; if d ≥ 3 then G is thick. that . knɛn < 1 ≤ (kn + 1) ɛn The graphs G for which the hunter has a If , consider points d(A0 A1) = ɛn Ai, 2 ≤ i ≤ kn + 1 strategy are thin trees. Equivalently, the hunter such that = for and d(Ai, Ai+1) ɛn 0 ≤ i ≤ kn d(A0, Ak +1) n does not have a strategy for graphs that = 1. We have d f(A ), f(A ) = 1 and thus ( 0 kn+1 ) contain some cycle, and also does not have a strategy for thick trees. In particular, the hunter does not have a strategy for the graph in Figure 3. If , is equal to , then , d f(A0) f(A1) 0 1 ≤ knɛn < 1 ( ) Notice first of all that if the hunter has a a contradiction, and so = , d(X, Y) ɛn ⇒ d f(X) f(Y) ( ) strategy, he can tell his strategy to the rabbit = . As before, we have , , ɛn d(X Y) = kɛn ⇒ d f(X) ( and still capture the rabbit. Next, notice that if G f(Y) = k for k ∈ . 0 ) ɛn is a subgraph of G1 and the hunter has a strategy Suppose now that there are points X, Y such for G1 then he also has a strategy for G0. that d f(X), f(Y) ≠ d(X, Y). Let n ∈ be such that ( ) Let us prove that if the graph is not a tree 4 < | d f(X), f(Y) – d(X, Y) | and P ∈ 2 with ɛn ( ) then there exists no strategy for the hunter ∈ , d(P, Y) < 2 . ɛn satisfying the conditions in the statement. Let 2 with = ; we have We need to prove that the hunter has no Q ∈ d(P, Y) d(P, Q) = ɛn d f(P), f(Q) = d f(Y), f(Q) = and therefore winning strategy if the graph the graph a cycle ( ) ( ) ɛn d f(P), f(Y) ≤ 2 . Since d(P, X) = d f(P), f(X) , {P , P ,..., P } (with n ≥ 3; P is connected to P for ( ) ɛn ( ) 1 2 n i i+1 we have | d f(X), f(Y) – d(X, Y) | ≤ d f(X), f(Y) – 1 ≤ i ≥ n, and we take the indices modulo n, that ( ) ( ) 73

Figure 2. A thin tree.

is, Pn is connected to Pi+1 = P1). Indeed, if the rabbit Figure 3. The smallest thick tree. is at Pi he can go to Pn -1 or Pi+1. If we assume the rabbit knows the strategy, we know where the hunter will play next and he may therefore go Finally, we prove that the hunter has no elsewhere. strategy for the thick tree in Figure 3 (and Thus, for the hunter to have a strategy, the therefore for no thick tree). In order to do this, graph has to be a tree and therefore bipartite. We we describe a valid path for the rabbit (assuming can paint vertices black and white, or, equivalently, he knows the strategy of the hunter). We again assign parities (0 or 1) to the vertices so that assume that the hunter knows the parity of the every edge joins vertices of opposite parities and initial position of the rabbit and always plays therefore the rabbit’s position changes color (or in the correct parity. Assume without loss of parity) at each step. We may assume that the generality that at even time t the rabbit is on hunter knows the parity of the position of the a white vertex (P or Yi). The rabbit constructs rabbit: he assumes one parity and plays; if that his path as follows. For any even value of t, if the hunter does not play on (i.e., ), the does not work he now assumes the opposite P vt ≠ P rabbit will be on w = P. The other even integers parity. t

