# 'I’ve Liked HSE and Moscow a Lot'

Carlos Cortez, a bronze medallist of the International Mathematical Olympiad (2011, 2012, 2013) and a Mathematics graduate of MIT, recently spent two months during his summer break completing a research internship at the HSE Faculty of Mathematics. He will soon be pursuing a Master’s degree at University Paris-Sud in France followed by a PhD at Northwestern University (USA).

The research internship was made possible through a cooperation agreement between MIT International Science and Technology Initiatives (MISTI) and HSE. This internship differs from standard exchange programmes first by its the duration – it is much shorter and not strictly tied to the academic calendar. Second, the tasks that participants complete over the course of their stay at a foreign university are primarily aimed at carrying out research projects rather than completing coursework, although participants are certainly welcome to attend courses during their internship. The internship at the Faculty of Mathematics usually involves the completion of an academic paper under the guidance of a faculty member.

## From Contests to Research

Carlos first got into mathematics around the age of 12 when he won a math contest without having really prepared for it. Enjoying the prize, he continued to take part in competitions, which eventually led him to pursue mathematics as a career and seek admission to MIT after completing high school in Ecuador.

In competitions, there is a lot of focus on problem-solving while in research there is often as much focus on theory-building – at least that’s what I realized that I liked. Sometimes it requires spending days, weeks or even months in order to understand the knowledge to get to some final result. But the final result might be so beautiful that it makes it all worth it

‘In mathematics competitions, the goal is quite well-defined – you have four and a half hours to solve some given problems’, Carlos says, recalling his experience. ‘You know that solutions exist to these problems, so you may want to think about how the person who created them thought when creating them. On the other hand, when doing mathematical research, you are not really sure what you are looking for. You may have an idea of what you want to explore, but you are grasping in the dark to see where you can find some light. Sometimes you don’t, sometimes you do, but you are not really sure what you are going to get out of it when you start’.

Recognizing the transition from ‘Olympiad thinking’ to mathematical research took Carlos about two to three years in college, which he’s not sure that he’s completed entirely.

‘I think I was in my third year of college when I realized that mathematical research will require a lot more patience than mathematical competitions do’, he says. ‘In competitions, there is a lot of focus on problem-solving while in research there is often as much focus on theory-building – at least that’s what I realized that I liked. Sometimes it requires spending days, weeks or even months in order to understand the knowledge to get to some final result. But the final result might be so beautiful that it makes it all worth it. Whereas in competitions after three hours you usually get all you can out of a problem. Also, the problems in competitions fall into four areas – algebra, geometry, number theory and combinatorics and the first three are actually quite far from what the real mathematics is like. Only combinatorics is somewhat close to its counterpart in the research world of mathematics’.

## Research Internship in Russia

Carlos’ journey to Russia began in his second year at MIT, when students are asked to develop a concentration apart from their math classes. Eight classes in the humanities are required for this, four of which must be in the same subject. He landed on Russian language as a subject, a decision motivated in part by the fact that math graduate schools often ask applicants to have some knowledge of a foreign language, such as German, French, Russian, or Chinese.

As Carlos became more immersed in Russian culture and literature, he also grew interested in the mathematical school of Russia and how it differs from the US, France, and the UK.

‘I initially heard about HSE from Alexandra Viktorova, whom I met at the International Summer School in Mathematics for Young Students in Lyon, 2012. She recently graduated from the HSE Faculty of Mathematics and is currently doing a PhD at Stony Brook University’, he said.

I think Russia has a considerably greater tendency towards early specialization into theoretical fields. In the US, students come to college not really knowing what they want to do – whether it should be math or computer science – they might decide in their second year. In Russia, it’s much more focused

‘When MIT and HSE made the offer of a research internship, there was a lot of flexibility in what it would entail’, Carlos said. ‘My project is of a combinatorial flavour, although it originated in knot theory. You can think of a knot as a shoe lace that is tangled in space but has its ends tied. It’s a very difficult problem to distinguish whether two knots are different, meaning whether we can turn one knot into another knot without separating the ends of the shoe lace. The original idea was to study some weight systems on chord diagrams, which are a combinatorial counterpart of the so-called Vassiliev invariants or finite type invariants on knots. Incidentally, Professor Viktor Vassiliev is now the Head of the Joint department with the Steklov Institute of Mathematics at HSE’.

In addition to doing research, Carlos has attended some lectures at the Russian-Chinese Conference on Knot Theory, which helped him to determine that he would keep the field as a side interest although not his main one. He also attended the summer school in algebra and geometry, which is organized jointly by HSE and Yaroslavl State Pedagogical University.

