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In dieser Arbeit wird die de-Rham-Kohomologie für die Quantengruppen zu den vier klassischen Serien von Lie-Gruppen bestimmt und es wird der Hodgeschen Zerlegungssatz gezeigt. Als entscheidendes Mittel wurde der Laplace-Beltrami-Operator L für Woronowicz’ äußere Algebren entwickelt. Für transzendente Werte von q und reguläre Kalkülparameter z ist L diagonalisierbar. Für die obigen Quantengruppen bestimmen wir die Eigenwerte von L, die neben q und z von zwei integralen dominanten Gewichten abhängen. Wie im klassischen Fall wird die de-Rham-Kohomologie durch harmonische Formen repräsentiert. Jedoch entspricht nur im Fall der A-Serie jeder harmonischen Form auch eine de-Rham-Kohomologieklasse. Im Falle der B-, C- und D-Serien sind biinvariante Formen nicht notwendig geschlossen. Es gilt aber, dass jede biinvariante Form harmonisch ist. Das zweite Hauptresultat ist die Hodge-Zerlegung für die Quantengruppen GLq(N) und SLq(N): Ist der Kalkülparameter z regulär, so lässt sich jede Form eindeutig zerlegen in die Summe aus einem Rand, einem Korand und einem Kohomologierepräsentanten. Ferner gilt, analog zum klassischen Fall, dass die folgenden drei Formenräume übereinstimmen: die biinvarianten Formen, die harmonischen Formen und die de-Rham-Kohomologie. Für die orthogonalen und symplektischen Quantengruppen gibt es keine vollständige Hodge-Zerlegung. Nur für die Elemente, die im Bild des Laplace-Beltrami-Operators liegen, gibt es eine eindeutige Zerlegung in Rand und Korand. Für die Standardkalküle auf den Quantengruppen GLq(N) und SLq(N) wird die Größe von Woronowicz’ äußerer Algebra bestimmt. Es wird gezeigt, dass der Raum der linksinvarianten k-Formen (N² über k)-dimensional ist. Die Algebra der biinvarianten Formen ist graduiert kommutativ. Ihre Poincaré-Reihe ist (1+t)(1+t³) ... (1+t^(2N-1)). Biinvariante Formen sind geschlossen. / Consider one of the standard bicovariant first order differential calculi for the quantum groups GLq(N), SLq(N), SOq(N), or SPq(N), where q is a transcendental complex number. It is shown that the de Rham cohomology of Woronowicz' external algebra coincides with the de Rham cohomologies of its left-invariant, its right-invariant and its bi-invariant subcomplexes. In the cases GLq(N) and SLq(N), the cohomology ring is isomorphic to the left-invariant external algebra and to the vector space of harmonic forms. We prove a Hodge decomposition theorem in these cases. The main technical tool is the spectral decomposition of the quantum Laplace-Beltrami operator. As in the classical case all three spaces of differential forms coincide: bi- invariant forms, harmonic forms and the de-Rham-cohomology. For orthog- onal and symplectic quantum groups there is no complete Hodge decompo- sition. In case of the standard calculi on the quantum groups GLq(N) and SLq(N), the size of exterior algebra is computed. The space of left-invariant k-forms has dimension C(N², k) (binomial coefficient). The algebra of bi-invariant forms is graded commutative with Poincaré series (1+t)(1+t³) ... (1+t^(2N-1)). Bi-invariant forms are closed.

