‘Our Subject Is Absolute Truth in the Literal Sense’

Valery Gritsenko

Photo courtesy of Valery Gritsenko

What path does one have to follow to become a theoretical mathematician? Is there such a thing as a ‘mathematical personality’? And is it really true that all mathematicians are a bit eccentric? Valery Gritsenko, Professor at the Faculty of Mathematics and Head of the International Laboratory for Mirror Symmetry and Automorphic Forms, talks about 13-year-long referee reports and good films about mathematics.

— How did your academic career develop before and after you joined HSE?

— I began my schooling in the scientifically romantic 1960s. I spent three years at the physics and mathematics boarding school affiliated with Leningrad University. After graduating from the Faculty of Mathematics and Mechanics of Leningrad State University, I spent two years as a postgraduate student at the Leningrad (now St Petersburg) branch of the Steklov Mathematical Institute of the Academy of Sciences (LOMI RAS). At the beginning of 1980, I defended my Candidate of Sciences dissertation on Siegel modular forms in the field of Mathematical Logic, Algebra, and Number Theory. Today, that legendary period is described by some as the era of Brezhnev’s stagnation, and by others as a time of flourishing. In any case, the rapid development of science worldwide at that time is a well-recognised fact. In 1979, I began working at LOMI.

In 1988, at the invitation of one of the world’s leading experts in number theory, Don Zagier—with whom I maintain scientific contact to this day—I spent three months as a visiting researcher at the Max Planck Institute for Mathematics in Bonn. It is one of the major mathematical centres in Europe.

The theory of automorphic forms, which is my main field of research, originated in Germany at the beginning of the 20th century. This visit allowed me to enter the research mainstream, to give talks at colloquia at leading German universities, to meet in Hamburg such legendary mathematicians as Ernst Witt and Erich Kähler, to work in Heidelberg with the patriarch of our discipline, Hans Maass, and to discuss new directions in Göttingen with Walter Bailey from the United States.

It was a discovery of the wider mathematical world, in which Soviet mathematicians were fully fledged participants.

In 1989, at the first USA–USSR Symposium on Algebraic Geometry in Chicago, I became closely acquainted with the leading algebraic geometers of Moscow. This proved to be the starting point of the path that eventually led me to HSE University. From 1990 onwards, I worked in Germany, Italy, Switzerland, England, and Japan. In 1998, I became Professor of Mathematics at the University of Lille. In Europe, I collaborate with specialists in automorphic forms, algebraic geometry, string theory (‘quantum gravity’), coding theory, and algorithmic complexity. Since 1993, I have been making fundamental joint projects with mathematicians from Russia, Germany, England, the United States, and China on infinite-dimensional Lie algebras and their applications (this series of joint works with Vyacheslav Nikulin of the Steklov Mathematical Institute and the University of Liverpool was nominated for the State Prize), on the algebraic geometry of moduli spaces, and on the theory of theta blocks and their applications in algebraic and arithmetic geometry. Together with Klaus Hulek (Hanover) and Gregory Sankaran (Bath, England), we managed to find a solution to the last open problem from André Weil’s famous programme on K3 surfaces, formulated in 1956. I continue to work jointly with my students in China, Russia, and England.

My first in-person contact with HSE took place in 2012 at a summer school organised by Fedor Bogomolov’s laboratory in Yaroslavl.

Photo courtesy of Valery Gritsenko

I liked the students of HSE’s Faculty of Mathematics so much that eventually I moved from France to Moscow to work here.

The history of the International Laboratory for Mirror Symmetry and Automorphic Forms (ILMSAF) began in 2016, when the well-known American mathematician and Moscow State University graduate Ludmil Katzarkov and I applied for a megagrant. The activities of the laboratory can be traced, at the very least, through the titles of popular-science articles about us: ‘Symmetry, Lunatics and Monsters: How the Theory of the Quantum World Is Built,’ ‘Hunters of Special Functions, or “Sirius” on the Eve of the Virus,’ ‘A Place Where Fantastic Ideas Are Proposed,’ ‘Young Stars of Mirror Symmetry Gather in Moscow,’ and others.

