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This thesis focuses on the connection between knot theory and Chern-Simons theory. In this regard, knot theory is explored and the Chern-Simons theory is discussed in terms of principle bundles. By quantising Chern-Simons theory a link invariant emerges: the vacuum expectation value, up to a factor of the partition function, of the product of a collection of Wilson loops. This link invariant depends on the gauge group and the representation, and relates to other link invariants, such as the Jones Polynomial, the HOMFLY-PT polynomial and the Kauffman polynomial. The discovery of this by Witten led to an intrinsically 3-dimensional way to calculate knot invariants. Moreover, by considering complexified tangent bundles and self-dual Lorentz connections on trivial complex tangent bundles, a solution to the Wheeler-De Witt equation was given, called the the Chern-Simons state. Furthermore, quantum states are considered as link invariants in terms of the components of self-dual Lorentz connections and their conjugate momentum., Denna avhandling fokuserar på kopplingen mellan knutteori och Chern-Simons teori. För att göra detta så utforskas knutteori och Chern-Simons teori samt G-buntar. Chern-Simons teori kvantiseras och då dyker det upp en länkinvariant: det onormaliserade vakuum väntevärdet av en samling Wilson loopar. Länkinvarianten beror på vilken gauge grupp och representation som används och har ett samband med andra länkinvarianter t.ex. Jones polynomet, HOMFLY-PT polynomet och Kauffman polynomet. Wittens upptäckt av detta ledde till ett 3-dimensionellt sätt att kalkylera länkinvarianter. Genom att fokusera på komplexifierade tangentbuntar och självduala Lorentz buntanslutning på triviala komplexa tangentbuntar så kan en lösning till Wheeler-De Witt ekvationen hittas. Denna lösning kallas för Chern-Simons tillståndet. I termer av komponenterna av självduala Lorentz buntanslutningar och deras generaliserade rörelsemängd så utforskar lösningarna till Wheeler-De Witt ekvationen i termer av länkinvarianter.

This thesis focuses on the connection between knot theory and Chern-Simons theory. In this regard, knot theory is explored and the Chern-Simons theory is discussed in terms of principle bundles. By quantising Chern-Simons theory a link invariant emerges: the vacuum expectation value, up to a factor of the partition function, of the product of a collection of Wilson loops. This link invariant depends on the gauge group and the representation, and relates to other link invariants, such as the Jones Polynomial, the HOMFLY-PT polynomial and the Kauffman polynomial. The discovery of this by Witten led to an intrinsically 3-dimensional way to calculate knot invariants. Moreover, by considering complexified tangent bundles and self-dual Lorentz connections on trivial complex tangent bundles, a solution to the Wheeler-De Witt equation was given, called the the Chern-Simons state. Furthermore, quantum states are considered as link invariants in terms of the components of self-dual Lorentz connections and their conjugate momentum., Denna avhandling fokuserar på kopplingen mellan knutteori och Chern-Simons teori. För att göra detta så utforskas knutteori och Chern-Simons teori samt G-buntar. Chern-Simons teori kvantiseras och då dyker det upp en länkinvariant: det onormaliserade vakuum väntevärdet av en samling Wilson loopar. Länkinvarianten beror på vilken gauge grupp och representation som används och har ett samband med andra länkinvarianter t.ex. Jones polynomet, HOMFLY-PT polynomet och Kauffman polynomet. Wittens upptäckt av detta ledde till ett 3-dimensionellt sätt att kalkylera länkinvarianter. Genom att fokusera på komplexifierade tangentbuntar och självduala Lorentz buntanslutning på triviala komplexa tangentbuntar så kan en lösning till Wheeler-De Witt ekvationen hittas. Denna lösning kallas för Chern-Simons tillståndet. I termer av komponenterna av självduala Lorentz buntanslutningar och deras generaliserade rörelsemängd så utforskar lösningarna till Wheeler-De Witt ekvationen i termer av länkinvarianter.

We give a simple mechanical motivation for a generalization of the classical Grassmannian considered as a space of m-dimensional linear subspaces of Rk to higher-orders cases. Our efforts are prolongated to the Weil functor theory, motivated by non-holonomic and semi-holonomic jets, applicable in the description of the Cosserat model.

