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Published: 14 March 2023

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1

Lunt, John Burnham. "Measurable perturbations of diffusions on manifolds." Thesis, University of Edinburgh, 1992. http://hdl.handle.net/1842/12486.

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2

Nazaikinskii, Vladimir, Anton Savin, Bert-Wolfgang Schulze, and Boris Sternin. "Differential operators on manifolds with singularities : analysis and topology : Chapter 5: Manifolds with isolated singularities." Universität Potsdam, 2003. http://opus.kobv.de/ubp/volltexte/2008/2665/.

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Contents: Chapter 5: Manifolds with Isolated Singularities 5.1. Differential Operators and the Geometry of Singularities 5.1.1. How do isolated singularities arise? Examples 5.1.2. Definition and methods for the description of manifolds with isolated singularities 5.1.3. Bundles. The cotangent bundle 5.2. Asymptotics of Solutions, Function Spaces,Conormal Symbols 5.2.1. Conical singularities 5.2.2. Cuspidal singularities 5.3. A Universal Representation of Degenerate Operators and the Finiteness Theorem 5.3.1. The cylindrical representation 5.3.2. Continuity and compactness 5.3.3. Ellipticity and the finiteness theorem 5.4. Calculus of ΨDO 5.4.1. General ΨDO 5.4.2. The subalgebra of stabilizing ΨDO 5.4.3. Ellipticity and the finiteness theorem
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3

Ma, L., and Bert-Wolfgang Schulze. "Operators on manifolds with conical singularities." Universität Potsdam, 2009. http://opus.kobv.de/ubp/volltexte/2009/3660/.

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We construct elliptic elements in the algebra of (classical pseudo-differential) operators on a manifold M with conical singularities. The ellipticity of any such operator A refers to a pair of principal symbols (σ0, σ1) where σ0 is the standard (degenerate) homogeneous principal symbol, and σ1 is the so-called conormal symbol, depending on the complex Mellin covariable z. The conormal symbol, responsible for the conical singularity, is operator-valued and acts in Sobolev spaces on the base X of the cone. The σ1-ellipticity is a bijectivity condition for all z of real part (n + 1)/2 − γ, n = dimX, for some weight γ. In general, we have to rule out a discrete set of exceptional weights that depends on A. We show that for every operator A which is elliptic with respect to σ0, and for any real weight γ there is a smoothing Mellin operator F in the cone algebra such that A + F is elliptic including σ1. Moreover, we apply the results to ellipticity and index of (operator-valued) edge symbols from the calculus on manifolds with edges.
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4

Nazaikinskii, Vladimir, Anton Savin, Bert-Wolfgang Schulze, and Boris Sternin. "Differential operators on manifolds with singularities : analysis and topology : Chapter 7: The index problem on manifolds with singularities." Universität Potsdam, 2004. http://opus.kobv.de/ubp/volltexte/2008/2670/.

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Contents: Chapter 7: The Index Problemon Manifolds with Singularities Preface 7.1. The Simplest Index Formulas 7.1.1. General properties of the index 7.1.2. The index of invariant operators on the cylinder 7.1.3. Relative index formulas 7.1.4. The index of general operators on the cylinder 7.1.5. The index of operators of the form 1 + G with a Green operator G 7.1.6. The index of operators of the form 1 + G on manifolds with edges 7.1.7. The index on bundles with smooth base and fiber having conical points 7.2. The Index Problem for Manifolds with Isolated Singularities 7.2.1. Statement of the index splitting problem 7.2.2. The obstruction to the index splitting 7.2.3. Computation of the obstruction in topological terms 7.2.4. Examples. Operators with symmetries 7.3. The Index Problem for Manifolds with Edges 7.3.1. The index excision property 7.3.2. The obstruction to the index splitting 7.4. Bibliographical Remarks
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5

Spathopoulos, Michael P. "Filtering and stochastic control for diffusions on manifolds." Thesis, Imperial College London, 1987. http://hdl.handle.net/10044/1/46638.

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6

Schulze, Bert-Wolfgang. "Pseudo-differential calculus on manifolds with geometric singularities." Universität Potsdam, 2006. http://opus.kobv.de/ubp/volltexte/2009/3020/.

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Differential and pseudo-differential operators on a manifold with (regular) geometric singularities can be studied within a calculus, inspired by the concept of classical pseudo-differential operators on a C1 manifold. In the singular case the operators form an algebra with a principal symbolic hierarchy σ = (σj)0≤j≤k, with k being the order of the singularity and σk operator-valued for k ≥ 1. The symbols determine ellipticity and the nature of parametrices. It is typical in this theory that, similarly as in boundary value problems (which are special edge problems, where the edge is just the boundary), there are trace, potential and Green operators, associated with the various strata of the configuration. The operators, obtained from the symbols by various quantisations, act in weighted distribution spaces with multiple weights. We outline some essential elements of this calculus, give examples and also comment on new challenges and interesting problems of the recent development.
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7

Mooers, Edith. "The heat kernel for manifolds with conic singularities." Thesis, Massachusetts Institute of Technology, 1996. http://hdl.handle.net/1721.1/38406.

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8

Schulze, Bert-Wolfgang, and Nikolai Tarkhanov. "The Riemann-Roch theorem for manifolds with conical singularities." Universität Potsdam, 1997. http://opus.kobv.de/ubp/volltexte/2008/2505/.

