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Legendrian submanifolds in high-dimensional contact topology
Strakoš, Filip ; Golovko, Roman (advisor) ; O'Buachalla, Re (referee)
This thesis deals with results concerning both flexible and rigid parts of contact topol- ogy. Basic notions of contact topology and constructions of higher-dimensional Legen- drian submanifolds are stated. There is proved the existence of infinite family of pair-wise Legendrian non-isotopic loose Legendrian embeddings of 3-torus so that each embedding is not a Legendrian product of lower-dimensional tori. In the rest of the text, the bilin- earized Legendrian contact homology invariant is described and the criterion for DGA- homotopy of augmentation of Chekanov-Eliashberg algebra for disconnected Legendrian submanifolds is proved. 1
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Symplectic Dirac operators on Gr2(C4)
Hudeček, Štěpán ; Krýsl, Svatopluk (advisor) ; Golovko, Roman (referee)
In this thesis we are presenting a construction of the symplectic Dirac operators as done by Katharina Habermann in 1995. We emphasize the differences with the classical Dirac operators. We are then computing the associated second order operator to the symplectic Dirac operators on the Kähler symmetric space Gr2(C4 ). We have also managed to find a way of inductive computing of its spectrum and we are presenting explicitly a part of the spectrum. 1
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Nocommutative structures in quantum field theory
Peksová, Lada ; Jurčo, Branislav (advisor) ; Sachs, Ivo (referee) ; Golovko, Roman (referee)
In this thesis, structures defined via modular operads and properads are generalized to their non-commutative analogs. We define the connected sum for modular operads. This way we are able to construct the graded commutative product on the algebra over Feynman transform of the modular operad. This forms a Batalin-Vilkovisky algebra with symmetry given by the modular operad. We transfer this structure to the cohomology via the Homological perturbation lemma. In particular, we consider these constructions for Quantum closed and Quantum open modular operad. As a parallel project we introduce associative analog of Frobenius properad, called Open Frobenius properad. We construct the cobar complex over it and in the spirit of Barannikov interpret algebras over cobar complex as homological differential operators. Furthermore we present the IBA∞-algebras as analog of well-known IBL∞-algebras. 1
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Vector fields on spheres
Strakoš, Filip ; Salač, Tomáš (advisor) ; Golovko, Roman (referee)
This thesis deals with partial results concerning the problem of existence of vector fields on spheres. The proof of the Hairy Ball Theorem is given using the tools of the the- ory of characteristic classes. Basic notions of algebraic topology are stated in order to define the Euler class. Its definition is followed by the computation of the Euler charac- teristic class for the tangent bundle of even-dimensional sphere. In the rest of the text, the method of construction of vector fields on spheres using the orthogonal multiplica- tion is explained and the Radon-Hurwitz-Eckmann Theorem is proved. A brief historical background of the existence of the finite-dimensional real division algebras is mentioned at the end.
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Reprezentace Chekanovových-Eliashbergových algeber
Poppr, Marián ; Golovko, Roman (advisor) ; Le, Hong Van (referee)
In this thesis, we study modern invariants of Legendrian knots on R3 with a standard contact structure. We introduce the notion of Chekanov-Eliashberg algebra (DGA) and Legendrian contact homology. Then we consider representa- tions of DGA as a way how to derive some computable invariants of Legendrian knots. Finally, we will find equivalence classes of graded 2-dimensional irreducible representations for a certain Legendrian knot. i
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