Winning the Young Russian Mathematicians Prize
Nikolay Bogachev, Research Scientist, Skoltech Center for Advanced Studies
Nikolay Bogachev comes from a family of mathematicians and grew to find his own path in this fascinating field. Recently he won a prize for Young Russian Mathematicians in the nomination "Young Scientists". This highly selective award was established by the educational foundation "Talent and Success" and is given for the first time ever. In addition to the award itself, the laureates also receive
a grant to participate in the International Mathematical Congress 2022 in St. Petersburg. It's expected that over time this prize will take the same place as
the prize of the European Mathematical Society.

To celebrate Nikolay's award, we asked him to share his story, what mathematicians do in their everyday life and how to achieve success in the field. Enjoy!
why mathematics
I became interested in mathematics in high school, when I was about 11-12 years old. At that time my parents, who are both mathematicians, brought me to a mathematical club at the Department of Mechanics and Mathematics (dubbed "MekhMat" by the locals) at the Lomonosov Moscow State University (MSU).

At the age of 13, I entered school N54 (now school N171) in Moscow which was particularly strong in maths and where most of the math classes were taught by MSU professors. I participated in several olympiads, and even got some prizes, though I was not very successful at first. I couldn't even make it to the final of the Russian National Math Olympiad. Needless to say, I felt disappointed. Now I can say that it is not so important, since one can become a strong mathematician without any olympiad achievements.

As far as I remember, I always liked geometry. And geometry is everywhere! The ability to draw or to imagine beautiful and complicated pictures has always given me great pleasure. And even more, I always enjoyed considering the same problem from different points of view, for example, to move from geometric language to algebraic and back.

In 2009, at the age of 16, I entered the actual MekhMat faculty of MSU, which is one of the top math departments in Russia and has many prominent professors who are world-class mathematicians. After math school, I was already familiar with some basic notions of higher mathematics, so it was not very difficult for me to be a freshman at MekhMat. This gave me a bit
of extra time, which I invested into attending advanced courses at the Independent University of Moscow (IUM).

In my sophomore year, we had to choose a scientific advisor. At that time, I already knew the late Professor Vinberg and had read his amazing textbook A Course in Algebra. I wanted to choose him as my advisor. However, just to be sure, I forced myself to doubt my choice, and attended various special seminars on algebra, geometry, number theory, and probability. Finally, all my doubts completely disappeared when in Spring 2011 Vinberg gave a fascinating special course "Discrete groups of motions". Thus, starting in Spring 2011 and until my PhD defense in 2019 at HSE, Ernest Borisovich Vinberg was my scientific advisor. It's worth mentioning that in 2014 I graduated with honors from MekhMat MSU, and in 2018 I finished my post-graduate study at MSU.

Let me say a few words about my advisor. Professor Ernest Borisovich Vinberg was a great mathematician and a kind person. He is widely known for his fundamental results and breakthrough discoveries in discrete subgroups of Lie groups, Lie groups and algebras, as well
as in invariant theory, representation theory, and algebraic geometry. To this day, I am very grateful to Vinberg, whose encouragement, constant help and precious advice were so important for me over the years. On the photo below he is playing with my son Sasha.
what mathematicians do
Let me tell you what it means to be a scientist or, more precisely, a mathematician. The main part
of our work is research. In mathematics, this means that we try to solve some open problems (unlike exam or olympiad problems, nobody knows how to solve the "open" ones). In general, we create and study mathematical objects. For example, we can try to find some new interesting properties of already existing mathematical notions or find unexpected relations between them. Sometimes we need to discover new objects or develop some new methods or algorithms.

An important part of the job is to publish papers and communicate our research to colleagues.
For these purposes, we use conferences, workshops, research visits to other universities in Russia and abroad. This part of the job is very exciting: you visit new countries and beautiful places, you meet new interesting people, and make new friends. Recently, Zoom meetings and email exchanges have become more widespread. It is also very important to work with co-authors
with different backgrounds and mathematical styles. This helps to understand and learn more new things, which brings you up to a new level and gives a broader outlook on the profession.

