@article{Lewandowski2022, author = {Lewandowski, Max}, title = {Hadamard states for bosonic quantum field theory on globally hyperbolic spacetimes}, series = {Journal of mathematical physics}, volume = {63}, journal = {Journal of mathematical physics}, number = {1}, publisher = {American Institute of Physics}, address = {Melville}, issn = {0022-2488}, doi = {10.1063/5.0055753}, pages = {34}, year = {2022}, abstract = {According to Radzikowski's celebrated results, bisolutions of a wave operator on a globally hyperbolic spacetime are of the Hadamard form iff they are given by a linear combination of distinguished parametrices i2(G˜aF-G˜F+G˜A-G˜R) in the sense of Duistermaat and H{\"o}rmander [Acta Math. 128, 183-269 (1972)] and Radzikowski [Commun. Math. Phys. 179, 529 (1996)]. Inspired by the construction of the corresponding advanced and retarded Green operator GA, GR as done by B{\"a}r, Ginoux, and Pf{\"a}ffle {Wave Equations on Lorentzian Manifolds and Quantization [European Mathematical Society (EMS), Z{\"u}rich, 2007]}, we construct the remaining two Green operators GF, GaF locally in terms of Hadamard series. Afterward, we provide the global construction of i2(G˜aF-G˜F), which relies on new techniques such as a well-posed Cauchy problem for bisolutions and a patching argument using Čech cohomology. This leads to global bisolutions of the Hadamard form, each of which can be chosen to be a Hadamard two-point-function, i.e., the smooth part can be adapted such that, additionally, the symmetry and the positivity condition are exactly satisfied.}, language = {en} } @article{Zoeller2022, author = {Z{\"o}ller, Gert}, title = {A note on the estimation of the maximum possible earthquake magnitude based on extreme value theory for the Groningen Gas Field}, series = {The bulletin of the Seismological Society of America : BSSA}, volume = {112}, journal = {The bulletin of the Seismological Society of America : BSSA}, number = {4}, publisher = {Seismological Society of America}, address = {El Cerito, Calif.}, issn = {0037-1106}, doi = {10.1785/0120210307}, pages = {1825 -- 1831}, year = {2022}, abstract = {Extreme value statistics is a popular and frequently used tool to model the occurrence of large earthquakes. The problem of poor statistics arising from rare events is addressed by taking advantage of the validity of general statistical properties in asymptotic regimes. In this note, I argue that the use of extreme value statistics for the purpose of practically modeling the tail of the frequency-magnitude distribution of earthquakes can produce biased and thus misleading results because it is unknown to what degree the tail of the true distribution is sampled by data. Using synthetic data allows to quantify this bias in detail. The implicit assumption that the true M-max is close to the maximum observed magnitude M-max,M-observed restricts the class of the potential models a priori to those with M-max = M-max,M-observed + Delta M with an increment Delta M approximate to 0.5... 1.2. This corresponds to the simple heuristic method suggested by Wheeler (2009) and labeled :M-max equals M-obs plus an increment." The incomplete consideration of the entire model family for the frequency-magnitude distribution neglects, however, the scenario of a large so far unobserved earthquake.}, language = {en} } @article{KayaFreitag2022, author = {Kaya, Adem and Freitag, Melina A.}, title = {Conditioning analysis for discrete Helmholtz problems}, series = {Computers and mathematics with applications : an international journal}, volume = {118}, journal = {Computers and mathematics with applications : an international journal}, publisher = {Elsevier Science}, address = {Amsterdam}, issn = {0898-1221}, doi = {10.1016/j.camwa.2022.05.016}, pages = {171 -- 182}, year = {2022}, abstract = {In this paper, we examine conditioning of the discretization of the Helmholtz problem. Although the discrete Helmholtz problem has been studied from different perspectives, to the best of our knowledge, there is no conditioning analysis for it. We aim to fill this gap in the literature. We propose a novel method in 1D to observe the near-zero eigenvalues of a symmetric indefinite matrix. Standard classification of ill-conditioning based on the matrix condition number is not true for the discrete Helmholtz problem. We relate the ill-conditioning of the discretization of the Helmholtz problem with the condition number of the matrix. We carry out analytical conditioning analysis in 1D and extend our observations to 2D with numerical observations. We examine several discretizations. We find different regions in which the condition number of the problem shows different characteristics. We also explain the general behavior of the solutions in these regions.