The International School for Students and Young Scientists 'Quaternions, Geometric Algebras and Applications' 2025 was held.

From November 14 to 16, 2025, the International School for Students and Young Scientists 'Quaternions, Geometric Algebras, and Applications' was held. It was dedicated to the 70th anniversary of Nikolay G. Marchuk. The school was organized by the Laboratory for Geometric Algebra and Applications (Faculty of Economic Sciences, HSE University, Moscow) jointly with the International Research Laboratory 'Multiscale Mathematical Modeling and Computing' (M.K. Ammosov North-Eastern Federal University, Yakutsk) as part of the HSE Mirror Laboratories project: 'Quaternions, Geometric Algebras, and Applications'.

School website: https://economics.hse.ru/gaaschool2025/

Video of the talks: https://vkvideo.ru/playlist/-227370571_3

Dmitry Shirokov

Head of Laboratory for Geometric Algebra and Applications

We are very pleased that this year we were able to not only repeat but also significantly expand the School's format. Regular participants from HSE and NEFU were joined by lecturers from universities and research centers in China, Japan, the USA, Peru, and India. The School was organized over three thematic days. The first day was devoted to the theoretical foundations — geometric algebras, quaternions, and other systems of hypercomplex numbers, as well as their application in classical geometry and image processing. The second day focused on applications of this mathematical apparatus in physics: field theory, quantum systems, and differential equations within an algebraic approach. On the third day, presentations were given on the applications of geometric algebras in geospatial intelligence and neural networks, and various modifications of Clifford algebras and quaternions were discussed. We were especially pleased that after each presentation, the participants actively asked questions and proposed new ideas — this kind of lively scientific exchange is precisely what we conceived the School for. We express our gratitude to all the speakers and participants of the School for their participation and contribution. We look forward to seeing you at the School in 2026!

The school was truly international: participants included researchers representing leading universities in Russia, China, Japan, the USA, Peru, and India. The daily program included four one-hour lectures. The opening lecture, 'Clifford’s geometric algebras in the context of Pauli algebra, Dirac algebra, quaternions, biquaternions, octonions and Okubo algebra', was given by Professor Eckhart Hitzer (International Christian University, Tokyo, Japan). Professor Hongbo Li (Chinese Academy of Sciences, Beijing, China) then presented a talk entitled 'Conformal Geometric Algebra with Geometric Applications'. Professor Carlos Castro Perelman (Bahamas Advanced Science Institute and Conferences, Long Island, Bahamas; Quantum Gravity Research, Los Angeles, USA) presented a report on 'Generalized Clifford Algebras and the N-th Root of Linear Differential Equations of Higher Order'. Nikolay G. Marchuk (Steklov Mathematical Institute of RAS; HSE University, Moscow, Russia) was also a keynote speaker, delivering a report on 'Lanczos' Biquaternionic Equation to Replace Dirac's Equation'. Professor Yu Zhaoyuan (Nanjing Normal University, Nanjing, China) spoke about the applications of geometric algebras in geoinformatics in his report 'Geometric Algebra: A Foundational Language for Geospatial Intelligence'. Pedro Amao (Pontifical Catholic University of Peru, Lima, Peru) presented a report on 'Two-State Quantum Systems Revisited: A Clifford Algebra Approach'. Wang Gang (Qufu Normal University, Qufu, China; NEFU, Yakutsk, Russia) presented a paper titled 'Multi-Scale Feature Extraction and Color Image Hashing Based on SVD of Split Quaternion Matrices'. Zhang Dong presented a paper titled 'Efficient Quaternion Matrix Decomposition Algorithms and Their Applications to Least-Squares Problems and Color Image Processing'. Zhenwei Guo (Liaocheng University, Liaocheng, China; NEFU Yakutsk, Russia) presented a paper titled 'Algebraic Algorithms for Eigen-Problems of a Reduced Biquaternion Matrix and Applications'. A key part of the school included presentations by young scientists—graduate and undergraduate students. Sofia Rumyantseva (HSE University) presented a talk entitled 'On the Real-Valued Analogue of the Multidimensional Dirac Equation', and Ekaterina Filimoshina (HSE University) presented a talk 'Equivariant Neural Networks Based on Clifford Geometric Algebras: Theory and Applications'. Heerak Sharma (Indian Institute of Science Education and Research, Pune, India) presented a talk entitled 'On commutative analogues of Clifford algebras'.

The 'Quaternions, Geometric Algebras, and Applications' school became an important platform for knowledge exchange and strengthening international scientific collaboration.