'Today, Human Existence Without Mathematics Is Difficult; Tomorrow, It Will Be Simply Impossible'
Mathematicians around the world share a common language and continue to collaborate despite the challenges of recent years. The hub of mathematical networking has been shifting to China, where scientists from various countries meet at conferences and other academic events. Partnerships with leading Chinese universities offer promising opportunities to strengthen existing ties and forge new ones. In this interview with the HSE News Service, Valery Gritsenko, Head of the HSE International Laboratory for Mirror Symmetry and Automorphic Forms, discusses this and other topics, including what AI is and why the state should engage with mathematicians.
— How was your laboratory established?
— It was founded in 2016, after we received a mega-grant from the Russian Government. Now, as an international laboratory at HSE University in our ninth year, we are a well-established centre operating on a global scale despite the limitations of recent years. Last year, we launched a new programme, Geometry and Physics, in partnership with China as part of HSE University’s International Academic Cooperation project. Our partners are the Laboratory for Algebraic Geometry and the Laboratory for Theoretical Physics at the Beijing Institute of Mathematical Sciences and Applications, Tsinghua University (BIMSA).
— Tell us about the key areas of your laboratory's work.
— From the outset, we have operated as a multidisciplinary laboratory. Mirror symmetry, discovered by theoretical physicists, is closely connected to algebraic geometry and mathematical physics.
My research focuses on automorphic forms, an important analytical tool for studying geometric, topological, and arithmetic problems. In mathematical physics, automorphic forms help reveal the hidden symmetries of various phenomena. As part of our laboratory’s work, we address problems across a wide range of mathematical fields. In this sense, the theory of automorphic forms in our research has an applied character within mathematics, as we use it to solve problems in other areas. Among our achievements, I would highlight the classification of infinite-dimensional Lie algebras—specifically, hyperbolic Kac–Moody algebras—used in the theory of microscopic black holes, as well as the recent proof of the irrationality of certain special four-dimensional algebraic varieties (more precisely, moduli of generalised Kummer varieties). Algebraic geometry studies the objects defined by the simplest and most fundamental functions: polynomials. Automorphic forms serve as a powerful transcendental tool within algebraic geometry.
— What are the prospects for applied use of your laboratory's research?
— One could argue that mathematics deals with absolute truth. That’s why we still refer to works from the 18th and 19th centuries, such as those by Leonhard Euler. Mathematics is certainly a conservative science: even school students study Euclid, who lived long before the modern era. However, this does not prevent mathematics from evolving beyond recognition every fifty years. Over such periods, the language transforms completely, fundamentally new scientific tools emerge, and entirely new objects are introduced into research. Yet, this does not invalidate earlier stages, as even ancient mathematics established absolutely true mathematical facts.
Why are mathematical results so well suited for practical application? This is a philosophical question that scientists from various fields actively debated throughout the twentieth century. Today, the new algorithmic era has effectively taken this question off the table and shifted the focus to the real-time use of mathematical algorithms across all devices—from personal smartphones to massive supercomputers to global computer networks. What is not digitised—that is, not mathematised—does not exist. We no longer just apply mathematics; we live within it. Today, human existence without mathematics is difficult; tomorrow, it will be simply impossible. Mathematics is the number one science of the 21st century. If a state lacks a guild of world-class mathematicians, it simply will not understand the digital (mathematical) processes shaping the world. All countries that have achieved technological and economic autonomy—such as China, South Korea, Brazil, Turkey, and Iran—have necessarily implemented large-scale mathematical education programmes at both school and university levels and built a strong community of professional mathematician-researchers. This is a historical fact of the late twentieth century. Since the second half of the nineteenth century, Russian mathematics has consistently held a prominent position in the global rankings. I would also highlight another historical fact: Russia’s first secular educational institution was the School of Mathematical and Navigational Sciences, established in Moscow in 1701. Today, the issue of mathematical education remains as relevant and urgent as ever—a matter of paramount importance. I can offer a concrete everyday example: the digital signature, without which many operations have become impossible. It is a cryptographic algorithm based on the theory of elliptic curves—objects from algebraic geometry that can be visualised at a basic level as a torus or a bagel. I would add that the proof of the Shimura-Taniyama conjecture, which linked the arithmetic properties of elliptic curves with modular forms, ultimately led to the solution of Fermat’s Last Theorem from the 17th century. Theoretical mathematics plays a far greater role in our everyday lives than we often realise—our world is becoming increasingly digital. It may seem unusual that no royalties apply to fundamental mathematical laws and discoveries. In a sense, they are priceless. Anyone who fails to understand them risks losing everything!