}, language = {en} } @article{HoudebertZass2022, author = {Houdebert, Pierre and Zass, Alexander}, title = {An explicit Dobrushin uniqueness region for Gibbs point processes with repulsive interactions}, series = {Journal of applied probability / Applied Probability Trust}, volume = {59}, journal = {Journal of applied probability / Applied Probability Trust}, number = {2}, publisher = {Cambridge Univ. Press}, address = {Cambridge}, issn = {0021-9002}, doi = {10.1017/jpr.2021.70}, pages = {541 -- 555}, year = {2022}, abstract = {We present a uniqueness result for Gibbs point processes with interactions that come from a non-negative pair potential; in particular, we provide an explicit uniqueness region in terms of activity z and inverse temperature beta. The technique used relies on applying to the continuous setting the classical Dobrushin criterion. We also present a comparison to the two other uniqueness methods of cluster expansion and disagreement percolation, which can also be applied for this type of interaction.}, language = {en} } @article{EvansHyde2022, author = {Evans, Myfanwy E. and Hyde, Stephen T.}, title = {Symmetric Tangling of Honeycomb Networks}, series = {Symmetry}, volume = {14}, journal = {Symmetry}, edition = {9}, publisher = {MDPI}, address = {Basel, Schweiz}, issn = {2073-8994}, doi = {10.3390/sym14091805}, pages = {1 -- 13}, year = {2022}, abstract = {Symmetric, elegantly entangled structures are a curious mathematical construction that has found their way into the heart of the chemistry lab and the toolbox of constructive geometry. Of particular interest are those structures—knots, links and weavings—which are composed locally of simple twisted strands and are globally symmetric. This paper considers the symmetric tangling of multiple 2-periodic honeycomb networks. We do this using a constructive methodology borrowing elements of graph theory, low-dimensional topology and geometry. The result is a wide-ranging enumeration of symmetric tangled honeycomb networks, providing a foundation for their exploration in both the chemistry lab and the geometers toolbox.}, language = {en} } @article{HydeEvans2022, author = {Hyde, Stephen T. and Evans, Myfanwy E.}, title = {Symmetric tangled Platonic polyhedra}, series = {Proceedings of the National Academy of Sciences of the United States of America}, volume = {119}, journal = {Proceedings of the National Academy of Sciences of the United States of America}, number = {1}, publisher = {National Acad. of Sciences}, address = {Washington}, issn = {0027-8424}, doi = {10.1073/pnas.2110345118}, pages = {10}, year = {2022}, abstract = {Conventional embeddings of the edge-graphs of Platonic polyhedra, {f,z}, where f,z denote the number of edges in each face and the edge-valence at each vertex, respectively, are untangled in that they can be placed on a sphere (S-2) such that distinct edges do not intersect, analogous to unknotted loops, which allow crossing-free drawings of S-1 on the sphere. The most symmetric (flag-transitive) realizations of those polyhedral graphs are those of the classical Platonic polyhedra, whose symmetries are *2fz, according to Conway's two-dimensional (2D) orbifold notation (equivalent to Schonflies symbols I-h, O-h, and T-d). Tangled Platonic {f,z} polyhedra-which cannot lie on the sphere without edge-crossings-are constructed as windings of helices with three, five, seven,... strands on multigenus surfaces formed by tubifying the edges of conventional Platonic polyhedra, have (chiral) symmetries 2fz (I, O, and T), whose vertices, edges, and faces are symmetrically identical, realized with two flags. The analysis extends to the "theta(z)" polyhedra, {2,z}. The vertices of these symmetric tangled polyhedra overlap with those of the Platonic polyhedra; however, their helicity requires curvilinear (or kinked) edges in all but one case. We show that these 2fz polyhedral tangles are maximally symmetric; more symmetric embeddings are necessarily untangled. On one hand, their topologies are very constrained: They are either self-entangled graphs (analogous to knots) or mutually catenated entangled compound polyhedra (analogous to links). On the other hand, an endless variety of entanglements can be realized for each topology. Simpler examples resemble patterns observed in synthetic organometallic materials and clathrin coats in vivo.}, language = {en} } @phdthesis{Hain2022, author = {Hain, Tobias Martin}, title = {Structure formation and identification in geometrically driven soft matter systems}, doi = {10.25932/publishup-55880}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-558808}, school = {Universit{\"a}t Potsdam}, pages = {xviii, 171}, year = {2022}, abstract = {Subdividing space through interfaces leads to many space partitions that are relevant to soft matter self-assembly. Prominent examples include cellular media, e.g. soap froths, which are bubbles of air separated by interfaces of soap and water, but also more complex partitions such as bicontinuous minimal surfaces. Using computer simulations, this thesis analyses soft matter systems in terms of the relationship between the physical forces between the system's constituents and the structure of the resulting interfaces or partitions. The focus is on two systems, copolymeric self-assembly and the so-called Quantizer problem, where the driving force of structure formation, the minimisation of the free-energy, is an interplay of surface area minimisation