Another part of our work is to give courses and advise students (which means that we should find some appropriate problems for them and show them around the "academic kitchen"). We also organize research seminars, workshops, and conferences.

At Skoltech, I am a Research Scientist without any teaching load, and this is a very comfortable job. I am also an Associate Professor at the Moscow Institute of Physics and Technology (MIPT,
or "Phystech") where I have two PhD students and four undergraduates under my supervision. Besides that, I also have a small teaching load: for example, in Spring 2021 I gave an advanced course "Geometry, arithmetic and dynamics of discrete groups" for students from IUM, MIPT
and HSE.
Research and the Young Russian Mathematicians Award
The award statement of my prize is "for a series of works that represent a significant contribution
to the modern theory of hyperbolic lattices and groups generated by reflections". The topic of this work is at the intersection of such expansive fields as discrete subgroups of Lie groups, geometric group theory, hyperbolic geometry, geometric topology, and number theory. This is a fascinating blend of many beautiful ideas and methods that meet and complement each other.

Let me try to describe what it means in more detail. We will start the discussion with reflections
and symmetries. Mirrors, reflections and mirror symmetries are part of our culture for several centuries or even millennia. Some of us (or maybe all of us) are familiar with a children's toy,
the kaleidoscope, in which colorful pieces of glass form an attractive and amazing pattern.
Such patterns are obtained by multiple reflections with respect to mirrors that have pairwise
angles of π/3. The value of these reflection angles is very important here: if it were different
from the numbers of the form π/k, then we would not obtain such a beautiful pattern.

Generalizing and formalizing this concept, we may consider a convex polytope P in a metric space X and the group Г generated by reflections in the supporting hyperplanes of the facets of P. We say that Г is a discrete reflection group, if the images of P under the action of Γ tessellate X
(i.e., X is entirely covered by copies of P, such that their interiors do not overlap). The polytope P,
in this case, is called the fundamental Coxeter polytope for Γ.

For example, we can tile the plane R^2 by triangles or squares, as shown in Picture 1a. In both cases, such tiling can be obtained from a single cell by the action of the group generated by reflections with respect to the sides of the corresponding triangle or square.

Similar tessellations can be considered not only for reflection groups, but also for many other groups, which we call 'discrete'. For example, the tessellation of the plane by squares in Picture 1b can be also obtained by the action of the discrete group Z^2 of translations by vectors with integer coordinates.

The concept of discrete groups and tessellations can be generalized to many other spaces, for example to Lobachevsky (or hyperbolic) space. Lobachevsky space is very important in mathematics and beyond. The discovery of this space (made by Lobachevsky, Bolyai, and Gauss independently) was probably one of the most important scientific discoveries of the 19th century. Hyperbolic space has many interesting features and plays an important role not only in mathematics but also in theoretical physics and computer science. Some reflective tilings of the hyperbolic plane are illustrated in Picture 2 (different models of the hyperbolic plane are represented there).

I should also add that such tilings of spaces always correspond to the so-called manifolds and orbifolds. For example, consider a tiling of the Euclidean plane by squares under the action of the group of integral translations. Using the vertical and horizontal generating translations we identify the opposite sides of the unit square and obtain a torus after such gluing (see Picture 3 below).

Another picture (see Picture 4) shows how to obtain a hyperbolic surface with one "cusp" (a small infinite end) using a tiling of the Lobachevsky plane. This surface is topologically equivalent to a torus with one puncture (a miniature "hole"). Moreover, every surface of genus g>1 (see Picture 5) is "hyperbolic" in the sense that it can be obtained from some tiling of the Lobachevsky plane.

The situation becomes much more complex when we consider the so-called arithmetic groups. For example, the group of integral automorphisms preserving the standard quadratic form of signature (n,1) naturally acts by isometries on hyperbolic n-space. Such arithmetic groups have many interesting geometric and topological properties.

Most of my results are devoted to building bridges between geometric topology and number theory of hyperbolic manifolds and orbifolds.
For example, in a series of papers I developed a new method for classification of arithmetic hyperbolic reflection groups. Namely, I proved that compact Coxeter polytopes have an edge such that the distance between its framing facets is small enough. This allows me to obtain a complete list of a particular class of the so-called sub-2-reflective arithmetic groups.