Purely scientific formulations of the results obtained in the laboratory are unlikely to be of much interest to you. However, I will say a few words about the most recent ILMSAF scientific conference, which took place in October 2025. It featured the presentation of an entirely new algebraic theory developed by associate members of the laboratory: one of the most highly acclaimed theoreticians in the world, Maxim Kontsevich; the aforementioned Ludmil Katzarkov, the scientific director of our laboratory; Tony Pantev; and Tony Yu. The project to construct a new theory of birational invariants was included in the laboratory’s research agenda in 2019, and on October 1–4, 2025, its world premiere took place as part of our joint project with China. During those days, a solution was presented to one of the principal classical problems of birational geometry with a century-long history—the problem of the rationality of a general four-dimensional cubic.

— What qualities make a person well suited to mathematics? Can one speak of a mathematical character?

— To become a mathematician, one needs mathematical abilities. But what exactly does that mean? Here is how an AI answered this question for me: ‘The concept of “mathematical abilities” is complex and multifaceted. It is not simply the ability to calculate quickly. It is a combination of cognitive processes that enable a person to work effectively with abstract concepts, spatial representations, logical structures, and symbols.’ A little dry and formal, but generally correct. For my part, I would particularly emphasise the capacity for systemic and structural thinking, the ability to extract the core idea from a general line of reasoning, to grasp and articulate a hierarchy of meanings, and to substantiate and reveal all possible connections between them. To this must be added the ability to solve problems where it is initially unclear how to approach them at all.

To identify other qualities of a mathematical character, let us briefly go through all the stages of training a mathematician.

We begin with specialised physics and mathematics schools, where students are required to constantly solve large numbers of numerical, combinatorial, geometrical, and logical problems. In a good school, over the course of a couple of years, students run through a fairly complex information field of ‘elementary’ mathematics that has taken shape over many centuries. It is precisely this introduction to mathematics as part of global culture that distinguishes the curriculum of Russian physics and mathematics schools, which are an absolutely unique product created in the USSR.

At the mathematics faculties of leading universities, the volume of material that needs to be mastered increases many times over. In compulsory undergraduate courses, students must assimilate the mathematical apparatus developed roughly between 1700 and the 1960s (differential and integral calculus was created as early as the late 17th century). At the same time, they are introduced to modern mathematics through specialised courses. The ability to absorb a large volume of material is an essential skill for a future mathematician.

Our discipline is absolute truth in the literal sense. Mathematical results remain true—absolutely—even a thousand years later, despite radical changes in the objects of mathematics, its methods, and even the mathematical language itself.

In master’s and doctoral programmes, students begin to tackle creative problems. The ability to be inspired by the difficulty of a question is another important quality for a mathematician. Many of our students go on to become successful in other fields. But if a person nevertheless decides to continue studying mathematics, they will have to spend a further two years in a master’s programme, four years in doctoral studies, and then another three years or so as a postdoc, provided they are successful. After that, they may teach mathematics at a university, but this still does not mean they will become a theoretical mathematician. And if they are lucky (or unlucky?), they will have to continue this entire process of learning indefinitely.

— How accurate is the idea that mathematicians are eccentrics, completely detached from the real world in their abstract reflections?

— Such examples certainly exist, and it is clear that they are the ones most widely publicised, but overall this is a myth. In reality, the majority of people who have successfully completed the full cycle of mathematical education are capable of carrying out any creative work (including managerial roles) involving the planning, organisation, and maintenance of the functioning of complex systems. This is a fact. For many years I have worked in the Paul Painlevé Mathematics Laboratory at the University of Lille. Who was Paul Painlevé? A major mathematician of the late 19th and first third of the 20th century, one of the founders of the analytic theory of differential equations, who at the same time served twice as Prime Minister of France. His government cabinet included another renowned mathematician, one of the founders of measure theory and modern probability theory, Émile Borel, who was Minister of the Navy.