and stretching contributions, favouring cells of uniform thickness. In the first part of the thesis we address copolymeric phase formation with sharp interfaces. We analyse a columnar copolymer system "forced" to assemble on a spherical surface, where the perfect solution, the hexagonal tiling, is topologically prohibited. For a system of three-armed copolymers, the resulting structure is described by solutions of the so-called Thomson problem, the search of minimal energy configurations of repelling charges on a sphere. We find three intertwined Thomson problem solutions on a single sphere, occurring at a probability depending on the radius of the substrate. We then investigate the formation of amorphous and crystalline structures in the Quantizer system, a particulate model with an energy functional without surface tension that favours spherical cells of equal size. We find that quasi-static equilibrium cooling allows the Quantizer system to crystallise into a BCC ground state, whereas quenching and non-equilibrium cooling, i.e. cooling at slower rates then quenching, leads to an approximately hyperuniform, amorphous state. The assumed universality of the latter, i.e. independence of energy minimisation method or initial configuration, is strengthened by our results. We expand the Quantizer system by introducing interface tension, creating a model that we find to mimic polymeric micelle systems: An order-disorder phase transition is observed with a stable Frank-Caspar phase. The second part considers bicontinuous partitions of space into two network-like domains, and introduces an open-source tool for the identification of structures in electron microscopy images. We expand a method of matching experimentally accessible projections with computed projections of potential structures, introduced by Deng and Mieczkowski (1998). The computed structures are modelled using nodal representations of constant-mean-curvature surfaces. A case study conducted on etioplast cell membranes in chloroplast precursors establishes the double Diamond surface structure to be dominant in these plant cells. We automate the matching process employing deep-learning methods, which manage to identify structures with excellent accuracy.}, language = {en} } @phdthesis{Fischer2022, author = {Fischer, Jens Walter}, title = {Random dynamics in collective behavior - consensus, clustering \& extinction of populations}, doi = {10.25932/publishup-55372}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-553725}, school = {Universit{\"a}t Potsdam}, pages = {242}, year = {2022}, abstract = {The echo chamber model describes the development of groups in heterogeneous social networks. By heterogeneous social network we mean a set of individuals, each of whom represents exactly one opinion. The existing relationships between individuals can then be represented by a graph. The echo chamber model is a time-discrete model which, like a board game, is played in rounds. In each round, an existing relationship is randomly and uniformly selected from the network and the two connected individuals interact. If the opinions of the individuals involved are sufficiently similar, they continue to move closer together in their opinions, whereas in the case of opinions that are too far apart, they break off their relationship and one of the individuals seeks a new relationship. In this paper we examine the building blocks of this model. We start from the observation that changes in the structure of relationships in the network can be described by a system of interacting particles in a more abstract space. These reflections lead to the definition of a new abstract graph that encompasses all possible relational configurations of the social network. This provides us with the geometric understanding necessary to analyse the dynamic components of the echo chamber model in Part III. As a first step, in Part 7, we leave aside the opinions of the inidividuals and assume that the position of the edges changes with each move as described above, in order to obtain a basic understanding of the underlying dynamics. Using Markov chain theory, we find upper bounds on the speed of convergence of an associated Markov chain to its unique stationary distribution and show that there are mutually identifiable networks that are not apparent in the dynamics under analysis, in the sense that the stationary distribution of the associated Markov chain gives equal weight to these networks. In the reversible cases, we focus in particular on the explicit form of the stationary distribution as well as on the lower bounds of the Cheeger constant to describe the convergence speed. The final result of Section 8, based on absorbing Markov chains, shows that in a reduced version of the echo chamber model, a hierarchical structure of the number of conflicting relations can be identified. We can use this structure to determine an upper bound on the expected absorption time, using a quasi-stationary distribution. This hierarchy of structure also provides a bridge to classical theories of pure death processes. We conclude by showing how