We give a simple mechanical motivation for a generalization of the classical Grassmannian considered as a space of m-dimensional linear subspaces of Rk to higher-orders cases. Our efforts are prolongated to the Weil functor theory, motivated by non-holonomic and semi-holonomic jets, applicable in the description of the Cosserat model.

We give a simple mechanical motivation for a generalization of the classical Grassmannian considered as a space of m-dimensional linear subspaces of Rk to higher-orders cases. Our efforts are prolongated to the Weil functor theory, motivated by non-holonomic and semi-holonomic jets, applicable in the description of the Cosserat model.

In this article, we solve the problem of constructing moduli spaces of semistable principal bundles (and singular versions of them) over smooth projective varieties over algebraically closed ground fields of positive characteristic., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

4th Iberoamerican Conference on Complex Geometry Location: Ouro Preto, BRAZIL Date: AUG 12-18, 2007., This is a survey of work done in the past two decades relating a basic equation of general relativity and quantum field theory with the theory of stable vector bundles and stable principal bundles, as well as providing moduli spaces for such objects, and compactifying them., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

We give a simple mechanical motivation for a generalization of the classical Grassmannian considered as a space of m-dimensional linear subspaces of Rk to higher-orders cases. Our efforts are prolongated to the Weil functor theory, motivated by non-holonomic and semi-holonomic jets, applicable in the description of the Cosserat model.

We give a simple mechanical motivation for a generalization of the classical Grassmannian considered as a space of m-dimensional linear subspaces of Rk to higher-orders cases. Our efforts are prolongated to the Weil functor theory, motivated by non-holonomic and semi-holonomic jets, applicable in the description of the Cosserat model.

For complex connected, reductive, affine, algebraic groups G, we give a Lie-theoretic characterization of the semistability of principal G-co-Higgs bundles on the complex projective line p in terms of the simple roots of a Borel subgroup of G. We describe a stratification of the moduli space in terms of the Harder-Narasimhan type of the underlying bundle.

This article is a written version of author's lecture on generalized Kepler problems at the XVII-th International Conference on Geometry, Integrability and Quantization, June 5-10, 2015 Varna, Bulgaria. It begins with a review of the Kepler problem for planetary motion and its magnetized cousins, from which a surprising relationship with Lorentz transformation emerges. Next, we give a review for euclidean Jordan algebra and the associated universal Kepler problem. Finally, we demonstrate that, via the universal Kepler problem, a suitable Poisson realization of the conformal algebra for a simple euclidean Jordan algebra gives rise to a super integrable model that resembles the Kepler problem. In particular we demonstrate how the Kepler problem and its magnetized cousins are obtained this way.

This article is a written version of author's lecture on generalized Kepler problems at the XVII-th International Conference on Geometry, Integrability and Quantization, June 5-10, 2015 Varna, Bulgaria. It begins with a review of the Kepler problem for planetary motion and its magnetized cousins, from which a surprising relationship with Lorentz transformation emerges. Next, we give a review for euclidean Jordan algebra and the associated universal Kepler problem. Finally, we demonstrate that, via the universal Kepler problem, a suitable Poisson realization of the conformal algebra for a simple euclidean Jordan algebra gives rise to a super integrable model that resembles the Kepler problem. In particular we demonstrate how the Kepler problem and its magnetized cousins are obtained this way.

This article is a written version of author's lecture on generalized Kepler problems at the XVII-th International Conference on Geometry, Integrability and Quantization, June 5-10, 2015 Varna, Bulgaria. It begins with a review of the Kepler problem for planetary motion and its magnetized cousins, from which a surprising relationship with Lorentz transformation emerges. Next, we give a review for euclidean Jordan algebra and the associated universal Kepler problem. Finally, we demonstrate that, via the universal Kepler problem, a suitable Poisson realization of the conformal algebra for a simple euclidean Jordan algebra gives rise to a super integrable model that resembles the Kepler problem. In particular we demonstrate how the Kepler problem and its magnetized cousins are obtained this way.