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We introduce the formalism of supersymmetric quantum mechanics, including super-symmetry charges,Z2-graded Hilbert spaces, the chirality operator and the Wittenindex. We show that there is a one to one correspondence of fermions and bosons forenergies different than zero, which implies that the Witten index measures the dif-ference of fermions and bosons at the ground state. We argue that the Witten indexis the index of an elliptic operator. Quantization of the supersymmetric non-linearsigma model shows that the Witten index equals the Euler characteristic of the un-derlying Riemannian manifold over which the theory is defined. Finally, the pathintegral representation of the Witten index is applied to derive the Gauss-Bonnettheorem. Apart from this we introduce elementary mathematical background in thesubjects of topological invariance, Riemannian manifolds and index theory / Vi introducucerar formalismen f ̈or supersymmetrisk kvantmekanik, d ̈aribland super-symmetryladdningar,Z2-graderade Hilbertrum, kiralitetsoperatorn och Wittenin-dexet. Vi visar att det r ̊ader en till en-korrespondens mellan fermioner och bosonervid energiniv ̊aer skillda fr ̊an noll, vilket medf ̈or att Wittenindexet m ̈ater skillnadeni antal fermioner och bosoner vid nolltillst ̊andet. Vi argumenterar f ̈or att Wittenin-dexet ̈ar indexet p ̊a en elliptisk operator. Kvantisering av den supersymmetriskaicke-linj ̈ara sigmamodellen visar att Wittenindexet ̈ar Eulerkarakteristiken f ̈or denunderliggande Riemannska m ̊angfald ̈over vilken teorin ̈ar definierad. Slutligenapplicerar vi v ̈agintegralrepresentationen av Wittenindexet f ̈or att h ̈arleda Gauss-Bonnets sats. Ut ̈over detta introduceras ocks ̊a grundl ̈aggande matematisk bakrundi ämnena topologisk invarians, Riemmanska m ̊angfalder och indexteori.

In dieser Arbeit wird die de-Rham-Kohomologie für die Quantengruppen zu den vier klassischen Serien von Lie-Gruppen bestimmt und es wird der Hodgeschen Zerlegungssatz gezeigt. Als entscheidendes Mittel wurde der Laplace-Beltrami-Operator L für Woronowicz’ äußere Algebren entwickelt. Für transzendente Werte von q und reguläre Kalkülparameter z ist L diagonalisierbar. Für die obigen Quantengruppen bestimmen wir die Eigenwerte von L, die neben q und z von zwei integralen dominanten Gewichten abhängen. Wie im klassischen Fall wird die de-Rham-Kohomologie durch harmonische Formen repräsentiert. Jedoch entspricht nur im Fall der A-Serie jeder harmonischen Form auch eine de-Rham-Kohomologieklasse. Im Falle der B-, C- und D-Serien sind biinvariante Formen nicht notwendig geschlossen. Es gilt aber, dass jede biinvariante Form harmonisch ist. Das zweite Hauptresultat ist die Hodge-Zerlegung für die Quantengruppen GLq(N) und SLq(N): Ist der Kalkülparameter z regulär, so lässt sich jede Form eindeutig zerlegen in die Summe aus einem Rand, einem Korand und einem Kohomologierepräsentanten. Ferner gilt, analog zum klassischen Fall, dass die folgenden drei Formenräume übereinstimmen: die biinvarianten Formen, die harmonischen Formen und die de-Rham-Kohomologie. Für die orthogonalen und symplektischen Quantengruppen gibt es keine vollständige Hodge-Zerlegung. Nur für die Elemente, die im Bild des Laplace-Beltrami-Operators liegen, gibt es eine eindeutige Zerlegung in Rand und Korand. Für die Standardkalküle auf den Quantengruppen GLq(N) und SLq(N) wird die Größe von Woronowicz’ äußerer Algebra bestimmt. Es wird gezeigt, dass der Raum der linksinvarianten k-Formen (N² über k)-dimensional ist. Die Algebra der biinvarianten Formen ist graduiert kommutativ. Ihre Poincaré-Reihe ist (1+t)(1+t³) ... (1+t^(2N-1)). Biinvariante Formen sind geschlossen. / Consider one of the standard bicovariant first order differential calculi for the quantum groups GLq(N), SLq(N), SOq(N), or SPq(N), where q is a transcendental complex number. It is shown that the de Rham cohomology of Woronowicz'' external algebra coincides with the de Rham cohomologies of its left-invariant, its right-invariant and its bi-invariant subcomplexes. In the cases GLq(N) and SLq(N), the cohomology ring is isomorphic to the left-invariant external algebra and to the vector space of harmonic forms. We prove a Hodge decomposition theorem in these cases. The main technical tool is the spectral decomposition of the quantum Laplace-Beltrami operator. As in the classical case all three spaces of differential forms coincide: bi- invariant forms, harmonic forms and the de-Rham-cohomology. For orthog- onal and symplectic quantum groups there is no complete Hodge decompo- sition. In case of the standard calculi on the quantum groups GLq(N) and SLq(N), the size of exterior algebra is computed. The space of left-invariant k-forms has dimension C(N², k) (binomial coefficient). The algebra of bi-invariant forms is graded commutative with Poincaré series (1+t)(1+t³) ... (1+t^(2N-1)). Bi-invariant forms are closed.