are then divided into blocks of the form to + 2,..., We now describe a strategy for the hunter if t1 – 2 so that to and t1 are both even, the position the graph is a thin tree. The hunter starts at, and of the rabbit at both t and t is W = W = P and o 1 to t1 will always be at a vertex of the same parity as the position of the hunter satisfiesv = ••• = v = to+2 t1-2 the rabbit. If Q , Q ,..., Q are the vertices that 11 12 1i1 P. For the even values of t inside each block the are not in the path C and are adjacent to P , 1 rabbit will be at a vertex w = ••• = v = Y where to+2 t1-2 1 the hunter starts with P , Q , P , Q ,..., P , Q , P 1 11 1 12, 1 1i1 1 iis chosen so that W {v , v }. The rabbit then i ∉ to+1 t1-1 guaranteeing that, if the rabbit starts in a branch completes his trajectory by defining its position that leaves P1, then the hunter catches it. If he for odd t. For transitions between blocks, set did not catch it, it moves to P2 and, if Q21, Q22..., w w X . For the other odd values of t we to+1 = t1-1 = i Q are the vertices that are not in the path C and 2i2 have wt-1 = wt+1 ∈ {P, Y1, Y2, Y3} so that there are at are adjacent to P , the hunter continues with 2 least two valid values for wt: just take one which P , Q , P , Q ,..., P , Q , P , guaranteeing that, if 2 21 2 22, 2 2i2 2 is different fromv t . This defines a trajectory for the rabbit starts in a branch that leaves P2 then the rabbit which satisfies the conditions in the the hunter catches it. If not, the hunter moves statement of the problem and also wt ≠ vt for all to P3 and so on. t, as desired. Tourist guide

Tourist guide

Brazil’s most famous mountain is one of Rio’s main postcards. At the foot of the Christ the Redeemer statue, 710 meters above sea level, visitors can enjoy an amazing 360-degree landscape of the city. On one side, there is Copacabana, the entire edge of the southern part of the city and the lagoon called Rodrigo de Freitas. Right in front, Guanabara Bay, the Sugar Loaf and the city of Niterói further in the distance. On the opposite side, the Maracanã Stadium and neighborhoods. In the background, the Tijuca National Park and its forested peaks. All around, one is wrapped by an aura of peace and harmony where tourists and nature integrate as elements of one of the most beautiful cities in the world. Inaugurated in 1931, the art deco style statue is 38 meters high and the width between the open arms measures 28 metres. The statue is one of the must-see tourist attractions of Rio de Janeiro and Brazil. Each year, 1 million CORCOVADO people visit the Corcovado. Tourists have two options to access the top of the mountain: by passenger trains leaving from the Cosme Velho station or in vans from Copacabana, Barra da Tijuca and Largo do Machado. Pedro Kirilos/Riotur Pedro

The cable car landing on top of the rock mountain, at the entrance to Guanabara Bay, is famous worldwide and was part of the scenery in one of the James Bond films. It offers a magnificent view, both for those SUGAR LOAF who look from the Enseada de Botafogo and to the visitors who enjoy the beauty of the Baía de Guanabara from above and the mountain line from the Floresta da Tijuca. The cable car parts from Avenida Pasteur, in Urca. The trip is divided into two stages. The first stop is at Morro da Urca, 220 meters high, whilst the second and final stop is at the Pão de Açucar peak, 400 metres above sea level. Each stretch of the ride lasts three minutes. The cable car has a capacity of 65 passengers. The entrance cost for adults is R$80; students pay half price. Alexandre Macieira/Riotur Alexandre MARACANÃ For decades, the stadium was the largest in the world, with a capacity of nearly 200,000 fans. It has been a sanctuary for world football, where icons like Pelé, Garrincha, Rivelino, Zico, Maradona, Beckenbauer, Nilton Santos, Jairzinho, Romário, Ronaldo, Ronaldinho and many others have played. Further to its modernization for the 2014 FIFA World Cup, the stadium reduced its total number of available seats whilst remaining one of the most visited tourist attractions in Rio. In August, Maracanã will host matches of the Brazilian Soccer Championship, thereby giving tourists the opportunity to see the stadium and watch a game. Guided tours are also available, allowing tourists to walk on the field when there are no matches. João Paulo Engelbrecht/Riotur João Paulo

Copacabana beach is the greatest icon of the Brazilian vast coast. Around the world, when one talks about Brazilian beaches, COPACABANA BEACH Copacabana is the first that comes to mind. The promenade is an integral part of the city of Rio, spanning through four kilometers of distance between Leme on the left side of the strip of sand