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9

Nazaikinskii, Vladimir, Bert-Wolfgang Schulze, Boris Sternin, and Victor Shatalov. "Quantization of symplectic transformations on manifolds with conical singularities." Universität Potsdam, 1997. http://opus.kobv.de/ubp/volltexte/2008/2508/.

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The structure of symplectic (canonical) transformations on manifolds with conical singularities is established. The operators associated with these transformations are defined in the weight spaces and their properties investigated.
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10

Abed, Jamil. "An iterative approach to operators on manifolds with singularities." Phd thesis, Universität Potsdam, 2010. http://opus.kobv.de/ubp/volltexte/2010/4475/.

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We establish elements of a new approach to ellipticity and parametrices within operator algebras on manifolds with higher singularities, only based on some general axiomatic requirements on parameter-dependent operators in suitable scales of spaes. The idea is to model an iterative process with new generations of parameter-dependent operator theories, together with new scales of spaces that satisfy analogous requirements as the original ones, now on a corresponding higher level. The "full" calculus involves two separate theories, one near the tip of the corner and another one at the conical exit to infinity. However, concerning the conical exit to infinity, we establish here a new concrete calculus of edge-degenerate operators which can be iterated to higher singularities.
Wir führen einen neuen Zugang ein zu Elliptizität und Parametrices in Operatorenalgebren auf Mannigfaltigkeiten mit höheren Singularitäten, nur basierend auf allgemeinen axiomatischen Voraussetzungen über parameter-abhängige Operatoren in geeigneten Skalen von Räumen. Die Idee besteht darin, ein iteratives Verfahren zu modellieren mit neuen Generationen von parameter-abhängigen Operatortheorien, zusammen mit neuen Skalen von Räumen, die analoge Voraussetzungen erfüllen wie die ursprünglichen Objekte, jetzt auf dem entsprechenden höheren Niveau. Der „volle“ Kalkül besteht aus zwei separaten Theorien, eine nahe der Spitze der Ecke und eine andere am konischen Ausgang nach Unendlich. Allerdings, bezüglich des konischen Ausgangs nach Unendlich, bauen wir hier einen neuen konkreten Kalkül von kanten-entarteten Operatoren auf, der für höhere Singularitäten iteriert werden kann.
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11

Schulze, Bert-Wolfgang. "Operator algebras with symbol hierarchies on manifolds with singularities." Universität Potsdam, 1999. http://opus.kobv.de/ubp/volltexte/2008/2564/.

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Problems for elliptic partial differential equations on manifolds M with singularities M' (here with piece-wise smooth geometry)are studied in terms of pseudo-differential algebras with hierarchies of symbols that consist of scalar and operator-valued components. Classical boundary value problems (with or without the transmission property) belong to the examples. They are a model for operator algebras on manifolds M with higher "polyhedral" singularities. The operators are block matrices that have upper left corners containing the pseudo-differential operators on the regular MM' (plus certain Mellin and Green summands) and are degenerate (in streched coordinates) in a typical way near M'. By definition M' is again a manifold with singularities. The same is true of M'', and so on. The block matrices consist of trace, potential and Mellin and Green operators, acting between weighted Sobolev spaces on M(j) and M(k), with 0 ≤ j, k ≤ ord M; here M(0) denotes M, M(1) denotes M', etc. We generate these algebras, including their symbol hierarchies, by iterating so-called "edgifications" and "conifications" os algebras that have already been constructed, and we study ellipicity, parametrics and Fredholm property within these algebras.
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12

Schulze, Bert-Wolfgang. "The structure of operators on manifolds with polyhedral singularities." Universität Potsdam, 2006. http://opus.kobv.de/ubp/volltexte/2009/3009/.

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13

Caspart, Sven [Verfasser], and F. [Akademischer Betreuer] Herrlich. "Singularities of Translation Manifolds / Sven Caspart ; Betreuer: F. Herrlich." Karlsruhe : KIT-Bibliothek, 2021. http://d-nb.info/122951466X/34.

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14

Suleymanova, Asilya. "On the spectral geometry of manifolds with conic singularities." Doctoral thesis, Humboldt-Universität zu Berlin, 2017. http://dx.doi.org/10.18452/18420.