If <i>E</i> is an exterior algebra on a finite dimensional vector space and <i>M</i> is a graded <i>E</i>-module, the relationship between the resonance varieties of <i>M</i> and the minimal free resolution of <i>M </i>is studied. In the context of Orlik–Solomon algebras, we give a condition under which elements of the second resonance variety can be obtained. We show that the resonance varieties of a general <i>M</i> are invariant under taking syzygy modules up to a shift. As corollary, it is shown that certain points in the resonance varieties of <i>M</i> can be determined from minimal syzygies of a special form, and we define syzygetic resonance varieties to be the subvarieties consisting of such points. The (depth one) syzygetic resonance varieties of a square-free module <i>M</i> over <i>E</i> are shown to be subspace arrangements whose components correspond to graded shifts in the minimal free resolution of <i>_SM</i>, the square-free module over a commutative polynomial ring <i>S </i>corresponding to <i>M</i>. Using this, a lower bound for the graded Betti numbers of the square-free module<i> M</i> is given. As another application, it is shown that the minimality of certain syzygies of Orlik–Solomon algebras yield linear subspaces of their (syzygetic) resonance varieties and lower bounds for their graded Betti numbers.

[pt] Arnold no seu trabalho The asymptotic Hopf Invariant and its applications de 1986, considerou sobre um domínio (ômega maiúsculo) compacto de R3 com bordo suave e homología trivial campos X e Y de divergência nula e tangentes ao bordo de (ômega maiúsculo) e definiu o índice de enlaçamento assintótico lk(X; Y ) e o invariante de Hopf associados a X e Y pela integral I(X; Y ) igual a (integral em ômega maiúsculo de alfa produto d-beta), onde (d-alfa) igual a iX-vol e (d-beta) igual a iy-vol, e mostrou que I(X; Y ) igual a lk(X; Y ). Agora, no presente trabalho estenderemos estas definições de índices de enlaçamento assintótico lk(fi maiúsculo,xi maiúsculo) e de invariante de Hopf I(fi maiúsculo,xi maiúsculo), onde (fi maiúsculo) e (xi maiúsculo) são ações de Rk e de Rs, k mais s igual a n-1, respectivamente de difeomorfismos que preservam volume em (ômega maiúsculo n) a bola unitária fechada em Rn e mostraremos que lk (fi maiúsculo, xi maiúsculo) igual a I(fi maiúsculo,xi maiúsculo). / [en] V.I. Arnold, in his paper The algebraic Hopf invariant and its applications published in 1986, considered a compact domain (ômega maiúsculo) in R3 with a smooth boundary and trivial homology and two divergence free vector fields X and Y in (ômega maiúsculo) tangent to the boundary. He defined an asymptotic linking invariant lk(X; Y ) and a Hopf invariant associated to X and Y by the integral I(X; Y ) equal (integral em ômega maiúsculo de alfa produto d-beta) where (d-alfa) equal iX-vol e (d-beta) equal iy- vol. He showed that que I(X; Y ) equal lk(X; Y ). In the present work we extend these definitions of the asymptotic linking invariant lk(fi capital letter,xi capital letter) and the Hopf invariant I(fi maiúsculo,xi capital letter) where (fi capital letter) and (xi capital letter) are actions Rk and Rs, k plus s equal n-1 by volume preserving diffeomorphisms, on the closed unit ball (ômega capital letter n) in and we show lk (fi capital letter, xi capital lette r equal I(ficapital letter ,xi capital letter).