future research can exploit this link and by discussing the importance of the results as building blocks for a full theoretical understanding of the echo chamber model. Finally, Part IV presents a published paper on the birth-death process with partial catastrophe. The paper is based on the explicit calculation of the first moment of a catastrophe. This first part is entirely based on an analytical approach to second degree recurrences with linear coefficients. The convergence to 0 of the resulting sequence as well as the speed of convergence are proved. On the other hand, the determination of the upper bounds of the expected value of the population size as well as its variance and the difference between the determined upper bound and the actual value of the expected value. For these results we use almost exclusively the theory of ordinary nonlinear differential equations.}, language = {en} } @phdthesis{Schanner2022, author = {Schanner, Maximilian Arthus}, title = {Correlation based modeling of the archeomagnetic field}, doi = {10.25932/publishup-55587}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-555875}, school = {Universit{\"a}t Potsdam}, pages = {vii, 146}, year = {2022}, abstract = {The geomagnetic main field is vital for live on Earth, as it shields our habitat against the solar wind and cosmic rays. It is generated by the geodynamo in the Earth's outer core and has a rich dynamic on various timescales. Global models of the field are used to study the interaction of the field and incoming charged particles, but also to infer core dynamics and to feed numerical simulations of the geodynamo. Modern satellite missions, such as the SWARM or the CHAMP mission, support high resolution reconstructions of the global field. From the 19 th century on, a global network of magnetic observatories has been established. It is growing ever since and global models can be constructed from the data it provides. Geomagnetic field models that extend further back in time rely on indirect observations of the field, i.e. thermoremanent records such as burnt clay or volcanic rocks and sediment records from lakes and seas. These indirect records come with (partially very large) uncertainties, introduced by the complex measurement methods and the dating procedure. Focusing on thermoremanent records only, the aim of this thesis is the development of a new modeling strategy for the global geomagnetic field during the Holocene, which takes the uncertainties into account and produces realistic estimates of the reliability of the model. This aim is approached by first considering snapshot models, in order to address the irregular spatial distribution of the records and the non-linear relation of the indirect observations to the field itself. In a Bayesian setting, a modeling algorithm based on Gaussian process regression is developed and applied to binned data. The modeling algorithm is then extended to the temporal domain and expanded to incorporate dating uncertainties. Finally, the algorithm is sequentialized to deal with numerical challenges arising from the size of the Holocene dataset. The central result of this thesis, including all of the aspects mentioned, is a new global geomagnetic field model. It covers the whole Holocene, back until 12000 BCE, and we call it ArchKalmag14k. When considering the uncertainties that are produced together with the model, it is evident that before 6000 BCE the thermoremanent database is not sufficient to support global models. For times more recent, ArchKalmag14k can be used to analyze features of the field under consideration of posterior uncertainties. The algorithm for generating ArchKalmag14k can be applied to different datasets and is provided to the community as an open source python package.}, language = {en} } @misc{EvansHyde2022, author = {Evans, Myfanwy E. and Hyde, Stephen T.}, title = {Symmetric Tangling of Honeycomb Networks}, series = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {1282}, issn = {1866-8372}, doi = {10.25932/publishup-57084}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-570842}, pages = {13}, year = {2022}, abstract = {Symmetric, elegantly entangled structures are a curious mathematical construction that has found their way into the heart of the chemistry lab and the toolbox of constructive geometry. Of particular interest are those structures—knots, links and weavings—which are composed locally of simple twisted strands and are globally symmetric. This paper considers the symmetric tangling of multiple 2-periodic honeycomb networks. We do this using a constructive methodology borrowing elements of graph theory, low-dimensional topology and geometry. The result is a wide-ranging enumeration of symmetric tangled honeycomb networks, providing a foundation for their exploration in both the chemistry lab and the geometers toolbox.