N E WS L E T T ER
OF THE EUROPEAN MATHEMATICAL SOCIETY

S E
M
M
E S

EuropeanMathematicalSocietyJune 2019
Issue 112ISSN 1027-488X

FeatureDeterminantal Point Processes
The Littlewood–Paley Theory
Artur Avila (photo courtesy of Daryan Dornelles/Divulgação IMPA)Interviews
Peter Scholze
Artur Avila

Research Center
Isaac Newton Institutefor Mathematical Sciences

Interview

A Discussion with Fields Medalist
Artur Avila
Michael Th. Rassias (University of Zürich, Switzerland)
for me, there was some kind of a random process of buying several books, some good and some not, which weremy original source of self-study. I was also very interested
early on in the concept of science, like Physics, chemistry,
etc., even in the primitive way I could understand it asa child. But at the time I certainly had no understanding that being a mathematician could be a career choice.
It somehow seemed a bit more detached from the othersciences. By the nature of the field, what you are usually
initially presented with is very old stuff. You don’t see
modern discoveries, whereas in physics for example, as a
kid you hear about relativity and things that you can associate with recent discoveries and thus feel that the field isactive. On the other hand, you might often get the impression that maths has been done by people in the past andthere is nothing new to do. So the concept of becoming amathematician did not initially exist.

Artur Avila, born in 1979 in Rio de Janeiro, Brazil, is aworld-renowned mathematician with outstanding contributions in a broad spectrum of topics, with his main
work focusing on dynamical systems. His problem-solving mastery, technical power and ingenuity have led him
to the proof of famous and long-standing open problems
like the “Ten Martini Problem” and the “Zorich–Kontsevich conjecture”, to name just a few. He was awarded
the Fields Medal in 2014, being the first Latin American
to win this award. He recently joined the University ofZürich, Switzerland, after spending years as a researcher in the CNRS (France), and remains associated with
IMPA (Brazil) where he got his PhD in 2001.
Honoured to be working under the same academicroof in Zürich, I had the great privilege of meeting Artura few months ago and spending some time with him discussing various topics. With this opportunity the following discussion arose, in which he talks about the pathof his mathematical trajectory, as well as his interesting
views on certain issues.
I know you participated in an IMO and won a gold
medal. From which age did you get interested in mathematics?
My particular interest for mathematics started pretty early on, from the age of 4 or 5. I tried to learn beyond what
was taught at school and I would ask for books in orderto read by myself. Since my parents were not academics
and would not know which books would be good to buy

EMS Newsletter June 2019

Do you think that this might be related to how mathematics is taught in Brazil or is it more general?
I think that this is a more general difficulty that mathematics has with respect to other sciences. This happens
because in mathematics we always build upon foundations. For example, what Euclid did is still valid and thatis why we study it until today, whereas we don’t try tolearn physics, say, according to Aristotle, as that was completely wrong. In physics we are being taught more modern things. Maybe we start with Galileo, but we don’t goback to Aristotle’s time, while in mathematics we do startwith Euclid. It is the nature of the subject to a certain extent. We go further back. So I don’t think that it is mainly
a matter of geography.
So how did you get more actively involved with mathematics?
At the age of 13, I heard about the existence of a national
mathematical olympiad in Brazil without knowing exactly what that was. At the time I was reading calculus books
and related things, but I was not familiar with this type ofproblem-solving. I got in touch with it and I thought that
it was interesting. My first encounters were difficult but
then I got really involved for a few years. That was mynew impression of what mathematics could be. In Brazil
olympiads are organised by IMPA to a large extent. Beingfrom Rio De Janeiro, my fist contact with IMPA occured
when I went there to collect a medal, as IMPA is also in
Rio. So that is how I found out about its existence and thefact that research was done there. Also, IMPA has a very
flexible system which even allows for high school students
to study there and obtain their master’s. So it can happen that a student starts a master’s at the age of 16 and27

