'We View Mathematics as a Universal Language for Natural Sciences, Economics, and Computer Science'

© iStock

The Laboratory for Geometric Algebra and Applications at the HSE Faculty of Economic Sciences is developing a universal language for mathematics, physics, and other exact and natural sciences. It is creating computational methods that are applied in geometry, physics, machine learning, engineering, computer science, and other fields. The laboratory’s staff, including doctoral students, have published numerous articles in leading scientific journals and have taken part in major international conferences. Dmitry Shirokov, Head of the Laboratory, spoke with the HSE News Service about their work.

— When was the laboratory established?

— Our Laboratory for Geometric Algebra and Applications was established in July 2024 after winning HSE University's Mirror Laboratories project competition.

Dmitry Shirokov

Photo courtesy of HSE University

The creation of the laboratory was a gradual process. One could say it all began with the course ‘Foundations of the Theory of Clifford Algebras and Spinors,' which I started teaching at HSE University in 2020 as both an open optional course and a master's-level elective course. After teaching this course, I was able to assemble a team of strong students who took up this promising and, in my opinion, fascinating topic. This small team went on to win the 2021 ‘Scientific Initiative’ project competition organised by the HSE Centre for Student Academic Development. In January 2022, we won a competition for research and study groups. Since July 2023, we have been implementing a collective grant from the Russian Science Foundation on related topics. In November of the same year, work began within the ‘Algebraic and Geometric Methods in Applied Sciences’ research group at the Faculty of Economic Sciences. The opening of the laboratory in July 2024 was therefore a natural outcome of several years of joint effort, and the HSE Mirror Laboratories competition served as an incentive for its institutionalisation. We plan to further pursue our research, and there is still much work ahead.

— What are the laboratory's main areas of focus?

— The primary focus of our research is on Clifford algebras—also known as geometric algebras. They can be viewed as a kind of universal language for mathematics, physics, and other exact sciences. Many widely used mathematical objects (matrices, vectors, tensors, quaternions, differential forms) can be interpreted as special cases of geometric algebras.

The objects we work with in geometric algebras are called multivectors. Their advantage is that they are not only convenient from an algebraic and computational perspective but also have a clear geometric interpretation. This is why such a formalism is often more suitable than a matrix-based one—for example, when describing rotations and other orthogonal transformations. In the laboratory, we develop algebraic, geometric, and computational methods based on this formalism and apply them across a wide range of fields: geometry, physics, computer science, engineering, economics, and other sciences.

— Can we describe this as interdisciplinary research, where mathematics and physics closely intersect?

— Absolutely, our work is interdisciplinary. We view mathematics as a universal language for the natural sciences, economics, and computer science. The twentieth century was the era of mathematical physics, while today fields such as machine learning, bioinformatics, and geoinformatics are coming to the forefront. Consequently, we do not restrict ourselves to physics but explore broader applications of our methods.

— Is it possible to explain, in simple terms for a layperson, the main aspects of your colleagues’ work?

— When people mention mathematical physics, they often think of its equations—partial differential equations, for example. But mathematical physics can also be understood more broadly as the study of mathematical models of physical phenomena. In describing and analysing these models, we do not always need to rely on differential equations; we can also use algebraic methods, differential geometry, and other approaches. In our laboratory, we are primarily interested in algebraic-geometric methods.

— In which fields can these methods be applied?

— We often tackle problems originating in physics, but we use purely algebraic methods without relying on differential equations. And sometimes, we work on problems not directly related to physics. For example, we have recently published papers on the implementation and study of certain Lie groups within Clifford algebras, on calculating elements of spinor groups from given elements of orthogonal groups in arbitrary dimensions, on designing a new architecture for equivariant neural networks, and on performing singular value decomposition in geometric algebras. We have published papers in mathematical physics, algebra, geometry, group theory and Lie algebras, computational mathematics, applied mathematics, machine learning, and more. Our main criterion for choosing problems is that they should be both interesting and applicable—if not immediately, then in the near future.

— Is your work more about basic science or applied research?

— I would describe our work as basic science with a clear emphasis on future applications. We prove theorems, but we always consider how these results might be used. We also regularly participate in conferences attended by leading applied scientists. For example, since 2020, I have participated annually in the ENGAGE workshop (Empowering Novel Geometric Algebra for Graphics & Engineering) as part of the Computer Graphics International conference. The workshop brings together specialists applying geometric algebra in computer graphics, geoinformation systems, motion detection, engineering, and machine learning—and our theoretical reports are received with genuine interest. We are planning joint projects with our international colleagues, where we will focus on the theoretical aspects while they handle the practical components. I would also like to take this opportunity to invite interested colleagues to participate in the next ENGAGE workshop, which will be held as part of CGI 2026 in London, where I will be one of the organisers.