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Wir beginnen mit der Herleitung der asymptotischen Entwicklung der Spur des Wärmeleitungskernes, $\tr e^{-t\Delta}$, für $t\to0+$, wobei $\Delta$ der Laplace-Beltrami-Operator auf einer Mannigfaltigkeit mit Kegel-Singularitäten ist; dabei folgen wir der Arbeit von Brüning und Seeley. Dann untersuchen wir, wie die Koeffizienten der Entwicklung mit der Geometrie der Mannigfaltigkeit zusammenhängen, insbesondere fragen wir, ob die (mögliche) Singularität der Mannigfaltigkeit aus den Koeffizienten - und damit aus dem Spektrum des Laplace-Beltrami-Operators - abgelesen werden kann. In wurde gezeigt, dass im zweidimensionalen Fall ein logarithmischer Term und ein nicht lokaler Term im konstanten Glied genau dann verschwinden, wenn die Kegelbasis ein Kreis der Länge $2\pi$ ist, die Mannigfaltigkeit also geschlossen ist. Dann untersuchen wir wir höhere Dimensionen. Im vier-dimensionalen Fall zeigen wir, dass der logarithmische Term genau dann verschwindet, wenn die Kegelbasis eine sphärische Raumform ist. Wir vermuten, dass das Verschwinden eines nicht lokalen Beitrags zum konstanten Term äquivalent ist dazu, dass die Kegelbasis die runde Sphäre ist; das kann aber bisher nur im zyklischen Fall gezeigt werden. Für geraddimensionale Mannigfaltigkeiten höherer Dimension und mit Kegelbasis von konstanter Krümmung zeigen wir weiter, dass der logarithmische Term ein Polynom in der Krümmung ist, das Wurzeln ungleich 1 haben kann, so dass erst das Verschwinden von mehreren Termen - die derzeit noch nicht explizit behandelt werden können - die Geschlossenheit der Mannigfaltigkeit zur Folge haben könnte.
We derive a detailed asymptotic expansion of the heat trace for the Laplace-Beltrami operator on functions on manifolds with one conic singularity, using the Singular Asymptotics Lemma of Jochen Bruening and Robert T. Seeley. Then we investigate how the terms in the expansion reflect the geometry of the manifold. Since the general expansion contains a logarithmic term, its vanishing is a necessary condition for smoothness of the manifold. It is shown in the paper by Bruening and Seeley that in the two-dimensional case this implies that the constant term of the expansion contains a non-local term that determines the length of the (circular) cross section and vanishes precisely if this length equals $2\pi$, that is, in the smooth case. We proceed to the study of higher dimensions. In the four-dimensional case, the logarithmic term in the expansion vanishes precisely when the cross section is a spherical space form, and we expect that the vanishing of a further singular term will imply again smoothness, but this is not yet clear beyond the case of cyclic space forms. In higher dimensions the situation is naturally more difficult. We illustrate this in the case of cross sections with constant curvature. Then the logarithmic term becomes a polynomial in the curvature with roots that are different from 1, which necessitates more vanishing of other terms, not isolated so far.
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15

Nazaikinskii, Vladimir, Bert-Wolfgang Schulze, Boris Sternin, and Victor Shatalov. "A Lefschetz fixed point theorem for manifolds with conical singularities." Universität Potsdam, 1997. http://opus.kobv.de/ubp/volltexte/2008/2507/.

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16

Nazaikinskii, Vladimir, Anton Savin, Bert-Wolfgang Schulze, and Boris Sternin. "Elliptic theory on manifolds with nonisolated singularities : II. Products in elliptic theory on manifolds with edges." Universität Potsdam, 2002. http://opus.kobv.de/ubp/volltexte/2008/2633/.

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Exterior tensor products of elliptic operators on smooth manifolds and manifolds with conical singularities are used to obtain examples of elliptic operators on manifolds with edges that do not admit well-posed edge boundary and coboundary conditions.
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17

Nazaikinskii, Vladimir, Anton Savin, Bert-Wolfgang Schulze, and Boris Sternin. "Elliptic theory on manifolds with nonisolated singularities : IV. Obstructions to elliptic problems on manifolds with edges." Universität Potsdam, 2002. http://opus.kobv.de/ubp/volltexte/2008/2641/.

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The obstruction to the existence of Fredholm problems for elliptic differentail operators on manifolds with edges is a topological invariant of the operator. We give an explicit general formula for this invariant. As an application we compute this obstruction for geometric operators.
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18

Nazaikinskii, Vladimir, Anton Savin, Bert-Wolfgang Schulze, and Boris Sternin. "Elliptic theory on manifolds with nonisolated singularities : V. Index formulas for elliptic problems on manifolds with edges." Universität Potsdam, 2003. http://opus.kobv.de/ubp/volltexte/2008/2650/.

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For elliptic problems on manifolds with edges, we construct index formulas in form of a sum of homotopy invariant contributions of the strata (the interior of the manifold and the edge). Both terms are the indices of elliptic operators, one of which acts in spaces of sections of finite-dimensional vector bundles on a compact closed manifold and the other in spaces of sections of infinite-dimensional vector bundles over the edge.
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19

Nazaikinskii, Vladimir, Anton Savin, Bert-Wolfgang Schulze, and Boris Sternin. "Differential operators on manifolds with singularities : analysis and topology : Chapter 6: Elliptic theory on manifolds with edges." Universität Potsdam, 2004. http://opus.kobv.de/ubp/volltexte/2008/2675/.

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Contents: Chapter 6: Elliptic Theory on Manifolds with Edges Introduction 6.1. Motivation and Main Constructions 6.1.1. Manifolds with edges 6.1.2. Edge-degenerate differential operators 6.1.3. Symbols 6.1.4. Elliptic problems 6.2. Pseudodifferential Operators 6.2.1. Edge symbols 6.2.2. Pseudodifferential operators 6.2.3. Quantization 6.3. Elliptic Morphisms and the Finiteness Theorem 6.3.1. Matrix Green operators 6.3.2. General morphisms 6.3.3. Ellipticity, Fredholm property, and smoothness Appendix A. Fiber Bundles and Direct Integrals A.1. Local theory A.2. Globalization A.3. Versions of the Definition of the Norm
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20

Schulze, Bert-Wolfgang, Boris Sternin, and Victor Shatalov. "On the index of differential operators on manifolds with conical singularities." Universität Potsdam, 1997. http://opus.kobv.de/ubp/volltexte/2008/2496/.

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The paper contains the proof of the index formula for manifolds with conical points. For operators subject to an additional condition of spectral symmetry, the index is expressed as the sum of multiplicities of spectral points of the conormal symbol (indicial family) and the integral from the Atiyah-Singer form over the smooth part of the manifold. The obtained formula is illustrated by the example of the Euler operator on a two-dimensional manifold with conical singular point.
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21

Schulze, Bert-Wolfgang, Vladimir Nazaikinskii, and Boris Sternin. "The index of quantized contact transformations on manifolds with conical singularities." Universität Potsdam, 1998. http://opus.kobv.de/ubp/volltexte/2008/2527/.