Interview
perficial understanding. It would be better to focus moreon something more specific and understand it well. Butof course, the question is where to focus. I advise students
to try to identify, as soon as they can, what attracts them
most and then put more energy into that, since you haveto concentrate on something in order to understand it
and you cannot achieve that by sharing time equally between several subjects. Unfortunately, the system is notdesigned to allow you to do this. Of course, it is easier
to observe where things go wrong than to suggest how
they could be done better. I would suggest to students to
identify a professor and try to get into contact with him/
her in order to get guidance.
Etienne Ghys (left) with Artur Avila (right) at the 2014 ICM in
Seoul. (Photo by Joachim Heinze, Archives of the Mathematisches
Forschungsinstitut Oberwolfach)finishes it at 18 and then obtains his PhD at the age of 21.
There are examples like that.
You are one such exceptional example. You started yourPhD studies at the very young age of 19 at IMPA, correct?
So yes, when I found out about the existence of IMPA I
thought it would be nice to get involved. In my mind, I
imagined that it could probably help me with my olympiads training. Once I got a gold medal at the IMO I got
an invitation from IMPA to take part in a pre-master’s
programme: a sort of scientific initiation program. It wasa special course called “Introduction to Topology”. I
got a fellowship, I did well and then started the master’s
programme while in the last year of high school. Subsequently I commenced my PhD there as well.
Do you believe that the strategic way of thinking you
had cultivated as an olympiad caliber problem-solver
helped you later when approaching problems on a research level?
The olympiads gave me confidence to start with. So when
I started the master’s programme I felt that it was something I could really do and I worked intensively for it. But
in research you learn how to think in a different manner.
Of course tricks and little techniques inherited by the olympiads may be helpful not to waste time occasionallyas you may recognise easily that something might fail towork, but in research you have to think about a problem
whose solution is unknown. In competitions, the problems are designed to be solved. There is of course creativity involved, but the problems are such that they can besolved by the use of a specific toolbox. In research you
deal with uncertainty. You also have to be able to recognise whether you should continue on a certain path ormaybe start researching a little bit to the side, and know
when to come back to the original problem again.
What is your advice to young talented students in Brazil
and elsewhere?
I feel that the students have to take so many courses
at the university that they have no time to engage at a
deeper level with any of them. With so many requiredclasses per week, they can only realistically strive for su-

28

You recently joined the University of Zürich (UZH),
while prior to that you were in Paris. How do you findthe new environment here?
The environment and the mathematical centre in Zürich
is very active. There is a very good synergy between the
UZH and ETH. An increasing number of good researchers stay here as well as just pass by, and thus the seminars
are very active. Even though Paris, for example, has many
more people, one can say that to some extent after a certain size of community you cannot really absorb everything that is going on. It is impossible to attend all of theseminars, meet with all of the people, etc. The nice thing
about Zurich is that the size is such that you can absorb
what happens and there is a good number of things happening. At the UZH we increased the dynamics groupas well. Apart from me, two other researchers from Bristol have joined. We also established an “Ergodic theory
and dynamical systems” seminar that is already becoming very active. We hope that it will be one of the central
seminars in dynamical systems in Europe.
Was there a specific paper, book, lecture, or even a theorem you came across that had a lasting impact on you
to the extent that it made you chose to become a mathematician?
In my case it was a long experience and a very gradualprocess. There were many things that grabbed my interest and several theorems that I liked while growing up.
Additionally, my mental understanding of what mathematics is progressed over the years and I realise now
that I was fortunate, since there were several stages in mydevelopment where the process might have not workedout so well. In general, a student might like olympiads
and really enjoy problem-solving but could not be so
attracted by the theory learning that happens at, say, inmaster’s program. Or they may like learning all thosewonderful theorems, but might not find it pleasurable totry and prove new theorems. Eventually ,what leads you
to becoming a researcher is a path that comes from manydifferent directions. But there are several little things in
the process that might not work out for someone andlead them elsewhere.
So in my case there was no particular theorem oropen problem that lead me to mathematics, such as Fermat’s last theorem, Poincaré’s conjecture, etc. It was along and gradual process.