4th Iberoamerican Conference on Complex Geometry Location: Ouro Preto, BRAZIL Date: AUG 12-18, 2007., This is a survey of work done in the past two decades relating a basic equation of general relativity and quantum field theory with the theory of stable vector bundles and stable principal bundles, as well as providing moduli spaces for such objects, and compactifying them., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

In this brief paper, we observe that basic results from rational homotopy theory provide formulas for the rational homotopy groups of gauge groups of principal bundles K -> P -> B in terms of the rational homotopy groups of K and cohomology groups of B alone.

In this article, we solve the problem of constructing moduli spaces of semistable principal bundles (and singular versions of them) over smooth projective varieties over algebraically closed ground fields of positive characteristic., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

In this article, we solve the problem of constructing moduli spaces of semistable principal bundles (and singular versions of them) over smooth projective varieties over algebraically closed ground fields of positive characteristic.

The structure of the algebra of gauge-invariant differential forms on the bundle $C\times _{M}E$ is determined, where $p\colon C\rightarrow M$ is the bundle of connections of a $U(1)$-principal bundle $\pi \colon P\rightarrow M$, and $E\rightarrow M$ is the associated bundle to $P$ by the representation $\lambda _{r}$, $r\in \mathbb{N}$, of $U(1)$ on $\mathbb{C}$ given by $\lambda _{r}(z)(w)=z^{r}\,w$, $z\in U(1)$, $w\in \mathbb{C}$.

Práce se zabývá Lieovou teorií z hlediska kinematiky a robotiky. V úvodní části je vybudován pojem variety jako základní pojem konfiguračního prostoru. Na konfiguračním prostoru je potom zavedena struktura, tj. Lieova grupa. K reprezentaci rychlostí je dále zaveden tečný prostor s vektorovým polem a na něm struktura Lieovy algebry. Tyto dvě struktury jsou propojeny exponenciálním zobrazením. Závěr práce se věnuje fibrovanému prostoru, zejména hlavnímu bandlu a hlavní konexi. V celé práci se objevují mnohé příklady, které dané pojmy ilustrují., This diploma thesis deals with the kinematic and robotic implications of Lie theory. In the introductory section, a manifold is defined as a basic element of configuration space. The main body of the thesis deals with the definition of a structure in the configuration space - Lie group. Tangent space with vector field including a structure of Lie algebra is defined to represent velocity. These two structures are connected using exponential mapping. The conclusion of the thesis focuses on fibre space, especially considering principal bundle and principal connection. Throughout the thesis, numerous examples are presented to illustrate the terms used.

Práce se zabývá Lieovou teorií z hlediska kinematiky a robotiky. V úvodní části je vybudován pojem variety jako základní pojem konfiguračního prostoru. Na konfiguračním prostoru je potom zavedena struktura, tj. Lieova grupa. K reprezentaci rychlostí je dále zaveden tečný prostor s vektorovým polem a na něm struktura Lieovy algebry. Tyto dvě struktury jsou propojeny exponenciálním zobrazením. Závěr práce se věnuje fibrovanému prostoru, zejména hlavnímu bandlu a hlavní konexi. V celé práci se objevují mnohé příklady, které dané pojmy ilustrují., This diploma thesis deals with the kinematic and robotic implications of Lie theory. In the introductory section, a manifold is defined as a basic element of configuration space. The main body of the thesis deals with the definition of a structure in the configuration space - Lie group. Tangent space with vector field including a structure of Lie algebra is defined to represent velocity. These two structures are connected using exponential mapping. The conclusion of the thesis focuses on fibre space, especially considering principal bundle and principal connection. Throughout the thesis, numerous examples are presented to illustrate the terms used.

