‘Algebraic Geometry Is the Geometry of Ideal Forms’

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Founded 15 years ago at HSE University, the Laboratory of Algebraic Geometry and Its Applications studies fundamental mathematics and helps to shape a unified language of mathematical science. The laboratory has become a well-known and respected research centre; recognition of its achievements includes presentations by its staff at international mathematical congresses and publications in leading global mathematical journals. The HSE News Service spoke with the head of the laboratory, Professor of the Russian Academy of Sciences Dmitry Kaledin, about its work.

— When was the laboratory established?

— We were one of the first two international laboratories at HSE in 2010, but I would start the story earlier, in 2008, with the creation of the Faculty of Mathematics and the period when HSE began to transform from a specialised economics university into a full-fledged research university. In 2010, the first government megagrant competition was held; we won it and gained the opportunity to carry out large-scale research, including fundamental research. We have been working in this format for 15 years now.

— Tell us about the academic supervisor of the laboratory.

— Fedor Bogomolov is an outstanding scholar, a true icon of science from the famous school of Igor Shafarevich, combining profound mathematical knowledge with broad intellectual horizons. It should be understood that geometry is often difficult to formalise and relies heavily on intuition, and Fedor Bogomolov possesses exceptional intuition. Under Shafarevich’s guidance and together with Andrei Tyurin, Vasily Iskovskikh and others, he created the Moscow school of algebraic geometry. Bogomolov’s PhD thesis was a breakthrough and became a true foundation for an entire branch of algebraic geometry. The theory of Calabi–Yau manifolds and string theory are largely based on his work. For his outstanding scientific achievements, he was elected a member of the Academia Europaea in July 2020. Fedor Bogomolov has been working at New York University since 1994, but he continues to collaborate actively with Russian colleagues. He readily accepted our invitation to work together and became our academic supervisor. We consider him our scientific and ideological leader.

I would also add that one of the key features of the Russian mathematical school has always been a broad outlook and a belief in the unity of mathematics, and this is something we strive to preserve. Fedor Bogomolov is a vivid example of this universality.

— For an ordinary person who rarely encounters mathematics in everyday life beyond basic calculations, algebra and geometry seem like completely different fields. How are they connected?

— Geometry is something that objectively exists in nature; geometric objects can certainly be studied scientifically. Algebra, on the other hand, operates more with symbols and involves a greater degree of arbitrariness. However, geometric intuition is difficult to formalise, while the foundation of modern mathematics lies in the fact that everything must be formally proven. So, in short, one can say that we study geometric objects using algebraic methods.

Generally speaking, mathematics has two main ‘technologies’: algebra and analysis. An analyst has a sense of infinitesimals, can make approximate estimates, and can indicate which terms in a formula may be neglected.

In algebra, everything is either equal or not equal; nothing can be ignored, and there are no shades in between. In a sense, algebraic objects are more rigid: they are harder to work with and harder to prove results about, but when one succeeds, the results are very powerful.

Dmitry Kaledin

Photo courtesy of Dmitry Kaledin

— What are the main areas of the laboratory’s work?

— Our division into areas is rather conditional. In modern theoretical mathematics, algebraic geometry occupies a central position and is connected with many other fields, so the scope of our work is quite broad. Roughly speaking, research in classical algebraic geometry is led by Yuri Prokhorov, who comes from the school of Vasily Iskovskikh. The study of symplectic manifolds, which overlaps with differential geometry and uses methods close to analysis, is carried out by a group headed by Ekaterina Amerik. There are also groups working on derived categories (a relatively new field that has been actively studied over the past 20–30 years), as well as a group focusing on noncommutative algebraic geometry.

— What does this term mean?