EMS Newsletter June 2019

Interview
What is the first thing that comes to mind when you arethinking of the word “mathematics”?
Generally, my way of thinking is that of an analyst. So,
when I think of “mathematics” the first things that cometo mind are “analysis” and “inequalities”. Inequalities
are really central to my way of thinking.
Is there a mathematician who influenced you the most?
Either through your mutual collaboration or interaction or even by studying his work?
There were several people who were important and influential. In research you have your own style, but it is very
good to be exposed to several other views and possibilities.
There is also a lot of advice you can get. You can of coursechoose to adopt some advice and some not. My PhD advisor Welington de Melo was influential to me. I was also
influenced by Mikhail Lyubich, with whom I worked a lot.
Later there was also Jean-Christophe Yoccoz, whose style
was very different. These mathematicians had their own
particular styles and I was exposed to that as I interactedwith them directly. Those were close to me, but of course
indirectly I was influenced by more distant people. For example, when I was shifting a bit between fields, I foundJean Bourgain’s work to be an inspiration. His style andabilities sparked some admiration for the corresponding
fields. His style of doing analysis is something I like verymuch and I have the highest respect for him.
You started your PhD studies at the very young age of
19 at IMPA. Your doctoral advisor there was Welington
de Melo who was working on dynamical systems andparticularly on one-dimensional dynamics, which is avery demanding field. Did he allure you somehow towards that dynamical systems or had you already chosen your path?
When I initially went to IMPA I had a very weak background and I was just finding out about the possibility of
doing research. I wanted to learn the basics at the timeand I was concentrated on trying to do well at my Master’s programme. During that period I attended a PhD
course in differential topology by de Melo. I was not very
keen on attending that course at first, as de Melo had thereputation of being very tough with students and it wasconsidered difficult to get a good mark in his courses.
But it went well and he liked me. After the completionof the course he approached me and he proposed that
we discuss some topics in complex analysis, quasiconformal mappings, etc. Through that process he naturally
became my advisor. At IMPA, dynamics was a top field
and they were strong at it. De Melo was a bit isolated,though. He was working on one-dimensional dynamics:
a field that had already become very important but was
and remains very hard. So one had to put all ones energy into this specific field and it made it hard to interact
with others working on different things. The techniquesand main problems are very difficult. An example of thecomplications involved is the following: we often have towork with fractal sets with very complicated combinatorics, and just to get the correct language to describe theirstructure is very hard. In analysis in some cases you might

EMS Newsletter June 2019

Artur Avila (left) with Ashkan Nikeghbali (right) at the University ofZürich.