‘It was a great school!’ Carlos recalls with enthusiasm. ‘It was a bit tricky because my Russian is still not good enough to follow all of the oral lecturers, but I could follow what was written on the board, which helped a lot. Besides, after listening to six hours of lectures each day, I feel that I understand Russian much better than I did when I first came here. I met a lot of interesting people at the school. Some of them are HSE graduates who are also going to France’.

## Mathematics in the US and Russia

In the US, students are expected to start college with little knowledge of math. There is a lot of flexibility, meaning that students can take harder classes, but the general guidelines of what students should know by the time they start university or by the time they graduate is not very high. This contrasts with what Carlos has seen in the first- and second-year students in the algebra and geometry school in Yaroslavl, which is that in Russia some people get into theoretical mathematics really early.

Having all mathematicians in one place, like at HSE, can help to “live mathematics” and specialize much more because you are immersed in this world

‘Third- and fourth-year Bachelor’s students, Master’s students and PhDs seem to be younger in Russia than in the US on average. It’s striking’, he says. ‘I think Russia has a considerably greater tendency towards early specialization into theoretical fields. In the US, students come to college not really knowing what they want to do – whether it should be math or computer science – they might decide in their second year. In Russia, it’s much more focused’.

Carlos has found that the difference between HSE and MIT in particular is that the Faculty of Mathematics at HSE is separate from the rest of the university, so there is a lot of interaction among math students, whereas at MIT the Math Department is quite close, geographically and academically, to the rest of the university, which gives students breadth to explore other disciplines.

‘Having all mathematicians in one place, like at HSE, can help to “live mathematics” and specialize much more because you are immersed in this world’ he says. ‘The Faculty of Mathematics seems great. It’s very active and in general Moscow is a very busy place in terms of having math talks and seminars. The classes at HSE seem quite high level and interesting. I had heard very good things about HSE before coming and I am quite pleased to see it in person. I’ve liked HSE and Moscow a lot. I would definitely enjoy spending more time here’.

## Personalized Guidance

Carlos’ academic advisor during his internship was Professor Sergei Lando. The two initially met during a visit by Professor Lando to MIT in March 2017 when he recommended literature for Carlos to study and formulated possible research tasks; this allowed him to prepare for meaningful discussions.

'The two months spent by Carlos in Russia were done so effectively, in my opinion’, Professor Lando says. ‘Choosing the summer months for such an internship comes with a natural disadvantage: people who are the same age as the intern are spread out all over the place, and as a result, the environment for potential interaction is diluted substantially. Carlos demonstrated a high level of maturity and ability to work independently. Summer time does have its own merits, though. For example, I was able to devote more time to him than I would have during the academic year’.

Vladlen Timorin, Dean of the Faculty of Mathematics, expanded, saying, ‘More often than not, in addition to immersion in some actual mathematical problem, studying the relevant literature and trying to achieve their first independent results, we try to introduce our interns to Moscow’s mathematical life, which remains quite active even during the summer. We also aim for them to participate in seminars and summer schools on topics close their area of research, as well as to interact with our students working in related fields’.

A formal outcome of the internship is most often a report that often forms the basis for a future scientific article, which is frequently one of the first for interns. For students and graduate students, the internship is a unique opportunity to learn about the work of world famous scientific schools from within and expand their mathematical horizons. For HSE’s faculty members, it is a way to talk about the university’s programmes to potential motivated applicants and an opportunity to start promising cooperation with future scholars.

‘In many Western universities – for example, in France – a long internship abroad is compulsory for students of a certain year of study’, notes Professor Lando. ‘Several years ago, I had a rather positive experience communicating with a student from the Lyon Higher Normal School. These students are natural candidates for similar internships. In recent years, similar requirements are spreading to Russian students. At the Higher School of Economics in particular, most students at the Master’s and post-graduate level go through similar internships in various countries around the world, although coverage doesn’t seem to be at 100% yet’.

Every teacher who leads his or her own research activities and directs similar activities of students and graduate students has a selection of tasks and subjects that are suitable for discussion with students. The same tasks are also offered to potential interns, who choose the right one for themselves. Tasks are then further specified and take on a concrete form, according to Professor Lando.

‘You can’t count on fundamental breakthroughs over a period of just two months’, he says, ‘so I try to focus the intern’s attention on mastering various approaches to solving the problem and on possible applications of the future result, even if it’s just a partial one. In the future – after the internship is completed – a former intern will be able to work on the task independently using the methods that have been mastered’.

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## International Laboratory for Mirror Symmetry and Automorphic Forms

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