— The basic algebraic concept in algebraic geometry is the ring—that is, a set whose elements can be added and multiplied. To a geometric object we associate a ring of functions defined on it. The multiplication of functions, like the multiplication of ordinary numbers, is commutative: the product A × B equals B × A. However, there also exist noncommutative rings, for example the ring of matrices. For a long time this seemed like a purely abstract intellectual exercise, but since the revolution in physics brought about by the emergence of quantum mechanics, it has become clear that noncommutative rings are also necessary for describing the real world. Relatively recently, it was discovered that even within pure mathematics this theory is important and interesting, and that surprisingly many results from classical commutative algebraic geometry also work in the noncommutative setting. This is how a new field—noncommutative algebraic geometry—emerged. At our laboratory, this area is pursued by my group and by the group of our Research Fellow, Corresponding Member of the Russian Academy of Sciences Alexander Kuznetsov. I would like to emphasise once again that, despite some division into areas, our discipline is unified, and many colleagues work in several directions at once. Excessive specialisation in mathematics is unproductive: one must clearly understand one’s own field while also having a good grasp of neighbouring areas.

— Do you mainly focus on fundamental research or applied studies?

— We primarily study fundamental principles. Our work is organised in such a way that what we research will eventually find practical application, but not immediately, and it is impossible to predict what will prove useful in the short, medium, or long term. We create the foundation without which colleagues engaged in applied research simply could not work. Unfortunately, I cannot explain this more concretely—and no one really can. The point is that mathematics develops as a language. There is a well-known phrase: ‘To understand is to simplify.’ Proofs of new theorems are often extremely complex. Over time, we come to understand how things work, invent an appropriate language, and in that language the theorem becomes almost obvious; at the same time, this language allows us to formulate new theorems and new complex proofs, and the process repeats. This is how progress in mathematics works, and it happens quite rapidly. Modern fundamental mathematics is about 200 years ahead of school mathematics and about 100 years ahead of ‘higher mathematics for engineers,’ and translating from the modern language into the old one is technically impossible. As I have already said, algebraic geometry is one of the central fields of contemporary mathematics—and not only mathematics. For example, it has applications in computer science and many applications in theoretical physics.

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Physicists, in fact, think geometrically but solve their problems algebraically, which is why our work is very much in demand among them. And sometimes it turns out that things once considered hopelessly abstract are precisely what is needed for very practical applications—cryptography and coding theory are the usual examples, though there are many others.

— In one of the laboratory’s presentations, a polynomial equation was mentioned as a relatively simple example. What is that?

— A polynomial is an expression consisting of a finite sum of powers of a variable, each multiplied by some coefficient. Curves on a plane defined by polynomial equations are, roughly speaking, the graphs of polynomials. If the polynomial is of degree 1, meaning it contains powers no higher than the first, its graph is a straight line—this is what school geometry deals with. Degree 2 gives circles, ellipses, and hyperbolas. This is also classical knowledge, known even to the ancient Greeks. It is possible and desirable to consider higher-degree polynomials, and not only in a plane but also in multidimensional spaces. At a basic level, this is what we work on. Of course, these are still idealised forms that do not exist in the real world. For instance, two straight lines may intersect at only one point in theory, but in reality, any curve can be slightly perturbed and the number of intersection points may increase. However, if you have, for example, a hurricane, what matters most is its idealised form—that it twists into a spiral; the exact shape of that spiral and the precise speed at which it moves is another question. One could say that algebraic geometry is the geometry of ideal forms.

— Which achievements of the laboratory and your colleagues are you most proud of?

— Mathematicians do not receive Nobel Prizes. The main, and essentially the only, formal sign of worldwide recognition is giving a talk at the International Congress of Mathematicians. These congresses are held every four years and summarise the developments in our field over that period. Here, things are going very well for us. For example, our associate researcher Alexander Kuznetsov gave one of the 20 plenary talks at the 2022 Congress; previously, in 2014, he gave a sectional talk. Two of our aforementioned colleagues, Yuri Prokhorov and Alexander Kuznetsov, are Corresponding Members of the Russian Academy of Sciences. Additionally, a number of our staff regularly publish articles in leading international mathematical journals.

These are formal measures. But there is also something harder to quantify: we are recognised and collaborate with colleagues both in Russia and abroad. Our laboratory has become a recognisable brand.

— Could you tell us about the laboratory’s seminars? Who participates? Are they mainly HSE students?