Let us take a more recent example. Nicușor Dan, elected President of Romania in 2025, won first place at the International Mathematical Olympiads in 1987 and 1988. He graduated from the most prestigious mathematics faculty in Paris—the École Normale Supérieure—wrote a strong dissertation in arithmetic algebraic geometry, and published several papers in leading mathematical journals. Another similar example is Cédric Villani, a recognised mathematical genius, winner of the 2010 Fields Medal, an intellectual and a well-known French politician. Alexander Gerko defended a Candidate of Sciences dissertation at Moscow State University on a topic as abstract as one could imagine—'Homological Dimensions and Semidualising Complexes’—and then went on to become one of Europe’s leading traders specialising in high-frequency algorithmic trading on the stock market. In 2023, The Sunday Times named him the largest taxpayer in the United Kingdom. Recently, according to The Times, he decided to open a mathematics school in London for children aged 11 and above.

— What is distinctive about the preparation of a mathematical publication? In one of your recent interviews, you said that a mathematical paper can be under review in an international journal for a year or even longer. What accounts for this?

— Yes, that is true. I have had cases where I received referees’ reports on a paper a year and a half to two years after submission to a journal. Professional peer review plays a formative role in mathematics. Understanding and checking every detail of someone else’s work is often difficult, especially if the paper contains genuinely new results. Sometimes such an assessment is carried out by an entire seminar over the course of one or two semesters, rather than by a single referee. So publication of a paper three years or so after submission to a good journal is a fairly common occurrence. This circumstance also has to be taken into account when drafting reports for a potential grant, which is usually awarded for two years.

There are even more complex examples. In 2012, the exceptionally gifted Japanese mathematician Shinichi Mochizuki, a recipient of many prestigious awards, presented to the public four substantial texts (over 700 pages) outlining his new theory of inter-universal Teichmüller geometry. One of the consequences of his theory is a solution to the famous abc conjecture, one of the central problems of modern number theory. The proposed texts are publicly available and are periodically revised by the author. Major scientific conferences have been devoted to these preprints, yet the validity of the proposed theory has still not been verified over the past 13 years, and work on it continues.

Incidentally, one of the authors of the abc conjecture, the French mathematician Joseph Oesterlé, delivered a series of lectures at our laboratory’s summer student school in 2019—which speaks to the level of our research and teaching.

Photo courtesy of Valery Gritsenko

— How does the HSE Faculty of Mathematics differ from other Russian and foreign universities in terms of teaching mathematics and organising mathematical research?

— This question would be better addressed to the founder and first dean of the Faculty of Mathematics, Sergei Lando. I can only share my own impressions of working at one of the youngest mathematics faculties in Russia (founded in 2007). In fact, its youth is its distinguishing feature.

The teaching staff represents a variety of Russian mathematical schools, which creates excellent conditions for the synthesis of research traditions and ensures openness to new talent from across the country.

This distinguishes the HSE Faculty of Mathematics from ‘classical’ faculties, where a professor is almost always a graduate of the same faculty.

The research potential of the Faculty of Mathematics includes four international and two Russian research and teaching laboratories, placing it on a par with the best European mathematics faculties.

— Recently, HSE has been actively cooperating with Chinese universities, including through the laboratory you head. How does the organisation of mathematical research in China differ, given that, according to many accounts, it is at the very highest level?

