'Quaternions, Geometric Algebras, and Applications'

Speaker: Carlos Castro Perelman (Bahamas Advanced Science Institute and Conferences, Long Island, Bahamas; Quantum Gravity Research, Los Angeles, USA)

Abstract: It is shown how generalized Clifford algebras allows to construct the N-th root of N-order linear differential equations involving massless and massive particles. The N-th higher-order linear differential equation is equivalent, after a factorization and cyclic permutation of the factors, to N first-order differential equations. Explicit solutions are found. We conclude with the study of the generalized Dirac equation and gauge theories in Generalized Clifford Spaces.

Speaker: Nikolay Marchuk (Steklov Mathematical Institute of RAS; HSE University, Moscow, Russia)

Language: Russian

Abstract: I plan to talk about the Lanczos equation (1929) and its conservative modification. Let's discuss the question of whether it is possible to use the Lanczos equation as a replacement for the Dirac equation in the modified Standard Model of Elementary Particles.

Speaker: Sofiia Rumiantseva (HSE University, Moscow, Russia)

Abstract: In the talk, we will introduce the multidimensional Dirac–Hestenes equation, a real-valued analogue of the Dirac equation formulated in the geometric algebra Cl(1,n), and discuss some of its fundamental properties. It is known that the classical four-dimensional Dirac equation is equivalent to the Dirac–Hestenes equation in geometric algebra Cl(1,3). It means that one might obtain a solution to the Dirac–Hestenes equation using a solution to the Dirac equation and vice versa. The Dirac–Hestenes equation may provide a deeper understanding of geometry in various tasks, as the considering wave function is entirely real. We will show that the theory is extended to the multidimensional case. Since the matrix representation of complex geometric algebra depends on the parity of n, the cases of even and odd n are analyzed separately. In the even-dimensional case, there are two types of spinors which are solutions to the Dirac equation: semi-spinors and double spinors. Additionally, we will demonstrate that the multidimensional Dirac–Hestenes equation exhibits both gauge invariance and Lorentz invariance.

Speaker: Pedro Amao (Pontifical Catholic University of Peru, Lima, Peru)

Abstract: We revisit the topic of two-state quantum systems using the Clifford algebra in three dimensions, denoted Cl3. In this formulation, both quantum states and operators are expressed entirely as elements of Cl3, due to the algebra isomorphism between the algebra of 2x2 complex matrices (the Pauli algebra) and Cl3. Although no new results are presented, the Clifford algebra framework allows us to uncover the intrinsic geometric structure of these quantum systems.

Speaker: Yu Zhaoyuan (Nanjing Normal University, Nanjing, China)

Abstract: This presentation redefines geospatial analysis via a formal 'geographical language' framework, centered on Geometric Algebra as its mathematical core. Derived from a tri-world ontology (Natural, Human, Information), the framework decomposes spatiotemporal phenomena into a '7x7 periodic table' (7 elements × 7 dimensions), transforming unstructured data into machine-readable knowledge. It enables seamless translation of geographical semantics into Geometric Algebra-based computations, empowering AI systems (e.g., LLMs) with a semantic scaffold for automated reasoning. This work establishes the foundational science for geospatial Intelligence, advancing beyond 'where' to explain 'why', with actionable applications in smart cities, environmental management, and sustainable development.

Speaker: Ekaterina Filimoshina (HSE University, Moscow, Russia)

Abstract: This talk presents geometric algebras as a foundational framework for designing neural networks that are equivariant with respect to any pseudo-orthogonal transformation, such as rotations and reflections. We focus on the key mathematical result that enables this: the fundamental relationship between pseudo-orthogonal matrix groups and Lipschitz groups in geometric algebras. We will provide the motivation, consider the mathematical details, demonstrate current state-of-the-art equivariant models based on geometric algebras, present our own results on this topic, and discuss open questions.

Speaker: Zhenwei Guo (Liaocheng University, Liaocheng, China; NEFU Yakutsk, Russia)

Abstract: This report investigates the eigen-problems of reduced biquaternion matrices using their complex representations. Constructive algebraic algorithms are proposed to compute eigenvalues and eigenvectors, addressing the challenges caused by zero divisors in reduced biquaternion algebra. The study reveals that such matrices possess infinitely many eigenvalues and that multiple eigenvalues may correspond to the same eigenvector.

Speaker: Heerak Sharma (Indian Institute of Science Education and Research, Pune, India)

Abstract: In this talk, we will introduce commutative analogues of Clifford algebras – algebras defined in the same way as Clifford algebras but their generators commute instead of anti-commuting. We will show that these algebras have two 'isomorphism classes’ – any commutative analogue of Clifford algebra is either isomorphic to the multicomplex numbers or the 'multi-split-complex numbers'. We will discuss a tensor product decomposition, a direct sum decomposition and a representation for these algebras. We will end the discussion by giving formulas for multiplicative inverse in these algebras.