— We organise them for everyone. Besides HSE students, participants include students and postgraduates from the Moscow Institute of Physics and Technology (MIPT), Moscow State University, and other universities, typically 10–15 people. Many continue attending and give their own talks, which shows that our efforts in this direction are worthwhile.

— How actively do students and postgraduates take part in the laboratory’s work?

— Very actively. One of the key goals of our laboratory is to serve as an interface between experienced, senior mathematicians and students. At the student seminar, only postgraduate and undergraduate students give talks, reviewing interesting scientific articles. I should note that our research assistants reach a high level quickly; some begin writing scientific articles in their first year, and by the third or fourth year, almost all of them are producing publications.

Dmitry Kaledin

Photo courtesy of Dmitry Kaledin

For us, it is common for research assistants to go on to postgraduate studies at another university, in another city, or abroad to broaden their horizons. Many then return, completing an important cycle that enriches the scientific community.

— Would you like to see more established scientists or early-career researchers in the laboratory?

— There is a certain problem in Moscow mathematics: there are not enough people aged 60 to 70—many left for abroad in the early 1990s. Beyond that gap, however, there is no major shortage. In our team, we have a combination of young people on one hand and experienced researchers on the other, who value the opportunity to work with the younger generation and pass on accumulated skills and knowledge. Working as a researcher without teaching obligations is, of course, enjoyable, but never interacting with young people is neither pleasant nor healthy; it leads to stagnation. Even today, in the age of the internet, mentors and experienced scientists are needed to prevent the young from following excessive scientific trends blindly and to help them gain a clearer overall picture of our field.

We sought a balance and, broadly speaking, we have found it; a ratio of roughly two-thirds early-career researchers to one-third established scientists seems optimal. Of course, this can be adjusted if necessary.

— How do you attract early-career researchers besides regular seminars?

— Summer schools play an important role. The first thing we did after receiving funding was to organise an annual school for postgraduates and advanced students in Yaroslavl, with 70–80 participants. These schools are now held in Suzdal. Attendees come not only from Moscow and St Petersburg, but also from Krasnoyarsk, Novosibirsk, Samara, and other cities. We also hold one- and two-day conferences on algebraic geometry.

— How are the results of your work used in the teaching process?

— Many of our staff teach courses at the Faculty of Mathematics. We assist in running seminars and colloquia, and our interns are simultaneously postgraduates, master’s students, and undergraduates. Finally, participants in our regular seminar receive evaluations, gain experience in public speaking, and develop presentation skills.

— Which HSE divisions do you collaborate with?

— With the Faculty of Computer Science, and mathematicians from our campuses in St Petersburg and Nizhny Novgorod.

— How is collaboration with international colleagues continuing under the current difficult circumstances?

— I have not heard of any cases in which an international mathematical journal has rejected an article because of nationality or affiliation. In many Western countries, institutional collaboration with Russian scientific organisations is discouraged, but cooperation continues on an individual level; in the USA the restrictions are fewer than in Europe. We still have foreign affiliated researchers. There is interest in collaboration, and we continue to maintain scientific contacts; the difficulties are mostly logistical.

We have established connections with several colleagues in China and have much to discuss with them, but we do not want to focus solely on China. There are also strong mathematical schools in India and Brazil, and we plan to establish contacts with them as well.

Since 2010, HSE University has been implementing the Centres of Excellence project, aimed at creating leading research schools under the guidance of prominent Russian and international scientists. The main idea of the project is to attract leading researchers, representing a wide range of research methods and traditions, to work collaboratively and establish new scientific schools within the university.

The project was launched following the success of two HSE research teams in a competition held under Russian Government Resolution No. 220, adopted on April 9, 2010. Today, 42 Centres of Excellence conduct research across a diverse range of disciplines, including mathematics, biology, sociology, and linguistics.

‘The implementation of this project has enriched HSE, expanded the scope of its research and educational programmes, and enabled us to attract world-renowned scientific leaders as well as talented early-career researchers. Over these years, we have created advanced research schools whose achievements HSE is proud of and which make a significant contribution to the university’s development strategy,’ emphasised Marina Litvintseva, HSE Director for Advanced Research.