In a joint paper with Misha Belolipetsky (IMPA, Brazil), Sasha Kolpakov (Univ. Neuchâtel, Switzerland), and Leone Slavich (Univ. Pavia, Italy) we obtained an effective arithmeticity criterion for hyperbolic manifolds and orbifolds in terms of their totally geodesic subspaces. For example, let us take a fundamental polygon of a reflection group acting in the plane. We can consider this polygon with reflection singularities in its sides as an orbifold. Then totally geodesic lines in this orbifold will correspond to closed billiards inside this polygon. Roughly speaking, our main theorem shows that the corresponding orbifold is arithmetic if and only if it has infinitely many such totally geodesic subspaces. We also show that in this case, all such subspaces have a simple algebraic description. This work results from a synergy between algebra, geometry, number theory, and topology.
Of course, I am very happy to win this prize. There are many strong young mathematicians in Russia and abroad, so it is a great honor for me that my work has received such attention. Indeed, I hoped (but did not expect) to get the prize. I feel that I am fortunate, so I should work with even more energy to meet new goals and expectations.

Also, I am very pleased that the fascinating research domain of reflection groups receives attention since I hope that this research area will attract other young mathematicians in Russia. The domain of discrete (and hyperbolic) groups was significantly developed by such prominent Russian and Soviet mathematicians as Gromov, Kazhdan, Margulis, and Vinberg, but unfortunately, very few researchers in this area have remained in Russia lately.

Finally, I am very grateful to all my friends and colleagues for the great experience of working together. Without their support, none of my present achievements would ever be possible.
influence of your research for the future
I hope that my methods will be used soon for classification of arithmetic hyperbolic reflection groups, which seems to be doable at least in Lobachevsky spaces of dimension 3 or 4. Besides that, together with Sasha Perepechko (IITP RAS and MIPT) and Rémi Bottinelli (Univ. Neuchâtel, Switzerland) we implemented the so-called Vinberg algorithm, which is widely used by experts
not only in discrete groups, but also in algebraic geometry. There have been efforts to produce
a robust and versatile computer implementation of Vinberg's algorithm since 1980's: one would wish for something that applies easily to a large variety of arithmetic groups to avoid creating custom code for each particular case. We started our project in 2017 and now it is virtually complete. This is not a very "mathematical" result, as it mostly concerns creating new software
and programming. However, I hope that our software will become a standard tool in the expert's toolbox because it can be used in a very general setting.

Finally, I expect that our new powerful arithmeticity criterion will shed some light on further open problems in the domain of arithmetic groups, since we created a new method with many interesting consequences.
advice to young scientists
The most important thing is to choose your scientific advisor who will be not only a strong mathematician but also just a good person. Besides that, you should be very patient and be
ready that you will not have any progress during several months or even a year. It is part of the job. It is not so easy to find a suitable research problem to start your PhD, and even more so later on in your career. Last year I tried to solve about 15 problems, and only 3 of them allowed good progress. Every researcher has a period when he or she has a hard time moving forward. It feels discouraging, but such times may and should be used for reading new books and papers, watching online courses, and learning from your colleagues. This is an important part of professional growth that leads to future achievements and allows you to do more interesting maths.
future plans and goals
I am going to work not only on the above-mentioned subjects but will also try to study and understand other branches of pure mathematics. Besides that, I am interested in learning more about possible applications of hyperbolic geometry in the real world. For example, very recently we submitted one paper to a top ML conference where we study nearest neighbor search algorithms in Lobachevsky spaces. Also, I would like to organize some research events in Russia including a conference and a research seminar, where I hope to invite some of the most renowned scientists.

Of course, all my success and future aspirations would not be possible without constant support and love from my family, whom I'm infinitely grateful for.
and now about Skoltech
We are Skoltech – a new international English-speaking STEM university that was founded by the group of world-renowned scientists in 2011 in Moscow, Russia. In just 8 years, we united dozens of researchers and globally renowned professors, built a stunning campus, set up world-class labs and made it to the top 100 young universities in the Nature Index. Read more >>
Discover Skoltech in less than 3 minutes