— Your laboratory seems quite compact, consisting of yourself, leading research fellow Nikolay Marchuk, and three research assistants. Are you planning to recruit new staff members, and what principles will guide your selection process?

— Currently, there are two Doctors of Sciences in the laboratory: Nikolay Marchuk and myself. We also have three doctoral students: Sofiia Rumiantseva, who is nearly ready to defend her PhD thesis, and Ekaterina Filimoshina and Kamron Abdulkhaev, who have just started their doctoral programmes but have already published eight and three papers, respectively. All five of us have been working in the laboratory since its foundation. We are a small but active team. Over the past nine months, we have published 18 articles and presented 11 reports at international conferences. We are always open to new team members; the most important qualities we look for are a commitment to long-term research and a genuine passion for science.

© iStock

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

— I would prefer our laboratory to nurture established scientists from aspiring early-career researchers. We actively engage with young talent. For the second consecutive year, we have been running a school for students and budding scientists. This year, it will be held in an international format, featuring speakers from China, Japan, and India. I invite all interested students and aspiring researchers to participate in the International School, which will be held online from November 14 to 16 (full details are available on the School’s website). Together with Ekaterina Filimoshina, I also lead the research project seminar 'Methods of Linear Algebra and Data Analysis in Economics' at the Faculty of Economic Sciences. This year, around 20 highly capable students (with an average GPA of 8.61) have joined us, and they will be writing their term papers under our academic supervision. Some of these students are already working on significant problems that are expected to lead to publications in scientific journals. We sincerely hope that many of them will become part of our team in the future.

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

— Our laboratory is still quite young, so it may be premature to sum up any major results. I believe our main achievements are yet to come. Today, it is common to measure success using scientometrics, and publications in leading journals as well as participation in top conferences play an important role in reporting on various projects. Therefore, I am particularly proud of our recent paper, which was accepted to ICML 2025, a leading international conference on machine learning, as well as our recent publications in Mathematical Methods in the Applied Sciences, Modern Physics Letters A, the Journal of Computational and Applied Mathematics, Linear and Nonlinear Algebra, and other journals. Of course, the relevance of our results is also a measure of success: several of our papers are already being actively cited by international colleagues. Our early-career researchers, Ekaterina Filimoshina and Kamron Abdulkhaev, won best paper awards at ICCA, a major international conference on Clifford algebras, in 2023 in Holon, Israel, and at the ENGAGE 2021 workshop in Geneva, Switzerland.

— How and on what topics do you collaborate with colleagues from North-Eastern Federal University within the framework of the Mirror Laboratories project?

— Together with our colleagues from NEFU, we are tackling problems as part of the joint project 'Quaternions, Geometric Algebras, and Applications.' The NEFU team, led by Prof. Vasiliev, has extensive experience working with quaternions and their various forms—split quaternions, commutative quaternions—and applying them in diverse fields such as image processing, electromagnetism, quantum mechanics, and quantum chemistry. We, in turn, have vast experience with geometric algebras, which generalise the concept of quaternions to higher dimensions. By combining our expertise, we are addressing more complex problems at the intersection of these fields. We hold a joint online seminar where we regularly discuss the latest results and explore approaches to new problems. Over the year and a half since our project began, we have already published nine papers with our NEFU colleagues, five of which are in A-list journals.

— Which departments of HSE University and other academic centres does the laboratory collaborate with?

— In addition to NEFU, we collaborate with the Department of Mathematical Physics at the Steklov Mathematical Institute of the Russian Academy of Sciences, with research teams at the Institute of Systems Science of the Chinese Academy of Sciences, and with Nanjing Normal University. Next month, we plan to give a presentation at the HSE Faculty of Mathematics. We are always open to dialogue with all interested colleagues.

— How are the results of your work applied in the educational process?

— In my view, science and teaching are inseparable. It is not enough to focus solely on research; active interaction with students is essential. I teach a linear algebra course to first-year bachelor's students, and the work we do in the laboratory serves as a kind of extension to this foundational course. As a result, strong students who have completed a course in linear algebra and wish to develop further in this area can enrol in our research workshop or optional course and begin solving scientific problems with us. In the optional course, I sometimes share our latest findings in this field, which may interest students and help engage them more deeply in research.