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The quantization of contact transformations of the cosphere bundle over a manifold with conical singularities is described. The index of Fredholm operators given by this quantization is calculated. The answer is given in terms of the Epstein-Melrose contact degree and the conormal symbol of the corresponding operator.
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22

Schulze, Bert-Wolfgang, Vladimir E. Nazaikinskii, and Boris Yu Sternin. "On the homotopy classification of elliptic operators on manifolds with singularities." Universität Potsdam, 1999. http://opus.kobv.de/ubp/volltexte/2008/2557/.

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We study the homotopy classification of elliptic operators on manifolds with singularities and establish necessary and sufficient conditions under which the classification splits into terms corresponding to the principal symbol and the conormal symbol.
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23

Coriasco, Sandro, Elmar Schrohe, and Jörg Seiler. "Bounded imaginary powers of differential operators on manifolds with conical singularities." Universität Potsdam, 2001. http://opus.kobv.de/ubp/volltexte/2008/2596/.

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We study the minimal and maximal closed extension of a differential operator A on a manifold B with conical singularities, when A acts as an unbounded operator on weighted Lp-spaces over B,1 < p < ∞. Under suitable ellipticity assumptions we can define a family of complex powers A up(z), z ∈ C. We also obtain sufficient information on the resolvent of A to show the boundedness of the pure imaginary powers. Examples concern unique solvability and maximal regularity of the solution of the Cauchy problem u' - Δu = f, u(0) = 0, for the Laplacian on conical manifolds.
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24

JANIGRO, AGNESE. "Compact 3-dimensional Anti-de Sitter manifolds with spin-cone singularities." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2022. https://hdl.handle.net/10281/402356.

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In questa tesi, ci occupiamo dello studio delle varietà anti-de Sitter compatte di dimensione 3 dotate di singolarità coniche con spin generalizzate. Data una superficie chiusa con metrica iperbolica e una mappa di contrazione tra il rivestimento universale della superficie e il semipiano iperbolico, è possibile costruire una varietà anti-de Sitter compatta di dimensione 3 come fibrato vettoriale con base la superficie iperbolica. Seguendo la stessa costruzione del caso non singolare, mostriamo che, a partire da una superficie equipaggiata di metrica iperbolica con singolarità coniche, è possibile costruire varietà compatte anti-de Sitter come fibrati vettoriali sulla superficie con fibre singolari. Queste fibre singolari sono localmente isometriche al cosiddetto Modello per singolarità conica con spin generalizzata. In particolare, dalla costruzione del modello, vengono fuori due invarianti che permettono di studiare tali varietà anti-de Sitter singolari. L’ultimo risultato trattato riguarda il calcolo del volume delle varietà anti-de Sitter con singolarità di questo tipo.
In this thesis, we study compact Anti-de Sitter manifolds of dimension 3 with generalized spin-cone singularities. Given a closed surface equipped with a hyperbolic metric and a contraction map between the universal cover of the surface and the hyperbolic plane, it is possible to construct a compact Anti-de Sitter manifold of dimension 3 as fiber bundle over the surface. We show that, when the surface has hyperbolic metric with conical singular points, the same construction of the non singular case leads to compact Anti-de Sitter manifolds as fiber bundle with singular fibers over the surface. These singular fibers over the singular conical points are locally isometric to what we defined Model for generalized spin-cone singularity. In particular, from the model come out two invariants that allows us to study the compact Anti-de Sitter manifolds of dimension 3 with spin-cone singularities. The last result of this work is about the computation of the volume of these compact Anti-de Sitter manifolds with spin-cone singularities.
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25

Högele, Michael, and Paulo Ruffino. "Averaging along Lévy diffusions in foliated spaces." Universität Potsdam, 2013. http://opus.kobv.de/ubp/volltexte/2013/6492/.

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We consider an SDE driven by a Lévy noise on a foliated manifold, whose trajectories stay on compact leaves. We determine the effective behavior of the system subject to a small smooth transversal perturbation of positive order epsilon. More precisely, we show that the average of the transversal component of the SDE converges to the solution of a deterministic ODE, according to the average of the perturbing vector field with respect to the invariant measures on the leaves (of the unpertubed system) as epsilon goes to 0. In particular we give upper bounds for the rates of convergence. The main results which are proved for pure jump Lévy processes complement the result by Gargate and Ruffino for Stratonovich SDEs to Lévy driven SDEs of Marcus type.
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26

Nazaikinskii, Vladimir, Anton Savin, Bert-Wolfgang Schulze, and Boris Sternin. "Differential operators on manifolds with singularities : analysis and topology : Chapter 4: Pseudodifferential operators." Universität Potsdam, 2003. http://opus.kobv.de/ubp/volltexte/2008/2658/.

