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

School for Students and Young Scientists 'Quaternions, Geometric Algebras, and Applications' took place from November 15 to 17, 2024. The School was organized by the Laboratory for Geometric Algebra and Applications (HSE University) in collaboration with the international research laboratory 'Multiscale Mathematical Modeling and Computer Calculations' of the Federal State Autonomous Educational Institution of Higher Education 'M.K. Ammosov North-Eastern Federal University' (NEFU, Yakutsk) as part of the project 'Mirror Laboratories (HSE Univeristy): 'Quaternions, Geometric Algebras, and Applications'.

School website: https://economics.hse.ru/en/gaaschool2024/

Video recordings of the classes are available in the playlist: https://vk.com/video/playlist/-227370571_

Dmitrii Shirokov

Head of Laboratory for Geometric Algebra and Applications

'This is our first experience in organizing such an event. We are glad that more than 60 participants registered for the school. There were 11 reports at the school: 5 reports were made by employees of our laboratory at the HSE University, 5 by our colleagues from NEFU, and 1 report by our Indian colleague. Video recordings of the reports are posted online and are available to everyone. We plan to make this event an annual event. In the following years, we want to hold it on a larger scale and in a mixed (offline-online) format. We already have plans to invite some of our foreign colleagues'.

The School program included lectures, practical classes, and scientific talks of the participants. The topics were diverse -- from the history of Clifford algebras to their modern applications in image processing, development of neural networks, and physics. The first lecture "History of Clifford Algebras" was given by N.G. Marchuk (Steklov Mathematical Institute, HSE University), who covered the historical development of this field. Vasil'ev V.I. (NEFU, Yakutsk) gave a lecture on "The Development of Mathematical Research in Yakutia". Professor Jiang Tongsong from Linyi University gave a lecture "Calculation and Application for Four Dimensional Algebraic Matrix Model", in which he considered in detail quaternions, quaternion matrices, and their applications. There was also a lecture on “Fundamental Operations in Clifford Algebras and Their Relation to Matrices” (Shirokov D.S., Filimoshina E.R. (HSE University)).

The practical part allowed the participants to get acquainted with computational methods and applications of quaternions and Clifford algebras. K.S. Abdulkhaev (HSE University) gave a practical session "Inverse Element in Geometric Algebra". The practical lesson "Dirac–Hestenes Equation and Its Invariance" by S.V. Rumyantseva (HSE University) covered applications of geometric algebras in physics. The lesson "Clifford Algebras and Equivariant Neural Networks: From Theory to Implementation" by E.R. Filimoshina (HSE University) considered the theoretical foundations of applications of geometric algebras for constructing equivariant neural networks and their implementation. Wang Gang (NEFU) gave a practical lesson on "Singular Value Decomposition of Split-Quaternion Matrices and Its Application to Color Image Processing". Guo Zhenwei (NEFU) shared methods for applying quaternions in image processing "Calculating the Eigenvalues ​​of Hermitian Quaternion Matrices and Face Recognition". Zhang Dong (NEFU) held a practical lesson on "Decomposition Algorithms for Commutative Quaternion Matrices and Color Image Processing".

Varvara Nechitailo

School participant

'I would like to thank the organizers and speakers for the classes, which were very interesting. Even with zero initial knowledge on the topic, I was able to perceive the information presented.'

The School's participants had the opportunity to present their own talks. A representative of the Indian Institute of Science Education and Research, Heerak Sharma, presented the talk "Commutative analogues of Clifford algebras".

Heerak Sharma

School participant, speaker, Indian Institute of Science Education and Research (IISER), Pune, India

'First of all I would like to thank the school for inviting me as a participant and especially for giving me the opportunity to give a talk. The school was well planned. It holistically covered the topic of Clifford algebras starting from their history. I found the contents of the school very interesting, in particular, I liked that the school covered nice applications of Clifford algebras in image processing, designing neural networks and in physics. I liked giving my talk in the school. Prof. Marchuk's question was very good and gave me a new perspective on representations of K_{p,q}s. In general I like giving talks and discussions with people and I enjoyed it this time as well.