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Twistor operator in symplectic spin geometry
Dostálová, Marie ; Krýsl, Svatopluk (advisor) ; Doubek, Martin (referee)
The topic of the diploma thesis is symplectic spinor geometry. Its re- search was started by D. Shale, B. Kostant and K. Habermann. We focus our attention to one of the so called symplectic twistor operators introduced by S. Kr'ysl. We investigate the action of this operator on real even dimensio- nal vector spaces considered as symplectic manifold, its invariance properties and regularity. We describe a part of the kernel of the symplectic twistor operator when acting on symplectic spinors on R2. The kernel forms a repre- sentation of the so called metaplectic group (double cover of the symplectic group). 1
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New Integral Formulae in Hypercomplex Analysis
Sikora, Martin ; Souček, Vladimír (advisor) ; Krýsl, Svatopluk (referee) ; Vanžura, Jiří (referee)
Title: New Integral Formulae in Hypercomplex Analysis Author: Mgr. Martin Sikora Department: Mathematical Institute of Charles University Supervisor: prof. RNDr. Vladimír Souček, DrSc., MÚ UK Supervisor's e-mail address: soucek@karlin.mff.cuni.cz Abstract: The Dirac equation for Clifford algebra-valued functions on the even-dimensional Minkowski space can be understood as a hyperbolic sys- tem of partial differential equations. We show how to reconstruct the solution from initial data given on the upper sheet of the hyperboloid. In particular, we derive an integral formula which expresses the value of a function in a chosen point as an integral over a compact cycle given by the intersection of the null cone with the upper sheet of the hyperboloid in the Minkowski space. We also treat the ultra-hyperbolic case where the Dirac equation gives the ultra-hyperbolic system of partial differential equations. An analogue of the second order Cauchy formula is proved for (n − 1)-vector-valued holo- morphic functions. It reconstructs values inside a bounded domain in the 2n-dimensional complex space by integrating over the characteristic boun- dary of the domain. 1
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Generalized Cartan geometries and invariant differential operators
Salač, Tomáš ; Krýsl, Svatopluk (referee) ; Souček, Vladimír (advisor)
We are getting familiar with difficulties with invariance of differential operators in case of parabolic geometries and fully characterize first order invariant operators. We define, so called curved Casimir operator. It is generalization of Casimir operator from representation theory. We give a new prove of characterization of first order invariant operators. We investigate more thoroughly behavior of curved Casimir operator on section of tractor bandle in conformal case and give list of various apllications
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