Symmetry groups in hypercomplex analysis
| Thesis title in Czech: | Grupy symetrií v hyperkomplexní analýze |
|---|---|
| Thesis title in English: | Symmetry groups in hypercomplex analysis |
| Academic year of topic announcement: | 2021/2022 |
| Thesis type: | dissertation |
| Thesis language: | angličtina |
| Department: | Mathematical Institute of Charles University (32-MUUK) |
| Supervisor: | doc. RNDr. Roman Lávička, Ph.D. |
| Author: |
| Guidelines |
| The main aim is to apply representation theory methods to problems studied in hypercomplex analysis. Hypercomplex analysis is a higher dimensional generalization of the classical complex analysis and it studies properties of solutions of the Dirac equation in the Euclidean space of dimension m. The Dirac equation is a basic example of PDE's of the first order which are invariant with respect to the orthogonal group of rotations. Recently, many other invariant PDE's have been investigated in hypercomplex analysis, for example, the Hodge-de Rham and the Moisil-Theodoresco equations, the hermitian Dirac equations and the quaternionic Dirac equations. Since more and more complex invariant PDE's are dealt with it seems to be very useful to apply advanced tools from representation theory, including symmetry given by Lie groups and Lie supergroups, Howe duality, Gelfand-Tsetlin bases. Even though this approach is very succesful there are still some interesting open problems, for example, in several Clifford variables, in hypercomplex analysis on superspace and others. |
| References |
| [1] R. Delanghe, F. Sommen, V. Souček, Clifford Algebra and Spinor-valued Functions, Mathematics and Its Applications 53, Kluwer Academic Publishers, 1992.
[2] J. E. Gilbert and M. A. M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge University Press, Cambridge, 1991. [3] R. Lávička, Hypercomplex Analysis - Selected Topics, habilitation thesis, Faculty of Mathematics and Physics, Charles University, Prague, 2011. [4] A. I. Molev, Gelfand-Tsetlin bases for classical Lie algebras, in "Handbook of Algebra", Vol. 4, (M. Hazewinkel, Ed.), Elsevier, 2006, 109-170. [5] N. Ja. Vilenkin, Special Functions and Theory of Group Representations, Izdat. Nauka, Moscow, 1965, Transl. Math. Monographs, Vol. 22, Amer. Math. Soc, Providence, R. I.,1968. |
| Preliminary scope of work |
| Využij symetrii v teorii funkcí ve vyšších dimenzích! |
| Preliminary scope of work in English |
| Use symmetry in function theory in higher dimensions! |