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Contents: Chapter 4: Pseudodifferential Operators 4.1. Preliminary Remarks 4.1.1. Why are pseudodifferential operators needed? 4.1.2. What is a pseudodifferential operator? 4.1.3. What properties should the pseudodifferential calculus possess? 4.2. Classical Pseudodifferential Operators on Smooth Manifolds 4.2.1. Definition of pseudodifferential operators on a manifold 4.2.2. Hörmander’s definition of pseudodifferential operators 4.2.3. Basic properties of pseudodifferential operators 4.3. Pseudodifferential Operators in Sections of Hilbert Bundles 4.3.1. Hilbert bundles 4.3.2. Operator-valued symbols. Specific features of the infinite-dimensional case 4.3.3. Symbols of compact fiber variation 4.3.4. Definition of pseudodifferential operators 4.3.5. The composition theorem 4.3.6. Ellipticity 4.3.7. The finiteness theorem 4.4. The Index Theorem 4.4.1. The Atiyah–Singer index theorem 4.4.2. The index theorem for pseudodifferential operators in sections of Hilbert bundles 4.4.3. Proof of the index theorem 4.5. Bibliographical Remarks
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27

Khalil, Sara [Verfasser], and Bert-Wolfgang [Akademischer Betreuer] Schulze. "Boundary Value Problems on Manifolds with Singularities / Sara Khalil ; Betreuer: Bert-Wolfgang Schulze." Potsdam : Universität Potsdam, 2017. http://d-nb.info/1218404108/34.

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28

Perlmutter, Nathan. "Linking Forms, Singularities, and Homological Stability for Diffeomorphism Groups of Odd Dimensional Manifolds." Thesis, University of Oregon, 2015. http://hdl.handle.net/1794/19241.

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Let n > 1. We prove a homological stability theorem for the diffeomorphism groups of (4n+1)-dimensional manifolds, with respect to forming the connected sum with (2n-1)-connected, (4n+1)-dimensional manifolds that are stably parallelizable. Our techniques involve the study of the action of the diffeomorphism group of a manifold M on the linking form associated to the homology groups of M. In order to study this action we construct a geometric model for the linking form using the intersections of embedded and immersed Z/k-manifolds. In addition to our main homological stability theorem, we prove several results regarding disjunction for embeddings and immersions of Z/k-manifolds that could be of independent interest.
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29

Schulze, Bert-Wolfgang, and Nikolai N. Tarkhanov. "Elliptic complexes of pseudodifferential operators on manifolds with edges." Universität Potsdam, 1998. http://opus.kobv.de/ubp/volltexte/2008/2525/.

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On a compact closed manifold with edges live pseudodifferential operators which are block matrices of operators with additional edge conditions like boundary conditions in boundary value problems. They include Green, trace and potential operators along the edges, act in a kind of Sobolev spaces and form an algebra with a wealthy symbolic structure. We consider complexes of Fréchet spaces whose differentials are given by operators in this algebra. Since the algebra in question is a microlocalization of the Lie algebra of typical vector fields on a manifold with edges, such complexes are of great geometric interest. In particular, the de Rham and Dolbeault complexes on manifolds with edges fit into this framework. To each complex there correspond two sequences of symbols, one of the two controls the interior ellipticity while the other sequence controls the ellipticity at the edges. The elliptic complexes prove to be Fredholm, i.e., have a finite-dimensional cohomology. Using specific tools in the algebra of pseudodifferential operators we develop a Hodge theory for elliptic complexes and outline a few applications thereof.
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30

Fedosov, Boris, Bert-Wolfgang Schulze, and Nikolai Tarkhanov. "The index of elliptic operators on manifolds with conical points." Universität Potsdam, 1997. http://opus.kobv.de/ubp/volltexte/2008/2509/.

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For general elliptic pseudodifferential operators on manifolds with singular points, we prove an algebraic index formula. In this formula the symbolic contributions from the interior and from the singular points are explicitly singled out. For two-dimensional manifolds, the interior contribution is reduced to the Atiyah-Singer integral over the cosphere bundle while two additional terms arise. The first of the two is one half of the 'eta' invariant associated to the conormal symbol of the operator at singular points. The second term is also completely determined by the conormal symbol. The example of the Cauchy-Riemann operator on the complex plane shows that all the three terms may be non-zero.
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31

Schulze, Bert-Wolfgang, Vladimir Nazaikinskii, and Boris Sternin. "A semiclassical quantization on manifolds with singularities and the Lefschetz Formula for Elliptic Operators." Universität Potsdam, 1998. http://opus.kobv.de/ubp/volltexte/2008/2529/.

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For general endomorphisms of elliptic complexes on manifolds with conical singularities, the semiclassical asymptotics of the Atiyah-Bott-Lefschetz number is calculated in terms of fixed points of the corresponding canonical transformation of the symplectic space.
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32

Kapanadze, David, and Bert-Wolfgang Schulze. "Symbolic calculus for boundary value problems on manifolds with edges." Universität Potsdam, 2001. http://opus.kobv.de/ubp/volltexte/2008/2604/.

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Boundary value problems for (pseudo-) differential operators on a manifold with edges can be characterised by a hierarchy of symbols. The symbol structure is responsible or ellipicity and for the nature of parametrices within an algebra of "edge-degenerate" pseudo-differential operators. The edge symbol component of that hierarchy takes values in boundary value problems on an infinite model cone, with edge variables and covariables as parameters. Edge symbols play a crucial role in this theory, in particular, the contribution with holomorphic operatot-valued Mellin symbols. We establish a calculus in s framework of "twisted homogenity" that refers to strongly continuous groups of isomorphisms on weighted cone Sobolev spaces. We then derive an equivalent representation with a particularly transparent composition behaviour.
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33

Behrndt, Tapio. "Generalized Lagrangian mean curvature flow in almost Calabi-Yau manifolds." Thesis, University of Oxford, 2011. http://ora.ox.ac.uk/objects/uuid:f8a490d4-5b7c-4709-96e5-65ad3fefe922.