and Forte de Copacabana, to the right. Maia/Riotur Fernando The well designed boardwalk replicating waves is one of its main characteristics. Recreation areas attract thousands of swimmers and sportsmen in the sunny days of summer. Copacabana is synonymous with celebration and fun. There is no shortage of good options of bars, restaurants and nightclubs in its neighborhood. 75 IPANEMA BEACH The view of Morro Dois Irmãos from Ipanema beach is breathtaking. However, if you ask the world around what the main reference for the neighborhood would be, the song “The girl from Ipanema” composed by Antonio Carlos Jobim and Vinícius de Moraes in 1962 would most likely be quoted. During a walk through Ipanema’s promenade, pedestrians may come across Tom Jobim himself – figuratively, of course. A statue of the maestro, based on his image with a guitar held on his back, recorded during the opening of Brasília in 1961, was inaugurated in 2014 at the edge of Arpoador. Photos next to Tom Jobim’s statue are better taken later in the day so that you can check out the most famous sunset of the city. It comes as no surprise that oftentimes the public applauds the scene. Located at Zona Sul, Ipanema is filled with bars and restaurants and also features the traditional Feira Hippie held on Sunday mornings at Praça General Osório where there is a subway station. Since the late 1960s, the fair brings together pieces of art and crafts and clothes. Hungry, adventurous guests may enjoy dropping by the tenda das baianas and experience the delicious typical food from the Brazilian northeastern state of Bahia. The food is spicy and for those unaccustomed to too much pepper, it is advisable to exercise caution and avoid the risk of falling sick. Alexandre Macieira/Riotur Alexandre

BOTANICAL GARDEN The main green area on the Zona Sul of Rio is a must-visit for tourists interested in a program beyond the city’s touristic circuit beach/nightlife/Corcovado/Pão de Açúcar. The tranquil setting of the garden is absolute when you walk through the tree alleys, surrounded by palm trees and imperial centenary vegetation. The botanical garden was created by the Portuguese Prince Regent Dom João VI in 1808. It occupies 54 acres of preserved area, where magnificent examples of Brazilian and foreign flora are kept. The institution also has the most important Brazilian library specializing in Botany and the largest Herbarium in the country, in addition to works of art and archaeological redoubts. In order to familiarize yourself with the garden prior to a visit, App Botanic is available for download as developed by the Visgraf laboratory and the IMPA, in partnership with Instituto Antonio Carlos Jobim. The app allows users to explore the beauties of the fauna, flora and architecture of the park along with poems written by the famous composer, who died in 1994, and was passionate about botanical gardens in his lifetime. Practical information on thematic trails, visitation and access are available via this powerful augmented reality navigation tool. It provides guidance to sights in the park and allows for data sharing. It is Danilo Schinke/Flickr accessible through the Apple Store.

In close proximity to the Riocentro, Barra da Tijuca offers options for those interested in exploring the natural marvels of the city, getting to know Brazilian art, shopping and even practising radical BARRA DA sports as well as going out in the area’s diverse bars and restaurants. For visitors who would like a dip in the sea, there are many options along Avenida Lúcio Costa in Barra da Tijuca. Before choosing where to go, one should be aware that Praia do Pepê usually has TIJUCA strong waves and intense currents. At Praia da Reserva, as the name suggests, there is an ecological reserve characterized by its tranquil setting. If the idea is to look for something more rustic, head to Praia do Grumari, surrounded by cliffs and hills with restinga vegetation, or Prainha. It is important to remember to drink plenty of water, possibly from the coconut fruit, not only delicious but also a great refreshment to hydrate from the carioca heat. Parque Nacional da Tijuca with its waterfalls, caves and trails belongs to Barra da Tijuca. Visitors can go to Pedra da Gávea, where it is possible to admire a breathtaking view of the city – in both directions. For more radical activities, it is possible to do ramp jumping, hang gliding and paragliding in the neighboring district of São Conrado’s Pedra Bonita, a mountain top. Even if you are not so adventurous, it is worthwhile to visit Pedra Bonita for its stunning view of the beach and encompassing areas.

