@phdthesis{Gehring2023, author = {Gehring, Penelope}, title = {Non-local boundary conditions for the spin Dirac operator on spacetimes with timelike boundary}, doi = {10.25932/publishup-57775}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-577755}, school = {Universit{\"a}t Potsdam}, pages = {100}, year = {2023}, abstract = {Non-local boundary conditions - for example the Atiyah-Patodi-Singer (APS) conditions - for Dirac operators on Riemannian manifolds are rather well-understood, while not much is known for such operators on Lorentzian manifolds. Recently, B{\"a}r and Strohmaier [15] and Drago, Große, and Murro [27] introduced APS-like conditions for the spin Dirac operator on Lorentzian manifolds with spacelike and timelike boundary, respectively. While B{\"a}r and Strohmaier [15] showed the Fredholmness of the Dirac operator with these boundary conditions, Drago, Große, and Murro [27] proved the well-posedness of the corresponding initial boundary value problem under certain geometric assumptions. In this thesis, we will follow the footsteps of the latter authors and discuss whether the APS-like conditions for Dirac operators on Lorentzian manifolds with timelike boundary can be replaced by more general conditions such that the associated initial boundary value problems are still wellposed. We consider boundary conditions that are local in time and non-local in the spatial directions. More precisely, we use the spacetime foliation arising from the Cauchy temporal function and split the Dirac operator along this foliation. This gives rise to a family of elliptic operators each acting on spinors of the spin bundle over the corresponding timeslice. The theory of elliptic operators then ensures that we can find families of non-local boundary conditions with respect to this family of operators. Proceeding, we use such a family of boundary conditions to define a Lorentzian boundary condition on the whole timelike boundary. By analyzing the properties of the Lorentzian boundary conditions, we then find sufficient conditions on the family of non-local boundary conditions that lead to the well-posedness of the corresponding Cauchy problems. The well-posedness itself will then be proven by using classical tools including energy estimates and approximation by solutions of the regularized problems. Moreover, we use this theory to construct explicit boundary conditions for the Lorentzian Dirac operator. More precisely, we will discuss two examples of boundary conditions - the analogue of the Atiyah-Patodi-Singer and the chirality conditions, respectively, in our setting. For doing this, we will have a closer look at the theory of non-local boundary conditions for elliptic operators and analyze the requirements on the family of non-local boundary conditions for these specific examples.}, language = {en} } @article{MauerbergerSchannerKorteetal.2020, author = {Mauerberger, Stefan and Schanner, Maximilian Arthus and Korte, Monika and Holschneider, Matthias}, title = {Correlation based snapshot models of the archeomagnetic field}, series = {Geophysical journal international}, volume = {223}, journal = {Geophysical journal international}, number = {1}, publisher = {Oxford Univ. Press}, address = {Oxford}, issn = {0956-540X}, doi = {10.1093/gji/ggaa336}, pages = {648 -- 665}, year = {2020}, abstract = {For the time stationary global geomagnetic field, a new modelling concept is presented. A Bayesian non-parametric approach provides realistic location dependent uncertainty estimates. Modelling related variabilities are dealt with systematically by making little subjective apriori assumptions. Rather than parametrizing the model by Gauss coefficients, a functional analytic approach is applied. The geomagnetic potential is assumed a Gaussian process to describe a distribution over functions. Apriori correlations are given by an explicit kernel function with non-informative dipole contribution. A refined modelling strategy is proposed that accommodates non-linearities of archeomagnetic observables: First, a rough field estimate is obtained considering only sites that provide full field vector records. Subsequently, this estimate supports the linearization that incorporates the remaining incomplete records. The comparison of results for the archeomagnetic field over the past 1000 yr is in general agreement with previous models while improved model uncertainty estimates are provided.}, language = {en} } @article{RoosOtoba2021, author = {Roos, Saskia and Otoba, Nobuhiko}, title = {Scalar curvature and the multiconformal class of a direct product Riemannian manifold}, series = {Geometriae dedicata}, volume = {214}, journal = {Geometriae dedicata}, number = {1}, publisher = {Springer}, address = {Dordrecht}, issn = {0046-5755}, doi = {10.1007/s10711-021-00636-9}, pages = {801 -- 829}, year = {2021}, abstract = {For a closed, connected direct product Riemannian manifold (M, g) = (M-1, g(1)) x ... x (M-l, g(l)), we define its multiconformal class [[g]] as the totality {integral(2)(1)g(1) circle plus center dot center dot center dot integral(2)(l)g(l)} of all Riemannian metrics obtained from multiplying the metric gi of each factor Mi by a positive function fi on the total space M. A multiconformal class [[ g]] contains not only all warped product type deformations of g but also the whole conformal class [(g) over tilde] of every (g) over tilde is an element of[[ g]]. In this article, we prove that [[g]] contains a metric of positive scalar curvature if and only if the conformal class of some factor (Mi, gi) does, under the technical assumption dim M-i = 2. We also show that, even in the case where every factor (M-i, g(i)) has positive scalar curvature, [[g]] contains a metric of scalar curvature constantly equal to -1 and with arbitrarily large volume, provided l = 2 and dim M = 3.