The aim of this thesis is to give a mathematical framework for scattering of electromagnetic waves by rough surfaces. We prove that the Maxwell transmission problem with a weakly Lipschitz interface,in finite energy norms, is well posed in Fredholm sense for real frequencies. Furthermore, we give precise conditions on the material constants ε, μ and σ and the frequency ω when this transmission problem is well posed. To solve the Maxwell transmission problem, we embed Maxwell’s equations in an elliptic Dirac equation. We develop a new boundary integral method to solve the Dirac transmission problem. This method uses a boundary integral operator, the rotation operator, which factorises the double layer potential operator. We prove spectral estimates for this rotation operator in finite energy norms using Hodge decompositions on weakly Lipschitz domains. To ensure that solutions to the Dirac transmission problem indeed solve Maxwell’s equations, we introduce an exterior/interior derivative operator acting in the trace space. By showing that this operator commutes with the two basic reflection operators, we are able to prove that the Maxwell transmission problem is well posed. We also prove well-posedness for a class of oblique Dirac transmission problems with a strongly Lipschitz interface, in the L_2 space on the interface. This is shown by employing the Rellich technique, which gives angular spectral estimates on the rotation operator.

This thesis deals with cohomologies on the symplectic quotient of a Frölicher space which is locally diffeomorphic to a Euclidean Frölicher subspace of Rn of constant dimension equal to n. The symplectic reduction under consideration in this thesis is an extension of the Marsden-Weinstein quotient (also called, the reduced space) well-known from the finite-dimensional smooth manifold case. That is, starting with a proper and free action of a Frölicher-Lie-group on a locally Euclidean Frölicher space of finite constant dimension, we study the smooth structure and the topology induced on a small subspace of the orbit space. It is on this topological space that we will construct selected cohomologies such as : sheaf cohomology, Alexander-Spanier cohomology, singular cohomology, ~Cech cohomology and de Rham cohomology. Some natural questions that will be investigated are for instance: the impact of the symplectic structure on these di erent cohomologies; the cohomology that will give a good description of the topology on the objects of category of Frölicher spaces; the extension of the de Rham cohomology theorem in order to establish an isomorphism between the five cohomologies. Beside the algebraic, topological and geometric study of these new objects, the thesis contains a modern formalism of Hamiltonian mechanics on the reduced space under symplectic and Poisson structures. / Mathematical Sciences / D. Phil. (Mathematics)

This thesis deals with cohomologies on the symplectic quotient of a Frölicher space which is locally diffeomorphic to a Euclidean Frölicher subspace of Rn of constant dimension equal to n. The symplectic reduction under consideration in this thesis is an extension of the Marsden-Weinstein quotient (also called, the reduced space) well-known from the finite-dimensional smooth manifold case. That is, starting with a proper and free action of a Frölicher-Lie-group on a locally Euclidean Frölicher space of finite constant dimension, we study the smooth structure and the topology induced on a small subspace of the orbit space. It is on this topological space that we will construct selected cohomologies such as : sheaf cohomology, Alexander-Spanier cohomology, singular cohomology, ~Cech cohomology and de Rham cohomology. Some natural questions that will be investigated are for instance: the impact of the symplectic structure on these di erent cohomologies; the cohomology that will give a good description of the topology on the objects of category of Frölicher spaces; the extension of the de Rham cohomology theorem in order to establish an isomorphism between the five cohomologies. Beside the algebraic, topological and geometric study of these new objects, the thesis contains a modern formalism of Hamiltonian mechanics on the reduced space under symplectic and Poisson structures. / Mathematical Sciences / D. Phil. (Mathematics)