Returning to my laboratory’s work in the context mentioned earlier, we are conducting research on topics such as the paramodular Brumer–Cramer conjecture, which is a two-dimensional generalisation of the Shimura–Taniyama–Weil conjecture. Elliptic curves are replaced by two-dimensional objects called Abelian surfaces. The conjecture involves Siegel modular forms of genus 2, which I constructed in 1993 to solve Siegel’s problem on moduli of Abelian surfaces, originally posed in the late 1950s.
— What is your attitude toward artificial intelligence?
— Artificial intelligence is a mathematical, algorithmic system that draws on numerous branches of mathematics, including linear algebra, differential geometry and topology, mathematical analysis, and probability theory. For example, the concept of a gradient, originating from mathematical analysis and geometry, has led to algorithmic methods for self-learning. AI is a highly complex mathematical and software product that processes various databases. There are many different artificial intelligences, each shaped by its developers, who cannot predict all of its features in advance. This is one of AI's miracles. In a sense, artificial intelligence represents modern experimental mathematics, with outcomes that are not entirely predictable. However, to operate successfully and perform its 'miracles,' AI requires storing vast amounts of digital information, demanding energy consumption comparable to that of some European countries. Miracles come at a cost. The results of AI’s work are already transforming the entire educational process. I am confident that we will witness tremendous changes in higher education over the next five years. Are we prepared for these unpredictable shifts in the system? There’s no time to 'learn to swim'—the new wave of information is already upon us.
— How long does it take to train a good mathematician?
— Of course, there are self-taught geniuses, but generally, it’s best to spend two final school years at a specialised math school, six years at university, followed by three to four years preparing and defending a dissertation, and then a similar period working as a postdoctoral fellow. It takes no less time than training a neurosurgeon. Moreover, mistakes in a mathematician’s work can be even more significant, making it difficult to shorten the path to becoming a scientist.
— Are there any researchers in your laboratory whose careers began there?
— Yes, currently four of our former research assistants have become permanent members of the laboratory staff after defending their PhD dissertations. For example, Alexander Kalmynin, now a senior research fellow, joined us as a third-year bachelor’s student. Next year, we hope to recruit two more early-career candidates of sciences. For example, Dmitry Adler, who graduated in 2021, is now a postdoctoral fellow at the Max Planck Institute for Mathematics in Germany. Our former research assistants are becoming promising early-career scientists. After defending their degrees, they typically move on to other universities and research centres, eg in St Petersburg or abroad. Young researchers shouldn’t be left to stew in their own juices under the sole care of their academic supervisors. They need to explore the world, become independent investigators, and discover their own scientific focus. A new generation of early-career mathematicians is emerging, and our laboratory hopes to expand. I look forward to securing increased funding specifically for recruiting new permanent staff. The working conditions in our laboratory are quite competitive compared to the European average. Additionally, researchers become part of a vibrant scientific community, as Moscow is one of the world’s capitals of mathematics—rivalled only by the very best mathematical centres globally. Many people may not realise that Russian is one of the four official languages of the International Congress of Mathematics. Russia has a fully developed national mathematical terminology, and while young people readily attend lectures in English, Russian textbooks, terminology, and traditions provide valuable support during the initial stages of learning. It’s hard to say who is the best, but in international rankings, Moscow’s mathematics community consistently reaches at least the semifinals—much like a top team in the Champions League of world football. We are recognised as one of the global leaders, making mathematical research and education an excellent area for investment by our major corporations.
— Is it possible to maintain previous and establish new international scientific contacts?
— It is clear that we have faced certain difficulties over the past three years. For example, the International Congress of Mathematicians in Moscow was cancelled; this event had included two satellite conferences organised by our laboratory However, differing socio-political views and sentiments cannot halt international mathematical collaboration. The foundation is unshakable: the multiplication table is the same in every country and on every continent. Our current work demonstrates that our laboratory is international not only in name, but also in its makeup. For the past three years, we have organised conferences on modular forms, algebraic geometry, and integrable systems at the Paul Painlevé Laboratory in Lille, France, as part of a joint scientific grant from the Russian Science Foundation (RSF) and the French National Research Agency (ANR), involving renowned Russian mathematicians working abroad. This enabled us to launch a new research project in the field of mathematical physics. I would like to note that under this grant, I led the French team, while Andrey Levin, a research fellow at our laboratory, headed the Russian group. Ending scientific cooperation will lead us nowhere, though the bureaucracy may see things differently. Unexpected restrictions are now emerging—for example, a young scientist was recently denied a U.S. visa, with one of the reasons cited being that he had once written a thesis on cryptography. Maxime scientia multa dolores (Latin for 'much knowledge, many sorrows'—Ed). But clashes with mathematicians can be costly for any state, as bureaucrats will never be able to figure out on their own 'how the signal gets through.'