just take a dyadic division, break everything into squares
and so on. But here you will need to do this in fractal sets,
and just the description of the pieces is complicated. You
have to describe the set and the topology is very intricate.
For this reason, until today, whenever I have to supervise
PhD students I tell them that if you do one-dimensional
dynamics you will have to put all your effort and energy
into that particular field, which is very specialised. Otherwise, we can cover almost everything else in dynamics. So
I often propose to them to follow the second option, as
it might not be a very good idea to get too specialised soearly on. Of course, when I started I did not know any ofthis, and I just said OK and worked on one-dimensional
dynamics. It turned out to be very interesting and very
hard as well. Later I learned more things and expanded
in other directions too. I still work on one-dimensional
dynamics, but now most of my work is on other things.
One may state that mathematics has witnessed great expansion during the last, say, one hundred years, withmany different areas emerging and various methods discovered, bridging seemingly different fields. How do you
see the future of mathematics in that respect? Do you
think that interdisciplinarity might be the theme of the
future for example?
Things tend to become more and more technical and thus
people get further apart as they don’t speak the same language. It has become clear that for an analyst it is moredifficult to understand what they are now doing in algebra. My view on this is that people should just do what
they want. If someone wants to think about prime numbers he should do that, if he wants to think about fractals then that is fine as well. Some researchers will makeprogress working on isolated topics and others mightwant to bridge together areas and examine the connections between discoveries. Not everyone has to do that.
Some people might just want to play on their own. You
cannot anticipate what is going to work out. It is also notnecessary to decide which fields of mathematics have to
be stimulated, since we don’t know whether the next discovery in geometry will depend on a discovery in algebra
and so on.
This is also my point of view regarding focusing on
pure or applied mathematics. What might be crucial in29Interview
the solution of some very specific problem (with industrial applications or whatever) often happens to be justsome beautiful object that someone studied for its own
sake. The motivation of the latter (often similar to themotivation a child has for playing a game) is irrelevantto the fact that what has been discovered turns out to be
useful to the former. And of course it works both ways:
much of my work deals with theories that first arose inconnection to physics, but for me what matters is thatthese theories are rich mathematically, and I study themfrom this perspective. So “pure’’ mathematicians have torecognise that many of the nice objects they have at theirdisposal with rich mathematical theorems that one works
on just because they are beautiful, without caring about
the motivation or the application, might have not been
discovered if it was not for some physical model thatsomeone was examining because they cared about thephysics of it. Thus, by admitting this and letting people
work on what they want, the whole community benefits
from the eventual (unpredictable) interactions.
It has happened many times in the past that great mathematicians moved from area to area, making contributions to various different mathematical domains. Stephen Smale is one such example, part of whose work
is on dynamical systems like yours, but who has also
worked in topology and mathematical economics withthe latter being fairly distant from the other two fields.
Do you see yourself exploring completely new researchterrains in the future?
With Smale there was a natural evolution from topology to dynamical systems, since part of his work towards
Poincare’s conjecture was already leading him to understand certain types of dynamical systems. Sometimes ofcourse there is an explicit desire for one to do something
different. So I believe that Smale actually forced himself
to turn to another direction. In my case, things happened
to evolve in directions that I had not anticipated. It has
happened that I have turned my attention to somethingwhich I found attractive and later it turned out to be related to things I had been doing. The connection proved
to be deeper than I would initially have expected. I assume that this will tend to continue to happen. Personally, I do not put in a great effort to work on different
areas, except if I find something attractive and there is anatural progression that leads me there.
On a humorous note, if – like in the case of Ramanujan– there were a supreme being that could hand you solutions to important problems in your sleep, and say thatyou could have only one such dream, which problem
would you like to see the solution for?
Mm… The first thing that comes to mind is the local connectivity of the Mandelbrot set. I don’t know whether I
would change my mind if I put more thought to into it,
but this is the first problem that I thought of when you
asked me this question.
I don’t actively work on it. I could possibly work on
it in the future if the corresponding techniques have30

evolved, but it is certainly a beautiful problem that is
captivating.
Many scientists from a broad spectrum of areas haveexpressed various views and opinions about the possible future consequences of the advancement of artificial
intelligence (AI) in connection to the so-called “AIcontrol problem”. Being a young scientist, during whoselife it is very likely that great advances in AI might beachieved, affecting our lives, do you have any views on
this?
Being a dynamicist, my training has taught me to a large
extent that there are a lot of limitations when we makepredictions. To my understanding, when it comes to AI
peoples’ hopes might be too high at the moment, as one
tends to get too optimistic following a few unexpectedsuccesses. It is of course correct to continue the effort,
as probably there is more success to come, but it is reasonable to imagine that obstacles will be identified sothat things will not advance as fast as it was hoped. Even
though it has happened many times in science that several things that we didn’t think could be done in a specific timeframe were actually achieved and we were very
surprised by the outcome, we should not get too carried
away. So, I am not particularly scared about the consequences of the advancement of AI, since expecting big
advancements in AI is already large assumption. Expecting that AI will become something applicable to almost
all situations involves too much wishful thinking, even in
the long run.
When we learn more about a corresponding domain
we are also able to better understand its limitations. The
point is to try to discover directions where the techniques
allow you to go. It is very important and at the same timedifficult to know what you cannot achieve. A lot of goodquality research is conducted to see exactly what is impossible to be done, in order to know what the barriersof the techniques are. Barriers that will not be overcome,since they are theoretical ones. That is, it is important toknow that it is not a matter of putting more effort towards a goal if that goal is really intractable. This is what
happened in mathematics when we learned that there
are unsolvable problems and all kind of limitations toformalism. Our understanding changed completely from
Hilbert to Gödel, could this have been predicted? Intechnology, while there was very fast progress with theclock speed of chips, we could anticipate that this needed
to stop at some level as there were known physical limitations. Maybe, with respect to AI, there are still unknown
theoretical limitations hiding in information theory oreven dynamical systems.
Michael Th. Rassias is on the Editorial Board of the EMS
Newsletter. His photo and CV can be found in previous
Newsletter issues.

EMS Newsletter June 2019