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In this work we study two problems about parabolic partial differential equations on Riemannian manifolds with conical singularities. The first problem we are concerned with is the existence and regularity of solutions to the Cauchy problem for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. By introducing so called weighted Hölder and Sobolev spaces with discrete asymptotics, we provide a complete existence and regularity theory for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. The second problem we study is the short time existence problem for the generalized Lagrangian mean curvature flow in almost Calabi-Yau manifolds, when the initial Lagrangian submanifold has isolated conical singularities that are modelled on stable special Lagrangian cones. First we use Lagrangian neighbourhood theorems for Lagrangian submanifolds with conical singularities to integrate the generalized Lagrangian mean curvature flow to a nonlinear parabolic equation of functions, and then, using the existence and regularity theory for the heat equation, we prove short time existence of the generalized Lagrangian mean curvature flow with isolated conical singularities by letting the conical singularities move around in the ambient space and the model cones to rotate by unitary transformations.
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34

Nazaikinskii, Vladimir, Anton Savin, Bert-Wolfgang Schulze, and Boris Sternin. "Elliptic theory on manifolds with nonisolated singularities : I. The index of families of cone-degenerate operators." Universität Potsdam, 2002. http://opus.kobv.de/ubp/volltexte/2008/2632/.

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We study the index problem for families of elliptic operators on manifolds with conical singularities. The relative index theorem concerning changes of the weight line is obtained. AN index theorem for families whose conormal symbols satisfy some symmetry conditions is derived.
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35

Nazaikinskii, Vladimir, Anton Savin, Bert-Wolfgang Schulze, and Boris Sternin. "Elliptic theory on manifolds with nonisolated singularities : III. The spectral flow of families of conormal symbols." Universität Potsdam, 2002. http://opus.kobv.de/ubp/volltexte/2008/2638/.

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When studyind elliptic operators on manifolds with nonisolated singularities one naturally encounters families of conormal symbols (i.e. operators elliptic with parameter p ∈ IR in the sense of Agranovich-Vishik) parametrized by the set of singular points. For homotopies of such families we define the notion of spectral flow, which in this case is an element of the K-group of the parameter space. We prove that the spectral flow is equal to the index of some family of operators on the infinite cone.
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36

Nazaikinskii, Vladimir, Anton Savin, Bert-Wolfgang Schulze, and Boris Sternin. "Differential operators on manifolds with singularities : analysis and topology : Chapter 1: Localization (surgery) in elliptic theory." Universität Potsdam, 2003. http://opus.kobv.de/ubp/volltexte/2008/2654/.

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Contents: Chapter 1: Localization (Surgery) in Elliptic Theory 1.1. The Index Locality Principle 1.1.1. What is locality? 1.1.2. A pilot example 1.1.3. Collar spaces 1.1.4. Elliptic operators 1.1.5. Surgery and the relative index theorem 1.2. Surgery in Index Theory on Smooth Manifolds 1.2.1. The Booß–Wojciechowski theorem 1.2.2. The Gromov–Lawson theorem 1.3. Surgery for Boundary Value Problems 1.3.1. Notation 1.3.2. General boundary value problems 1.3.3. A model boundary value problem on a cylinder 1.3.4. The Agranovich–Dynin theorem 1.3.5. The Agranovich theorem 1.3.6. Bojarski’s theorem and its generalizations 1.4. (Micro)localization in Lefschetz theory 1.4.1. The Lefschetz number 1.4.2. Localization and the contributions of singular points 1.4.3. The semiclassical method and microlocalization 1.4.4. The classical Atiyah–Bott–Lefschetz theorem
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37

Nazaikinskii, Vladimir, Anton Savin, Bert-Wolfgang Schulze, and Boris Sternin. "Differential operators on manifolds with singularities : analysis and topology : Chapter 3: Eta invariant and the spectral flow." Universität Potsdam, 2003. http://opus.kobv.de/ubp/volltexte/2008/2659/.

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Contents: Chapter 3: Eta Invariant and the Spectral Flow 3.1. Introduction 3.2. The Classical Spectral Flow 3.2.1. Definition and main properties 3.2.2. The spectral flow formula for periodic families 3.3. The Atiyah–Patodi–Singer Eta Invariant 3.3.1. Definition of the eta invariant 3.3.2. Variation under deformations of the operator 3.3.3. Homotopy invariance. Examples 3.4. The Eta Invariant of Families with Parameter (Melrose’s Theory) 3.4.1. A trace on the algebra of parameter-dependent operators 3.4.2. Definition of the Melrose eta invariant 3.4.3. Relationship with the Atiyah–Patodi–Singer eta invariant 3.4.4. Locality of the derivative of the eta invariant. Examples 3.5. The Spectral Flow of Families of Parameter-Dependent Operators 3.5.1. Meromorphic operator functions. Multiplicities of singular points 3.5.2. Definition of the spectral flow 3.6. Higher Spectral Flows 3.6.1. Spectral sections 3.6.2. Spectral flow of homotopies of families of self-adjoint operators 3.6.3. Spectral flow of homotopies of families of parameter-dependent operators 3.7. Bibliographical Remarks
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38

Suleymanova, Asilya [Verfasser], Jochen [Gutachter] Bruening, Julie [Gutachter] Rowlett, and Klaus [Gutachter] Kirsten. "On the spectral geometry of manifolds with conic singularities / Asilya Suleymanova ; Gutachter: Jochen Bruening, Julie Rowlett, Klaus Kirsten." Berlin : Humboldt-Universität zu Berlin, 2017. http://d-nb.info/1185579370/34.