— Today, there is a popular view that some achievements of Soviet science were made possible by the privileged conditions created for scientists in secret institutes.
— In the USSR of the 1950s and 60s, there were indeed restricted-access mathematical departments at academic institutions, but mathematics thrives on openness and international competition. These drive results far more effectively than any form of secrecy or isolation. Fundamental science depends on consistent exchange of ideas, personal communication, and professional expertise—and halting them can quite literally bring its development to a standstill. Overall, I see no serious limitations to open research; openness is the strength of science in generating objective knowledge. Openness enables transparent monitoring of the honesty and objectivity of papers, allowing for thorough review and evaluation, although distinguishing genuine proofs from false ones is not always easy. It’s not uncommon for a mathematical article to undergo peer review for a year or even longer before being published. Restricting access to fundamental scientific subjects risks their decline and eventual disappearance within a historically short period.
— Tell us how you established cooperation with Chinese scientists.
— Last year, as part of HSE University's International Academic Cooperation project, we signed an agreement with the Beijing Institute of Mathematical Sciences and Applications (BIMSA), founded by one of the greatest mathematicians of our time, Shing-Tung Yau. An academician of both the Chinese and American Academies, he is renowned for his fundamental work in various fields of mathematics, especially geometry. He worked in the United States for many years before returning to China, where he founded the Yau Mathematical Sciences Centre and later BIMSA. Alongside established researchers, BIMSA hosts postdoctoral fellows and doctoral students. We collaborate with the Laboratory for Algebraic Geometry and the Laboratory for Mathematical Physics at this institute, led by the talented young mathematician Artan Sheshmani, who previously worked at Harvard, and my colleague from the St Petersburg Department of the Steklov Institute of Mathematics (PDMI), Nicolai Reshetikhin, who relocated to Beijing from Berkeley.
— You recently attended a conference in China. What are your impressions?
— The conference took place at Wuhan University, where the Faculty of Mathematics is undergoing rapid modernisation: a state-of-the-art faculty building has been completed, the faculty staff is actively expanding, student enrolment has increased, and a doctoral school has been established. This representative international conference was dedicated to modular forms and their applications—the very topic I have been working on for the past 35 years.
The conference, held as part of our joint project, continued last year's inaugural Russian-Chinese conference in Moscow, which was attended by three Fields Medallists (the Fields Medal is the equivalent of the Nobel Prize for mathematicians—Ed). The conference in Wuhan was attended by 90 participants, and 27 presentations were made on applications of modular form theory in algebraic geometry, topology, number theory, mathematical physics, and string theory—areas closely aligned with my laboratory’s scientific interests. It proved to be a large-scale and highly engaging conference. The main organiser was my student Haowu Wang, a member of the official team of our project with BIMSA. He completed his thesis under my supervision in France in 2019 and, within six years, advanced from postdoctoral fellow to professor. The conference was supported by three major Chinese scientific foundations. I would like to note that scientists from many European countries, the United States, and Japan attended the conference in Wuhan. In particular, several mathematicians who had participated in our laboratory’s events in Moscow were present. I would also like to add that two speakers from Taiwan participated, demonstrating that normal scientific exchange between China and Taiwan continues. I was impressed by the high level of organisation, as well as the comfortable work environment and accommodation provided to all participants. Even the Japanese, who regard their scientific events as exemplary, were surprised that the organisers of this conference significantly outperformed their standards.
Many foreign scientists are currently working in China. Local universities attract not only young talent but also renowned retired professors from Europe and America. Many Chinese mathematicians who earned their degrees and worked abroad are now returning home, attracted by compelling offers and excellent working conditions.
— How important, in your opinion, is the development of cooperation with Chinese universities for HSE University?
— The establishment of strong partnerships—particularly through programmes creating mirror laboratories with major regional universities in China—could become a key area of international cooperation over the next five to ten years. Chinese universities are ambitious, building their own brands, attracting dynamic mathematicians, and cultivating a distinctive academic atmosphere. Regional governments allocate impressive budgets for the development of science. I believe the next five to seven years will be very fruitful, as many of the new professors in China represent international mathematics, having studied or defended their theses in the United States and Europe. Our doctoral students and postdoctoral fellows will immediately be exposed to top-level events with large international participation, and we should take full advantage of this opportunity. For example, in July, three research assistants from our laboratory participated in a large-scale international scientific summer school organised by Nicolai Reshetikhin (BIMSA) in China.