Alexandre Macieira/Riotur Alexandre At Casa do Pontal, a collection of folk art brings together works from various regions of the country. In a space of 5000 square meters, guests can contemplate the art work and enjoy the lush nature around. Not to be missed is neighboring Sítio Roberto Burle Marx where the distinguished Brazilian landscape designer kept an address for more than two decades amid native vegetation species of restinga, Atlantic forest and mangrove forest. Burle Marx was responsible for the architecture of Aterro do Flamengo. Visitors with time for shopping will find it difficult to pick from the neighborhood’s many malls: Barra Shopping, Downtown, River Bar Design, Garden Bar, Barra Square Mall, Village Mall and Barra Point. LAGOA RODRIGO DE FREITAS Those who want to take a break and rest in one of the scenes favored by cariocas should go to the beautiful Lagoa Rodrigo de Freitas, in the Zona Sul of Rio. There, it is pleasant to simply stretch out on the lawn or sip drinks and eat something from one of the kiosks surrounding the water mirror. Additionally, if you feel more active, there are other entertaining options such as bicycle rental pedal boats that permit boaters to explore more than seven kilometres around the lagoon. Early mornings and evenings, the water is busy with practitioners of racing and water sports like rowing and canoeing. Nearby, a rollerblading park offers refreshment stations and toys for children; and there is Lagoon, a recreational complex that brings together restaurants, cinema theatres and a show house. For nature lovers, next to Lagoa Rodrigo de Freitas, there is Parque da Catacumba (Avenida Epitácio Pessoa, 3,000), an ecological reserve with walking trails and a gazebo from where the lagoon can be admired from a vantage point. Opening hours are from 8:00 am to 5:00 pm. Ricardo Zerrenner/Riotur Ricardo

PARQUE LAGE For anyone interested in learning about Rio’s nature, a visit to Parque Lage is highly recommended. The park is located in an old sugar plantation from colonial Brazil. In 1957, Parque Lage was declared a public landmark by the Instituto do Patrimônio Histórico e Artístico Nacional (IPHAN) for its historic and cultural value. The footpath to the mansion built as a 19th century replica of an Italian palazzo is encircled by alleys of royal palms. The school of Visual Arts of Parque Lage (EAV) operates on site and offers training for artists and art training courses for young people, in addition to an intense program of exhibitions, seminars, lectures and video shows. In the central courtyard of the classic Villa, under the internal arches and by the pool, you can take a break for a coffee, a snack or a warm meal. If you prefer, you can also extend a mat in the green area of the Park and have a picnic. In its 52 acres of forest, filled with species of the Atlantic forest and at the foot of the massif of the Corcovado, you will find a lake, artificial islands, bridges, caves, a gazebo and a cave, as well as aquariums with fish species from Brazilian rivers. At Parque Lage (located at Rua Jardim Botânico, 414, Jardim Botânico), there is no charge for entry. Transit options to get there are buses, subway lines and a bike station. The park is open from 8:00 am to 5:00 pm. The restaurant opens for coffee at 9:00 am except for Saturdays when it starts

serving at 1:00 pm. Macieira/Riotur Alexandre

One day does not suffice to enjoy all that Rio’s port region has to offer. Originally a fishing village, this area then became the main entry point for the intake of slaves who ar- rived in the country. The area has undergone a revitalization process initiated in 2010 and brings together historic sights displayed along modern cultural equipment. For a tour of the art collections, one should go to the Museu de Arte do Rio (MAR). Installed at Praça Mauá, 5, the museum joins, through a square and a walkway, two build- ings of different styles and time periods. Its exhibition halls are located at Palacete Dom João VI, a heritage landmark, and the neighboring modernist building, once a bus station, hosts Escola do Olhar, a training center for public school educators. Open from Tuesday to Sunday, from 10:00 am to 5:00 pm. You can buy tickets online at http://museudeartedorio. org.br/pt-br/visite/compre-online Tuesdays, admission is free for everyone. After getting to know a little more about Rio, it is a good idea to head to the Museu PORT REGION do Amanhã, which, by means of science and a lot of interactivity, offers a narrative on Alexandre Macieira/Riotur Alexandre how mankind will live in the next 50 years. There are big questions like: “Where did we come from? Who are we? Where are we going? How do we want to get there? “The main exhibition hall is divided into five major areas – Cosmos, Earth, Anthropocene, Tomorrows and us – and bring together over 40 experiences available in Portuguese, Spanish and English. Opening hours are from Tuesday to Sunday, from 10:00 am to 6:00 pm – last entry is at 5:00 pm. Tickets online can be found at https://ingressos.museudoamanha.org.br/#!/home If it gets late in the evening, it may be advisable to come back the next day for a visit to the largest Marine Aquarium of South America. With 26000 m² of built area and 4.5 million litres of salt water distributed over 28 venues, the Aquarium has about 8000 animals from 350 different species on display. Through the underwater tunnel, there is a view of nearby fish, stingrays and sharks. Located at Praça Muhammad Ali in Gamboa, the Aquarium is open daily from 10:00 am to 6:00 pm, the last entry in the circuit is at 5:00 pm. For tickets online, visit http:// www.aquariomarinhodorio.com.br/explore-o-aquario/ingressos-tickets?utm_source=insite&utm_medium=bannerhome&utm_campaign=In- gressoAvulso In case of some free time to explore the region, one tip is to go to places that explain the African diaspora in Brazil and the beginnings of Brazilian society. A starting point would be the Cais do Valongo e Imperatriz, representing the arrival of Africans and their descendants in the country. Go to the Cemitério dos Pretos Novos, where newcomers who perished on the trip to Brazil were buried; afterwards, a visit to Largo do Depósito where slaves were sold; and the Jardim do Valongo, a symbol of the official story to erase traces of the slave trade. Pedra do Sal, a point of resistance and celebration, is considered the cradle of samba carioca; and no tour should be complete without a stop at Centro Cultural José Bonifácio, a reference point in black culture, based in the building where Latin America’s first public school opened in 1877. More details and archaeological history of the Circuit’s celebration of African heritage can be found at http://portomaravilha.com.br/circuito. Tourist guide 77