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39

Fedosov, Boris, Bert-Wolfgang Schulze, and Nikolai N. Tarkhanov. "A remark on the index of symmetric operators." Universität Potsdam, 1998. http://opus.kobv.de/ubp/volltexte/2008/2516/.

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We introduce a natural symmetry condition for a pseudodifferential operator on a manifold with cylindrical ends ensuring that the operator admits a doubling across the boundary. For such operators we prove an explicit index formula containing, apart from the Atiyah-Singer integral, a finite number of residues of the logarithmic derivative of the conormal symbol.
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40

Fedosov, Boris, Bert-Wolfgang Schulze, and Nikolai Tarkhanov. "On the index formula for singular surfaces." Universität Potsdam, 1997. http://opus.kobv.de/ubp/volltexte/2008/2511/.

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In the preceding paper we proved an explicit index formula for elliptic pseudodifferential operators on a two-dimensional manifold with conical points. Apart from the Atiyah-Singer integral, it contains two additional terms, one of the two being the 'eta' invariant defined by the conormal symbol. In this paper we clarify the meaning of the additional terms for differential operators.
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41

Fedosov, Boris, Bert-Wolfgang Schulze, and Nikolai N. Tarkhanov. "The index of higher order operators on singular surfaces." Universität Potsdam, 1998. http://opus.kobv.de/ubp/volltexte/2008/2512/.

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The index formula for elliptic pseudodifferential operators on a two-dimensional manifold with conical points contains the Atiyah-Singer integral as well as two additional terms. One of the two is the 'eta' invariant defined by the conormal symbol, and the other term is explicitly expressed via the principal and subprincipal symbols of the operator at conical points. In the preceding paper we clarified the meaning of the additional terms for first-order differential operators. The aim of this paper is an explicit description of the contribution of a conical point for higher-order differential operators. We show that changing the origin in the complex plane reduces the entire contribution of the conical point to the shifted 'eta' invariant. In turn this latter is expressed in terms of the monodromy matrix for an ordinary differential equation defined by the conormal symbol.
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42

Enders, Joerg. "Generalizations of the reduced distance in the Ricci flow - monotonicity and applications." Diss., Connect to online resource - MSU authorized users, 2008.

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43

Ashley, Michael John Siew Leung, and ashley@gravity psu edu. "Singularity theorems and the abstract boundary construction." The Australian National University. Faculty of Science, 2002. http://thesis.anu.edu.au./public/adt-ANU20050209.165310.

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The abstract boundary construction of Scott and Szekeres has proven a practical classification scheme for boundary points of pseudo-Riemannian manifolds. It has also proved its utility in problems associated with the re-embedding of exact solutions containing directional singularities in space-time. Moreover it provides a model for singularities in space-time - essential singularities. However the literature has been devoid of abstract boundary results which have results of direct physical applicability.¶ This thesis presents several theorems on the existence of essential singularities in space-time and on how the abstract boundary allows definition of optimal em- beddings for depicting space-time. Firstly, a review of other boundary constructions for space-time is made with particular emphasis on the deficiencies they possess for describing singularities. The abstract boundary construction is then pedagogically defined and an overview of previous research provided.¶ We prove that strongly causal, maximally extended space-times possess essential singularities if and only if they possess incomplete causal geodesics. This result creates a link between the Hawking-Penrose incompleteness theorems and the existence of essential singularities. Using this result again together with the work of Beem on the stability of geodesic incompleteness it is possible to prove the stability of existence for essential singularities.¶ Invariant topological contact properties of abstract boundary points are presented for the first time and used to define partial cross sections, which are an generalization of the notion of embedding for boundary points. Partial cross sections are then used to define a model for an optimal embedding of space-time.¶ Finally we end with a presentation of the current research into the relationship between curvature singularities and the abstract boundary. This work proposes that the abstract boundary may provide the correct framework to prove curvature singularity theorems for General Relativity. This exciting development would culminate over 30 years of research into the physical conditions required for curvature singularities in space-time.
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44

Binotto, Rosane Rossato. "Projetivos de curvatura." [s.n.], 2007. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306624.

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Orientadores: Sueli Irene Rodrigues Costa, Maria del Carmen Romero-Fuster
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
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Resumo: O projetivo de curvatura em um ponto de uma 3-variedade M de classe 'C POT. 2' imersa em 'IR POT. ?' , n >-4, é o lugar geométrico de todos os extremos dos vetores curvatura de secções normais ao longo de todas as direções tangentes a M em p. Mostramos que o projetivo de curvatura em p é isomorfo (difeomorfo) à superfície de Veronese clássica de ordem 2, composta com uma transformação linear. Conforme o posto desta transformação linear, o projetivo de curvatura será dado por projeções da superfície de Veronese em subespaços do espaço normal da variedade M. Quanto menor o posto, maior será a umbilicidade da variedade no ponto em questão. Também estudamos a natureza geométrica e singularidades para os diferentes casos de projetivos de curvatura em pontos de M, os quais incluem a superfície Romana de Steiner, a Cross-Cap, a superfície de Steiner de Tipo 5 e a Cross-Cup. Além disso, analisamos os pontos singulares de segunda ordem da imersão, no sentido de Feldman e estabelecemos condições relacionadas à natureza do projetivo de curvatura, para que uma 3-variedade imersa em 'IR POT. ?', n >_ 9, tenha contato de ordem _ 2 com k-planos e k-esferas de IRn, 3 _ k _ 8
Abstract: The curvature projective plane at each point p of three-manifolds M immersed in 'IR POT. ?', n _ 4, is the geometric locus of all end points of the curvature vectors of normal sections along of all tangent directions of M at p. In this study, we show that the curvature projective plane is isomorphic (diffeomorphic) to the classical Veronese surface of order two, composed with a linear transformation, and that according to the rank of this mapping, the curvature projective plane will be given by projections of the Veronese surface into subspaces of the normal space of M at p. Thus, the smaller the rank the greater the umbilicity of the manifold at this point. We also study the geometric nature and singularities of the curvature projective planes considering different possibilities, which include the Roman Steiner surface, the Cross-Cap, the Steiner surface of five-type, and the Cross-Cup. In addition, we analyze the order-two singularities of the immersion in the Feldman¿s sense and establish conditions related to the nature of the curvature projective plane for the existence of contacts of the three-manifolds in 'IR POT. ?', n _ 9, with k-planes and k-spheres, 3 _ k _ 8
Doutorado
Geometria
Doutor em Matemática
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45