The idea of China hosting scientific events is very appealing to our early-career researchers, who often face difficulties travelling to the West. In China, these young scientists can connect with colleagues from many countries and quickly immerse themselves in the mainstream of modern mathematical research. Our current Russian-Chinese programme allows us greater freedom and helps us forget about the restrictions imposed by the West. We meet not only with Chinese scientists but also with colleagues from the United States and Europe who come to China, enabling us to continue normal international cooperation. Most importantly, early-career researchers begin working at a high level—what’s often called 'starting from high C'—which is less common in Europe.
— How are your laboratory's achievements applied in the educational process?
— Almost all the members of the laboratory staff teach at the HSE Faculty of Mathematics. We also regularly organise a summer school at the HSE Study Centre 'Voronovo,' which attracts around 70 students from various universities, mostly in the senior years of their bachelor’s programmes.
A key feature of the school is that all students are required to attend a daily scientific seminar related to one of the three lecture courses. Students are expected to select one seminar to better master new scientific material. Lectures and seminars run throughout the day, from 9:30 am to 6:30 pm. The goal of the school is for students to develop as scientists here and now. The programme also includes an engaging component—evening lectures by leading Russian mathematicians, held after dinner from 9 to 11 pm. The school welcomes students with varying levels of preparation. Those who are less advanced work to catch up with stronger peers, while the more advanced students push themselves further when faced with real-time competition. Students from regional universities can understand the value of studying mathematics by attending original courses taught by leading researchers and beginning to solve interesting problems themselves. In my opinion, this is the best way to immerse students in science—by learning to swim on their own. Mathematics is a complex science, and attending students come to realise that they need to understand it here and now. This is the true purpose of the school in Voronovo.
Members of our laboratory staff conduct research seminars at the faculty, and new technologies make it easier to hold them at times convenient for everyone. Mathematics is best taught line by line. Chalk and blackboard remain the most effective tools for explaining complex concepts. Adding the element of personal presence helps transform a sequence of symbols into a clear understanding of the proof. A good low-tech chalkboard is already much more expensive than an electronic one, while a slide—a two-dimensional object—does not create the same sense of presence as writing out lines of characters by hand. A good lecture for students is always a mini-performance, because they must grasp the content here and now—and leave the audience a little smarter after 80 minutes.
We also work actively with master’s and doctoral students. I believe that during their studies, they should have the opportunity to connect with talented peers from St Petersburg and other universities. It is important for them to become aware of the competition early and to get to know potential future colleagues. This is precisely what is encouraged by the annual Siberian Summer School co-organised by our laboratory in Novosibirsk, and the winter student schools at the Faculty of Mathematics. In the future, we hope to establish a regular student scientific seminar connecting Moscow and St Petersburg.
— How actively are young people involved in the work of the laboratory?
— The laboratory includes 14 research assistants—students, including doctoral students—from the HSE Faculty of Mathematics. We also recruit students from earlier years of study. We have already employed second-year students, and this year, for the first time, we hired a first-year (!) student who has published a paper in a well-known mathematical journal. He is our first underage employee, and we find this youthful energy truly inspiring!
Earlier, I mentioned four staff members who completed their dissertations while working in our laboratory: Alexander Kalmynin, Alexey Golota, Artem Prikhodko, and Pavel Osipov. One of Alexander Kalmynin’s students is currently a research assistant in our laboratory. We are growing! We plan to hold a reporting conference for all research assistants in late autumn, featuring 14 presentations—a strong programme.
— Which subdivisions of HSE University do you collaborate with?
— The Faculty of Mathematics has four international laboratories, and collaboration among them is quite active. We are also establishing cooperation with the HSE Faculty of Computer Science (FCS), which hosted a well-received seminar on number theory. This year, our laboratory is successfully conducting a joint seminar on geometry with the FCS. Some of our research seminars share a similar focus with those of the FCS, and work on modular forms often involves extensive formal calculations. The exchange of ideas and students has begun, which is very positive, as we need to build a common mathematical space.
See also:
'We Are Now Nearing Practical Application of a Stimulus-Free Brain-Mapping System'
Neural interfaces developed by scientists at HSE University in collaboration with clinicians make it possible to communicate with the brain and decode its signals. The use of such interfaces opens up opportunities to stimulate brain activity, restore and normalise muscle control in patients who have suffered a stroke, heart attack, or other neurological disorders, and support the rehabilitation of individuals with traumatic brain injuries or limb loss. Alexey Ossadtchi, Director of the Centre for Bioelectric Interfaces at the HSE Institute for Cognitive Neuroscience, discusses the centre and its work.