BÚZIOS Rumor has it that ever since the French actress Brigitte Bardot visited Búzios in the sixties, the city has been a fashionable destination. Located in the State of Rio’s Região dos Lagos (Lake Region) and one hundred and seventy kilometers from the capital, Rio de Janeiro, the formerly quiet fish- ermen’s town offers over a dozen outstanding beaches. Geribá is one of the favorite beaches with its extended sand strip and bars in the proximity. João Fernandinho, whilst less accessible, has a wilder panorama. For visitors who enjoy a calm sea, Ferradura is the one to go to. Beyond its postcard views, Búzios’ nighttime is full of excitement. The renowned Rua das Pedras is filled with bars, fine dining restaurants and popular accommodations. Robinson dos Santos/Flickr/CC BY-SA 2.0 BY-SA dos Santos/Flickr/CC Robinson

PETRÓPOLIS

What about getting to know a little more about Brazilian history by taking a short trip to Petrópolis? A mountainous city located just seventy kilometers away from Rio, Petrópolis was favored by the im- perial family because of its mild weather. The summer residence of King Pedro II has become the city’s main attraction under the name of Museu Imperial, where you can see relics like the golden fountain pen used by Princess Isabel to sign the Áurea law abolishing slavery. Once in the historic center, it is definitely worthwhile to visit Santos Du- mont’s house and the Crystal Palace (Palácio de Cristal). The São Pe- dro de Alcântara Cathedral is where King Pedro II’s mortal remains are kept; the Palácio Negro; and the Praça da Liberdade where, in the old days, freed slaves held meetings to acquire the freedom of those who

Rodrigo Soldon/Flickr/CC BY-ND 2.0 BY-ND Soldon/Flickr/CC Rodrigo were still held captive.

PARATY If you do not mind a longer road trip, of approximately four hours, Paraty is a city not to be missed. Shaped by the gold and sugar cane cycles, this her- itage colonial town preserves its bucolic ambience and is an open invitation for walkers. The old houses, the cobblestoned and crooked streets and surrounding beaches attract tourists from all over the world. In July, Paraty hosts the Festival Literário de Paraty (FLIP), an international literary event that has been an integral part of the city’s calendar and counts with the participation by Brazilian and for- eign writers from all over. In addition to the long walks, boat tours are a great way to get to know the idyllic beaches of that region. Horse drawn cart rides are available for swiftly traveling in time. Daniel Raposo/Wikimedia Commons/CC BY-SA 4.0 BY-SA Commons/CC Daniel Raposo/Wikimedia