}, language = {en} } @phdthesis{Goebel2019, author = {G{\"o}bel, Franziska}, title = {Contributions to multiscale statistical analysis on random geometric graphs}, school = {Universit{\"a}t Potsdam}, pages = {127}, year = {2019}, language = {en} } @article{BeckusBellissardDeNittis2020, author = {Beckus, Siegfried and Bellissard, Jean and De Nittis, Giuseppe}, title = {Spectral continuity for aperiodic quantum systems}, series = {Journal of mathematical physics}, volume = {61}, journal = {Journal of mathematical physics}, number = {12}, publisher = {American Institute of Physics}, address = {Melville, NY}, issn = {0022-2488}, doi = {10.1063/5.0011488}, pages = {19}, year = {2020}, abstract = {This work provides a necessary and sufficient condition for a symbolic dynamical system to admit a sequence of periodic approximations in the Hausdorff topology. The key result proved and applied here uses graphs that are called De Bruijn graphs, Rauzy graphs, or Anderson-Putnam complex, depending on the community. Combining this with a previous result, the present work justifies rigorously the accuracy and reliability of algorithmic methods used to compute numerically the spectra of a large class of self-adjoint operators. The so-called Hamiltonians describe the effective dynamic of a quantum particle in aperiodic media. No restrictions on the structure of these operators other than general regularity assumptions are imposed. In particular, nearest-neighbor correlation is not necessary. Examples for the Fibonacci and the Golay-Rudin-Shapiro sequences are explicitly provided illustrating this discussion. While the first sequence has been thoroughly studied by physicists and mathematicians alike, a shroud of mystery still surrounds the latter when it comes to spectral properties. In light of this, the present paper gives a new result here that might help uncovering a solution.}, language = {en} } @article{KolbeEvans2020, author = {Kolbe, Benedikt Maximilian and Evans, Myfanwy E.}, title = {Isotopic tiling theory for hyperbolic surfaces}, series = {Geometriae dedicata}, volume = {212}, journal = {Geometriae dedicata}, number = {1}, publisher = {Springer}, address = {Dordrecht}, issn = {0046-5755}, doi = {10.1007/s10711-020-00554-2}, pages = {177 -- 204}, year = {2020}, abstract = {In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More specifically, we generalize results on Delaney-Dress combinatorial tiling theory using an extension of mapping class groups to orbifolds, in turn using this to study tilings of covering spaces of orbifolds. Moreover, we study finite subgroups of these mapping class groups. Our results can be used to extend the Delaney-Dress combinatorial encoding of a tiling to yield a finite symbol encoding the complexity of an isotopy class of tilings. The results of this paper provide the basis for a complete and unambiguous enumeration of isotopically distinct tilings of hyperbolic surfaces.}, language = {en} } @article{RedmannFreitag2021, author = {Redmann, Martin and Freitag, Melina A.}, title = {Optimization based model order reduction for stochastic systems}, series = {Applied mathematics and computation}, volume = {398}, journal = {Applied mathematics and computation}, publisher = {Elsevier}, address = {New York}, issn = {0096-3003}, doi = {10.1016/j.amc.2020.125783}, pages = {18}, year = {2021}, abstract = {In this paper, we bring together the worlds of model order reduction for stochastic linear systems and H-2-optimal model order reduction for deterministic systems. In particular, we supplement and complete the theory of error bounds for model order reduction of stochastic differential equations. With these error bounds, we establish a link between the output error for stochastic systems (with additive and multiplicative noise) and modified versions of the H-2-norm for both linear and bilinear deterministic systems. When deriving the respective optimality conditions for minimizing the error bounds, we see that model order reduction techniques related to iterative rational Krylov algorithms (IRKA) are very natural and effective methods for reducing the dimension of large-scale stochastic systems with additive and/or multiplicative noise. We apply modified versions of (linear and bilinear) IRKA to stochastic linear systems and show their efficiency in numerical experiments.}, language = {en} } @phdthesis{LopezValencia2023, author = {Lopez Valencia, Diego Andres}, title = {The Milnor-Moore and Poincar{\´e}-Birkhoff-Witt theorems in the locality set up and the polar structure of Shintani zeta functions}, doi = {10.25932/publishup-59421}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-594213}, school = {Universit{\"a}t Potsdam}, pages = {147}, year = {2023}, abstract = {This thesis bridges two areas of mathematics, algebra on the one hand with the Milnor-Moore theorem (also called Cartier-Quillen-Milnor-Moore theorem) as well as the Poincar{\´e}-Birkhoff-Witt theorem, and analysis on the other hand with Shintani zeta functions which generalise multiple zeta functions. The first part is devoted to an algebraic formulation of the locality principle in physics and generalisations of classification theorems such as Milnor-Moore and Poincar{\´e}-Birkhoff-Witt