Centre for Language and Brain Conducts First Neurolinguistic Field Study of Reading in Yakut
In July, a team from the HSE Centre for Language and Brain, in collaboration with the Centre for the Study, Preservation, and Development of Native Languages of the Academy of Sciences of the Republic of Sakha (Yakutia), conducted the first-ever neurolinguistic expedition to the village of Churapcha to study reading in the Yakut language using electroencephalography (EEG). For the first time, EEG data from 43 adults and behavioural data from 40 children was collected during the two-week expedition.
‘We Describe Unwritten Languages’
Chiara Naccarato, Research Fellow at the HSE Linguistic Convergence Laboratory, graduated from university in Italy and came to Russia to study the languages of Dahgestan and the speech patterns of bilingual speakers. She notes the friendly atmosphere of the laboratory and the hospitality of the people of Daghestan.
HSE Neurolinguists Reveal What Makes Apps Effective for Aphasia Rehabilitation
Scientists at the HSE Centre for Language and Brain have identified key factors that increase the effectiveness of mobile and computer-based applications for aphasia rehabilitation. These key factors include automated feedback, a variety of tasks within the application, extended treatment duration, and ongoing interaction between the user and the clinician. The article has been published in NeuroRehabilitation.
'Our Goal Is Not to Determine Which Version Is Correct but to Explore the Variability'
The International Linguistic Convergence Laboratory at the HSE Faculty of Humanities studies the processes of convergence among languages spoken in regions with mixed, multiethnic populations. Research conducted by linguists at HSE University contributes to understanding the history of language development and explores how languages are perceived and used in multilingual environments. George Moroz, head of the laboratory, shares more details in an interview with the HSE News Service.
Scientists Reveal Cognitive Mechanisms Involved in Bipolar Disorder
An international team of researchers including scientists from HSE University has experimentally demonstrated that individuals with bipolar disorder tend to perceive the world as more volatile than it actually is, which often leads them to make irrational decisions. The scientists suggest that their findings could lead to the development of more accurate methods for diagnosing and treating bipolar disorder in the future. The article has been published in Translational Psychiatry.
HSE Linguists Study How Bilinguals Use Phrases with Numerals in Russian
Researchers at HSE University analysed over 4,000 examples of Russian spoken by bilinguals for whom Russian is a second language, collected from seven regions of Russia. They found that most non-standard numeral constructions are influenced not only by the speakers’ native languages but also by how frequently these expressions occur in everyday speech. For example, common phrases like 'two hours' or 'five kilometres’ almost always match the standard literary form, while less familiar expressions—especially those involving the numerals two to four or collective forms like dvoe and troe (used for referring to people)—often differ from the norm. The study has been published in Journal of Bilingualism.
Mathematicians from HSE Campus in Nizhny Novgorod Prove Existence of Robust Chaos in Complex Systems
Researchers from the International Laboratory of Dynamical Systems and Applications at the HSE Campus in Nizhny Novgorod have developed a theory that enables a mathematical proof of robust chaotic dynamics in networks of interacting elements. This research opens up new possibilities for exploring complex dynamical processes in neuroscience, biology, medicine, chemistry, optics, and other fields. The study findings have been accepted for publication in Physical Review Letters, a leading international journal. The findings are available on arXiv.org.
Mathematicians from HSE University–Nizhny Novgorod Solve 57-Year-Old Problem
In 1968, American mathematician Paul Chernoff proposed a theorem that allows for the approximate calculation of operator semigroups, complex but useful mathematical constructions that describe how the states of multiparticle systems change over time. The method is based on a sequence of approximations—steps which make the result increasingly accurate. But until now it was unclear how quickly these steps lead to the result and what exactly influences this speed. This problem has been fully solved for the first time by mathematicians Oleg Galkin and Ivan Remizov from the Nizhny Novgorod campus of HSE University. Their work paves the way for more reliable calculations in various fields of science. The results were published in the Israel Journal of Mathematics (Q1).
From Neural Networks to Stock Markets: Advancing Computer Science Research at HSE University in Nizhny Novgorod
The International Laboratory of Algorithms and Technologies for Network Analysis (LATNA), established in 2011 at HSE University in Nizhny Novgorod, conducts a wide range of fundamental and applied research, including joint projects with large companies: Sberbank, Yandex, and other leaders of the IT industry. The methods developed by the university's researchers not only enrich science, but also make it possible to improve the work of transport companies and conduct medical and genetic research more successfully. HSE News Service discussed work of the laboratory with its head, Professor Valery Kalyagin.