Kapanadze, David, and Bert-Wolfgang Schulze. "Pseudo-differential crack theory." Universität Potsdam, 2000. http://opus.kobv.de/ubp/volltexte/2008/2575/.

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Crack problems are regarded as elements in a pseudo-differential algbra, where the two sdes int S± of the crack S are treated as interior boundaries and the boundary Y of the crack as an edge singularity. We employ the pseudo-differential calculus of boundary value problems with the transmission property near int S± and the edge pseudo-differential calculus (in a variant with Douglis-Nirenberg orders) to construct parametrices od elliptic crack problems (with extra trace and potential conditions along Y) and to characterise asymptotics of solutions near Y (expressed in the framework of continuous asymptotics). Our operator algebra with boundary and edge symbols contains new weight and order conventions that are necessary also for the more general calculus on manifolds with boundary and edges.
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46

Kapanadze, David, and Bert-Wolfgang Schulze. "Asymptotics of potentials in the edge calculus." Universität Potsdam, 2003. http://opus.kobv.de/ubp/volltexte/2008/2653/.

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Boundary value problems on manifolds with conical singularities or edges contain potential operators as well as trace and Green operators which play a similar role as the corresponding operators in (pseudo-differential) boundary value problems on a smooth manifold. There is then a specific asymptotic behaviour of these operators close to the singularities. We characterise potential operators in terms of actions of cone or edge pseudo-differential operators (in the neighbouring space) on densities supported by sbmanifolds which also have conical or edge singularities. As a byproduct we show the continuity of such potentials as continuous perators between cone or edge Sobolev spaces and subspaces with asymptotics.
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47

Calvo, D., and Bert-Wolfgang Schulze. "Edge symbolic structures of second generation." Universität Potsdam, 2005. http://opus.kobv.de/ubp/volltexte/2009/2994/.

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Operators on a manifold with (geometric) singularities are degenerate in a natural way. They have a principal symbolic structure with contributions from the different strata of the configuration. We study the calculus of such operators on the level of edge symbols of second generation, based on specific quantizations of the corner-degenerate interior symbols, and show that this structure is preserved under compositions.
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48

Pester, Cornelia. "A posteriori error estimation for non-linear eigenvalue problems for differential operators of second order with focus on 3D vertex singularities." Doctoral thesis, Logos Verlag Berlin, 2005. https://monarch.qucosa.de/id/qucosa%3A18520.

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This thesis is concerned with the finite element analysis and the a posteriori error estimation for eigenvalue problems for general operator pencils on two-dimensional manifolds. A specific application of the presented theory is the computation of corner singularities. Engineers use the knowledge of the so-called singularity exponents to predict the onset and the propagation of cracks. All results of this thesis are explained for two model problems, the Laplace and the linear elasticity problem, and verified by numerous numerical results.
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49

Pester, Cornelia. "A posteriori error estimation for non-linear eigenvalue problems for differential operators of second order with focus on 3D vertex singularities." Doctoral thesis, Berlin Logos-Verl, 2006. http://deposit.ddb.de/cgi-bin/dokserv?id=2806614&prov=M&dok_var=1&dok_ext=htm.

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50

Callander, Brian 1986. "Lefschetz fibrations = Fibrações de Lefschetz." [s.n.], 2013. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307041.

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Orientador: Elizabeth Terezinha Gasparim
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
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Resumo: O propósito desta tese é estudar fibrações de Lefschetz simpléticas, nas quais os ciclos evanescentes são subvariedades Lagrangianas das fibras. Para a descrição da teoria de interseção dos ciclos evanescentes utilizamos cohomologia de Floer Lagrangiana, cujo conceito revemos nesta tese. Apresentamos três exemplos principais e de caráteres distintos: (1) twists de Dehn generalizados, (2) o "espelho" da reta projetiva, e (3) uma fibração numa órbita adjunta de sl(3,C). O terceiro destes exemplos é original e utiliza um teorema recente de Gasparim- Grama-San Martin
Abstract: The objective of this thesis is to study symplectic Lefschetz fibrations, in which the vanishing cycles are Lagrangian submanifolds of the fibres. In order to describe the intersection theory of vanishing cycles we use Lagrangian intersection Floer cohomology, which we review. We present three main examples of distinct characters: (1) generalized Dehn twists, (2) the "mirror" of the projective line, and (3) a fibration on an adjoint orbit of sl(3,C). The third of these examples is original and uses a recent theorem of Gasparim- Grama-San Martin
Mestrado
Matematica
Mestre em Matemática
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