theorems to the locality framework. The locality principle roughly says that events that take place far apart in spacetime do not infuence each other. The algebraic formulation of this principle discussed here is useful when analysing singularities which arise from events located far apart in space, in order to renormalise them while keeping a memory of the fact that they do not influence each other. We start by endowing a vector space with a symmetric relation, named the locality relation, which keeps track of elements that are "locally independent". The pair of a vector space together with such relation is called a pre-locality vector space. This concept is extended to tensor products allowing only tensors made of locally independent elements. We extend this concept to the locality tensor algebra, and locality symmetric algebra of a pre-locality vector space and prove the universal properties of each of such structures. We also introduce the pre-locality Lie algebras, together with their associated locality universal enveloping algebras and prove their universal property. We later upgrade all such structures and results from the pre-locality to the locality context, requiring the locality relation to be compatible with the linear structure of the vector space. This allows us to define locality coalgebras, locality bialgebras, and locality Hopf algebras. Finally, all the previous results are used to prove the locality version of the Milnor-Moore and the Poincar{\´e}-Birkhoff-Witt theorems. It is worth noticing that the proofs presented, not only generalise the results in the usual (non-locality) setup, but also often use less tools than their counterparts in their non-locality counterparts. The second part is devoted to study the polar structure of the Shintani zeta functions. Such functions, which generalise the Riemman zeta function, multiple zeta functions, Mordell-Tornheim zeta functions, among others, are parametrised by matrices with real non-negative arguments. It is known that Shintani zeta functions extend to meromorphic functions with poles on afine hyperplanes. We refine this result in showing that the poles lie on hyperplanes parallel to the facets of certain convex polyhedra associated to the defining matrix for the Shintani zeta function. Explicitly, the latter are the Newton polytopes of the polynomials induced by the columns of the underlying matrix. We then prove that the coeficients of the equation which describes the hyperplanes in the canonical basis are either zero or one, similar to the poles arising when renormalising generic Feynman amplitudes. For that purpose, we introduce an algorithm to distribute weight over a graph such that the weight at each vertex satisfies a given lower bound.}, language = {en} } @article{ShlapunovTarchanov2021, author = {Shlapunov, Alexander and Tarchanov, Nikolaj Nikolaevič}, title = {An open mapping theorem for the Navier-Stokes type equations associated with the de Rham complex over R-n}, series = {Siberian electronic mathematical reports = Sibirskie ėlektronnye matematičeskie izvestija}, volume = {18}, journal = {Siberian electronic mathematical reports = Sibirskie ėlektronnye matematičeskie izvestija}, number = {2}, publisher = {Institut Matematiki Imeni S. L. Soboleva}, address = {Novosibirsk}, issn = {1813-3304}, doi = {10.33048/semi.2021.18.108}, pages = {1433 -- 1466}, year = {2021}, abstract = {We consider an initial problem for the Navier-Stokes type equations associated with the de Rham complex over R-n x[0, T], n >= 3, with a positive time T. We prove that the problem induces an open injective mappings on the scales of specially constructed function spaces of Bochner-Sobolev type. In particular, the corresponding statement on the intersection of these classes gives an open mapping theorem for smooth solutions to the Navier-Stokes equations.}, language = {en} } @article{RoppLesurBaerenzungetal.2020, author = {Ropp, Guillaume and Lesur, Vincent and B{\"a}renzung, Julien and Holschneider, Matthias}, title = {Sequential modelling of the Earth's core magnetic field}, series = {Earth, Planets and Space}, volume = {72}, journal = {Earth, Planets and Space}, number = {1}, publisher = {Springer}, address = {New York}, issn = {1880-5981}, doi = {10.1186/s40623-020-01230-1}, pages = {15}, year = {2020}, abstract = {We describe a new, original approach to the modelling of the Earth's magnetic field. The overall objective of this study is to reliably render fast variations of the core field and its secular variation. This method combines a sequential modelling approach, a Kalman filter, and a correlation-based modelling step. Sources that most significantly contribute to the field measured at the surface of the Earth are modelled. Their separation is based on strong prior information on their spatial and temporal behaviours. We obtain a time series of model distributions which display behaviours similar to those of recent models based on more classic approaches, particularly at large temporal and spatial scales. Interesting new features and periodicities are visible in our models at smaller time and spatial scales. An important aspect of our method is to yield reliable error bars for all model parameters. These errors, however, are only as reliable as the description of the different sources and the prior information used are realistic. Finally, we used a slightly different version of our method to produce candidate models for the thirteenth edition of the International Geomagnetic Reference Field.}, language = {en} }