Práce se zabývá Lieovou teorií z hlediska kinematiky a robotiky. V úvodní části je vybudován pojem variety jako základní pojem konfiguračního prostoru. Na konfiguračním prostoru je potom zavedena struktura, tj. Lieova grupa. K reprezentaci rychlostí je dále zaveden tečný prostor s vektorovým polem a na něm struktura Lieovy algebry. Tyto dvě struktury jsou propojeny exponenciálním zobrazením. Závěr práce se věnuje fibrovanému prostoru, zejména hlavnímu bandlu a hlavní konexi. V celé práci se objevují mnohé příklady, které dané pojmy ilustrují., This diploma thesis deals with the kinematic and robotic implications of Lie theory. In the introductory section, a manifold is defined as a basic element of configuration space. The main body of the thesis deals with the definition of a structure in the configuration space - Lie group. Tangent space with vector field including a structure of Lie algebra is defined to represent velocity. These two structures are connected using exponential mapping. The conclusion of the thesis focuses on fibre space, especially considering principal bundle and principal connection. Throughout the thesis, numerous examples are presented to illustrate the terms used.

Práce se zabývá Lieovou teorií z hlediska kinematiky a robotiky. V úvodní části je vybudován pojem variety jako základní pojem konfiguračního prostoru. Na konfiguračním prostoru je potom zavedena struktura, tj. Lieova grupa. K reprezentaci rychlostí je dále zaveden tečný prostor s vektorovým polem a na něm struktura Lieovy algebry. Tyto dvě struktury jsou propojeny exponenciálním zobrazením. Závěr práce se věnuje fibrovanému prostoru, zejména hlavnímu bandlu a hlavní konexi. V celé práci se objevují mnohé příklady, které dané pojmy ilustrují., This diploma thesis deals with the kinematic and robotic implications of Lie theory. In the introductory section, a manifold is defined as a basic element of configuration space. The main body of the thesis deals with the definition of a structure in the configuration space - Lie group. Tangent space with vector field including a structure of Lie algebra is defined to represent velocity. These two structures are connected using exponential mapping. The conclusion of the thesis focuses on fibre space, especially considering principal bundle and principal connection. Throughout the thesis, numerous examples are presented to illustrate the terms used.

Práce se zabývá Lieovou teorií z hlediska kinematiky a robotiky. V úvodní části je vybudován pojem variety jako základní pojem konfiguračního prostoru. Na konfiguračním prostoru je potom zavedena struktura, tj. Lieova grupa. K reprezentaci rychlostí je dále zaveden tečný prostor s vektorovým polem a na něm struktura Lieovy algebry. Tyto dvě struktury jsou propojeny exponenciálním zobrazením. Závěr práce se věnuje fibrovanému prostoru, zejména hlavnímu bandlu a hlavní konexi. V celé práci se objevují mnohé příklady, které dané pojmy ilustrují., This diploma thesis deals with the kinematic and robotic implications of Lie theory. In the introductory section, a manifold is defined as a basic element of configuration space. The main body of the thesis deals with the definition of a structure in the configuration space - Lie group. Tangent space with vector field including a structure of Lie algebra is defined to represent velocity. These two structures are connected using exponential mapping. The conclusion of the thesis focuses on fibre space, especially considering principal bundle and principal connection. Throughout the thesis, numerous examples are presented to illustrate the terms used.

Práce se zabývá Lieovou teorií z hlediska kinematiky a robotiky. V úvodní části je vybudován pojem variety jako základní pojem konfiguračního prostoru. Na konfiguračním prostoru je potom zavedena struktura, tj. Lieova grupa. K reprezentaci rychlostí je dále zaveden tečný prostor s vektorovým polem a na něm struktura Lieovy algebry. Tyto dvě struktury jsou propojeny exponenciálním zobrazením. Závěr práce se věnuje fibrovanému prostoru, zejména hlavnímu bandlu a hlavní konexi. V celé práci se objevují mnohé příklady, které dané pojmy ilustrují., This diploma thesis deals with the kinematic and robotic implications of Lie theory. In the introductory section, a manifold is defined as a basic element of configuration space. The main body of the thesis deals with the definition of a structure in the configuration space - Lie group. Tangent space with vector field including a structure of Lie algebra is defined to represent velocity. These two structures are connected using exponential mapping. The conclusion of the thesis focuses on fibre space, especially considering principal bundle and principal connection. Throughout the thesis, numerous examples are presented to illustrate the terms used.