}, language = {en} } @phdthesis{Hannes2022, author = {Hannes, Sebastian}, title = {Boundary Value Problems for the Lorentzian Dirac Operator}, doi = {10.25932/publishup-54839}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-548391}, school = {Universit{\"a}t Potsdam}, pages = {67}, year = {2022}, abstract = {The index theorem for elliptic operators on a closed Riemannian manifold by Atiyah and Singer has many applications in analysis, geometry and topology, but it is not suitable for a generalization to a Lorentzian setting. In the case where a boundary is present Atiyah, Patodi and Singer provide an index theorem for compact Riemannian manifolds by introducing non-local boundary conditions obtained via the spectral decomposition of an induced boundary operator, so called APS boundary conditions. B{\"a}r and Strohmaier prove a Lorentzian version of this index theorem for the Dirac operator on a manifold with boundary by utilizing results from APS and the characterization of the spectral flow by Phillips. In their case the Lorentzian manifold is assumed to be globally hyperbolic and spatially compact, and the induced boundary operator is given by the Riemannian Dirac operator on a spacelike Cauchy hypersurface. Their results show that imposing APS boundary conditions for these boundary operator will yield a Fredholm operator with a smooth kernel and its index can be calculated by a formula similar to the Riemannian case. Back in the Riemannian setting, B{\"a}r and Ballmann provide an analysis of the most general kind of boundary conditions that can be imposed on a first order elliptic differential operator that will still yield regularity for solutions as well as Fredholm property for the resulting operator. These boundary conditions can be thought of as deformations to the graph of a suitable operator mapping APS boundary conditions to their orthogonal complement. This thesis aims at applying the boundary conditions found by B{\"a}r and Ballmann to a Lorentzian setting to understand more general types of boundary conditions for the Dirac operator, conserving Fredholm property as well as providing regularity results and relative index formulas for the resulting operators. As it turns out, there are some differences in applying these graph-type boundary conditions to the Lorentzian Dirac operator when compared to the Riemannian setting. It will be shown that in contrast to the Riemannian case, going from a Fredholm boundary condition to its orthogonal complement works out fine in the Lorentzian setting. On the other hand, in order to deduce Fredholm property and regularity of solutions for graph-type boundary conditions, additional assumptions for the deformation maps need to be made. The thesis is organized as follows. In chapter 1 basic facts about Lorentzian and Riemannian spin manifolds, their spinor bundles and the Dirac operator are listed. These will serve as a foundation to define the setting and prove the results of later chapters. Chapter 2 defines the general notion of boundary conditions for the Dirac operator used in this thesis and introduces the APS boundary conditions as well as their graph type deformations. Also the role of the wave evolution operator in finding Fredholm boundary conditions is analyzed and these boundary conditions are connected to notion of Fredholm pairs in a given Hilbert space. Chapter 3 focuses on the principal symbol calculation of the wave evolution operator and the results are used to proof Fredholm property as well as regularity of solutions for suitable graph-type boundary conditions. Also sufficient conditions are derived for (pseudo-)local boundary conditions imposed on the Dirac operator to yield a Fredholm operator with a smooth solution space. In the last chapter 4, a few examples of boundary conditions are calculated applying the results of previous chapters. Restricting to special geometries and/or boundary conditions, results can be obtained that are not covered by the more general statements, and it is shown that so-called transmission conditions behave very differently than in the Riemannian setting.}, language = {en} } @article{BaerBandara2022, author = {B{\"a}r, Christian and Bandara, Lashi}, title = {Boundary value problems for general first-order elliptic differential operators}, series = {Journal of functional analysis}, volume = {282}, journal = {Journal of functional analysis}, number = {12}, publisher = {Elsevier}, address = {Amsterdam [u.a.]}, issn = {0022-1236}, doi = {10.1016/j.jfa.2022.109445}, pages = {69}, year = {2022}, abstract = {We study boundary value problems for first-order elliptic differential operators on manifolds with compact boundary. The adapted boundary operator need not be selfadjoint and the boundary condition need not be pseudo-local.We show the equivalence of various characterisations of elliptic boundary conditions and demonstrate how the boundary conditions traditionally considered in the literature fit in our framework. The regularity of the solutions up to the boundary is proven. We show that imposing elliptic boundary conditions yields a Fredholm operator if the manifold is compact. We provide examples which are conveniently treated by our methods.