— This is the most important new direction of our international cooperation. Scientific contacts with the West have largely been frozen, while China is rapidly becoming a centre of attraction for the global mathematical community. After the International Congress of Mathematicians (ICM) scheduled for Moscow in 2022 was cancelled (the congress is held every four years), China organised the International Congress of Chinese Mathematicians (ICCM), whose scale surpassed that of the ICM. As part of HSE’s International Academic Cooperation initiative, our laboratory is implementing a joint project with the Beijing Institute of Mathematical Sciences and Applications (BIMSA) at Tsinghua University. The head of the institute and of this programme on the Chinese side is Shing-Tung Yau, one of the most renowned and influential mathematicians in the world. The institute itself was established several years ago as a major international mathematical centre.

Photo courtesy of Valery Gritsenko

— What place does interdisciplinarity occupy in contemporary mathematical science?

— Many people perceive mathematics as a completely abstract science. This is true only in form.

Mathematics is the foundation of practical activity.

In fact, in the 21st century practical activity itself often turns out to be engineering mathematics. It is sometimes not so easy to distinguish between the two. And what lies ahead in the near future?

From the outset, our laboratory was conceived as multidisciplinary. The so-called mirror symmetry effect, or a new phenomenon of duality between superconformal field theories, was discovered by theoretical physicists. This new effect provided physical solutions to several serious mathematical problems in enumerative geometry. An adequate description of this phenomenon in theoretical physics required a new mathematical apparatus, which indeed emerged in the 21st century: categorical geometry, non-commutative Hodge theory, and new functional and arithmetic objects. Science is entering an era of a new synthesis. Fundamental mathematics is now finding new problems not only in physics, chemistry, and biology, but also in computer science. In the latter, theorems are usually not proved; instead, statistical information is obtained and processed for virtually all mass processes—weather, sociology, elections, the stock market, the economy, and so on. Everything has become mathematics.

The formulations of great mathematical problems have reflected new structures of the numerical, geometrical, logical, and physical world. Their solution has required the creation of new mathematical disciplines and theories, which in turn have contributed to breakthroughs in physics and technology. Today, society is confronted with entirely new informational challenges. How can we comprehend and describe the structures within this global flow of information? How can we work with it without drowning in its sheer volume? This is the task of fundamental mathematics and of mathematicians, whose work consists precisely in formalising and finding solutions to the most complex problems in any field.

Today, mathematics is moving beyond its traditional boundaries and becoming a subject of broad public interest (much as physics did in the second half of the 20th century!). I have found around fifteen full-length feature films about mathematics made over the past ten years. In addition to Oppenheimer, these include Marguerite’s Theorem (2023), about an attempt to solve Goldbach’s conjecture; Adventures of a Mathematician (2020), about Stanisław Ulam, one of the founders of the scientific group at Los Alamos; and The Man Who Knew Infinity (2015), a large-scale historical film about Srinivasa Ramanujan.

— How do you assess the potential and prospects of online mathematics programmes?

— Online programmes significantly expand students’ opportunities. In effect, they provide access to supplementary education. All the students I work with in the laboratory have experience of online learning. At the same time, the first compulsory in-person courses are an absolutely essential foundation of genuine education. One can only grasp mathematics here and now, in real time, through face-to-face lectures. However, university courses cover only part of mathematical education—its basic minimum. Moreover, compulsory courses inevitably carry a host of historical and administrative constraints (mandatory tests and exams), as well as social ones (professors want students to learn exactly what they themselves once studied), which do not always help either learning or research.

Online learning programmes are an accomplished fact. Our new partner and tool is artificial intelligence systems. AI solves all the mathematical problems of the first semesters, olympiad problems, and so on, more than successfully. Not to mention writing essays on any topic in many languages. This is not the invention of writing, printing, or the internet. It is something entirely different—almost like a new fairy tale or a unique magic wand for teachers, students, and anyone who is not too lazy to think.

The entire global education system is beginning to change before our eyes. And this is not the future tense—it is our present.

I derive enormous human pleasure from interacting with AI programmes that are beginning to take into account my personal mathematical preferences. Human capabilities in teaching and learning are reaching a completely new level. But that is a topic for a separate conversation.

Read the interview in Russian on Schola