FOZ DO IGUAÇU There are few sceneries as impressive as those seen from the observation decks at the Iguaçu River Falls on the Argentinean-Brazilian border. The Falls are a remarkable sight for tourists. There are 275 waterfalls constituting an un- forgettable view. The Iguaçu Falls, that include the Iguaçu National Park lands (Brazil) and the Iguazu National Park (Argentina) is one of the country’s main attractions. They are a must see for Brazilians and foreigners alike. Garganta do Diabo (Devil’s Throat) is one of the most widely known parts of the falls that are spread over almost three kilometers of the Iguaçu River and form a U shape with a height of 82 meters, 150 meters in width and 700 meters in length. To visit the Falls, the route of choice by tourists is through the town of Foz do Iguaçu in the

SF Brit (C.N.Battson)/Flickr/CC BY 2.0 BY SF Brit (C.N.Battson)/Flickr/CC Brazilian State of Paraná. Other attractions of the region: Itaipu’s artificial lake and the free trade zone on the frontier of Paraguay in Ciudad del Este. Tourist guide 78

SALVADOR

The first capital of Brazil has become a great metropolis with its preserved build- ings, landmarks and streets that keep alive a memory of colonial Brazil in the eigh- teenth and nineteenth centuries. Worldwide famous Pelourinho is the best example of a well-preserved historical city center in the midst of an urban area that took its time to modernize. The Cidade Alta and Cidade Baixa (the Upper and Lower Cities) intercommunicated by the Lacerda Elevator, the churches made of gold, the hills, the mansions, the Mercado Modelo (a traditional market), the Castro Alves Square and the Terreiro de Jesus are some of Salvador’s highlights. There are many historical attractions in Salvador, which is a beautiful city portrayed in postcards also through Farol da Barra (Barra’s Beacon), the Itapoã beach, the sand dunes of Abaeté and Barra’s Harbor, a beach area where the sundown has become an imperative for tourists. Marlon Bianco Matias/Wikimedia Matias/Wikimedia Marlon Bianco 3.0 BY Commons/CC

LENÇÓIS MARANHENSES An enormous preservation area in the State of Maranhão, Parque Nacional dos Lençóis Maranhenses is formed by dunes and lakes encompassed by the territory of three cities. It is a unique ecological paradise. Even today, it is difficult to access this region of the country. Tourists normally take a bus from São Luís do Maranhão to Barreirinhas in a 260 kilometers road trip. Given the sheer size of the dunes and sands, it is mandatory that tourists hire specialized tour guides and agencies. The local climate is divided into two well-defined seasons. During the rainy season, from January to July, thousands of lakes are formed in 156 thousand hectares of parkland. During the period of drought, from August to December, the rains stop and lakes tend to dry. It is ideal to visit when the lakes are pristine for luscious bathing. Ilias Bartolini/Flickr/CC BY-SA 2.0 BY-SA Ilias Bartolini/Flickr/CC

THE AMAZON FOREST

It is challenging to describe in words how gigantic the largest tropical forest in the world is. Suffice it to say that, despite the deforestation process, it is possible to fly over the Amazon forest for endless hours and its scenery remains unchanged: a green carpet crisscrossed by rivers. To get to know the forest, it is advisable for tourists to start in Manaus, the capital city of the State of Amazonas. From there, boat rides can be or- ganized for any length of time. Vessels navigate over the rivers Amazonas, Solimões, Negro and Madeira, some of the rivers that form the Bacia Amazônica (the Amazon basin). One of the most popular tours leaves from Manaus and covers the Anavilhanas Archipelago. Beyond the forest, visitors will also have the opportunity to know the river flows and the amazing fauna. Robinson dos Santos/Flickr/CC BY-SA 2.0 BY-SA dos Santos/Flickr/CC Robinson

COLONIAL CITIES OF MINAS GERAIS

Ouro Preto is one the main architectonic attractions of a region that pre- serves urban areas dated from the seventeenth century. In order to discover colonial Brazil’s way of life, tourists should visit this mountainous area of the country that prospered as a result of gold mining. Ouro Preto’s churches are temples of beauty. The main square, on top of a hill, charms viewers by Marcello Casal Jr./Agência Brasil Casal Jr./Agência Marcello the simplicity of its scenario. Also worth a visit: the neighboring city of Mar- iana; Congonhas and its Santuário de Bom Jesus de Matosinhos, impressive statues of the 12 prophets sculpted by Antonio Francisco Lisboa, known as “Aleijadinho”; São João del Rey and its seventeenth century palace; and Ti- radentes, the grand jewel of the Brazilian baroque, a small and lovely town.

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