}, language = {en} } @article{MeraTarkhanov2022, author = {Mera, Azal Jaafar Musa and Tarkhanov, Nikolai}, title = {An elliptic equation of finite index in a domain}, series = {Boletin de la Sociedad Matem{\´a}tica Mexicana}, volume = {28}, journal = {Boletin de la Sociedad Matem{\´a}tica Mexicana}, number = {2}, publisher = {Springer International}, address = {New York [u.a.]}, issn = {1405-213X}, doi = {10.1007/s40590-022-00442-7}, pages = {10}, year = {2022}, abstract = {We give an example of first order elliptic equation for a complex-valued function in a plane domain which has a finite number of linearly independent solutions for any right-hand side. No boundary value conditions are thus required.}, language = {en} } @article{EngbertRabeSchwetlicketal.2022, author = {Engbert, Ralf and Rabe, Maximilian Michael and Schwetlick, Lisa and Seelig, Stefan A. and Reich, Sebastian and Vasishth, Shravan}, title = {Data assimilation in dynamical cognitive science}, series = {Trends in cognitive sciences}, volume = {26}, journal = {Trends in cognitive sciences}, number = {2}, publisher = {Elsevier}, address = {Amsterdam}, issn = {1364-6613}, doi = {10.1016/j.tics.2021.11.006}, pages = {99 -- 102}, year = {2022}, abstract = {Dynamical models make specific assumptions about cognitive processes that generate human behavior. In data assimilation, these models are tested against timeordered data. Recent progress on Bayesian data assimilation demonstrates that this approach combines the strengths of statistical modeling of individual differences with the those of dynamical cognitive models.}, language = {en} } @article{HinzSchwarz2022, author = {Hinz, Michael and Schwarz, Michael}, title = {A note on Neumann problems on graphs}, series = {Positivity}, volume = {26}, journal = {Positivity}, number = {4}, publisher = {Springer}, address = {Dordrecht}, issn = {1385-1292}, doi = {10.1007/s11117-022-00930-0}, pages = {23}, year = {2022}, abstract = {We discuss Neumann problems for self-adjoint Laplacians on (possibly infinite) graphs. Under the assumption that the heat semigroup is ultracontractive we discuss the unique solvability for non-empty subgraphs with respect to the vertex boundary and provide analytic and probabilistic representations for Neumann solutions. A second result deals with Neumann problems on canonically compactifiable graphs with respect to the Royden boundary and provides conditions for unique solvability and analytic and probabilistic representations.}, language = {en} } @article{BaerHanke2022, author = {B{\"a}r, Christian and Hanke, Bernhard}, title = {Local flexibility for open partial differential relations}, series = {Communications on pure and applied mathematics / issued by the Courant Institute of Mathematical Sciences, New York Univ.}, volume = {75}, journal = {Communications on pure and applied mathematics / issued by the Courant Institute of Mathematical Sciences, New York Univ.}, number = {6}, publisher = {Wiley}, address = {Hoboken}, issn = {0010-3640}, doi = {10.1002/cpa.21982}, pages = {1377 -- 1415}, year = {2022}, abstract = {We show that local deformations, near closed subsets, of solutions to open partial differential relations can be extended to global deformations, provided all but the highest derivatives stay constant along the subset. The applicability of this general result is illustrated by a number of examples, dealing with convex embeddings of hypersurfaces, differential forms, and lapse functions in Lorentzian geometry. The main application is a general approximation result by sections that have very restrictive local properties on open dense subsets. This shows, for instance, that given any K is an element of Double-struck capital R every manifold of dimension at least 2 carries a complete C-1,C- 1-metric which, on a dense open subset, is smooth with constant sectional curvature K. Of course, this is impossible for C-2-metrics in general.}, language = {en} } @article{HanischLudewig2022, author = {Hanisch, Florian and Ludewig, Matthias}, title = {A rigorous construction of the supersymmetric path integral associated to a compact spin manifold}, series = {Communications in mathematical physics}, volume = {391}, journal = {Communications in mathematical physics}, number = {3}, publisher = {Springer}, address = {Berlin ; Heidelberg}, issn = {0010-3616}, doi = {10.1007/s00220-022-04336-7}, pages = {1209 -- 1239}, year = {2022}, abstract = {We give a rigorous construction of the path integral in N = 1/2 supersymmetry as an integral map for differential forms on the loop space of a compact spin manifold. It is defined on the space of differential forms which can be represented by extended iterated integrals in the sense of Chen and Getzler-Jones-Petrack. Via the iterated integral map, we compare our path integral to the non-commutative loop space Chern character of Guneysu and the second author. Our theory provides a rigorous background to various formal proofs of the Atiyah-Singer index theorem for twisted Dirac operators using supersymmetric path integrals, as investigated by Alvarez-Gaume, Atiyah, Bismut and Witten.}, language = {en} } @article{SchannerKorteHolschneider2022, author = {Schanner, Maximilian and Korte, Monika and Holschneider, Matthias}, title = {ArchKalmag14k: A kalman-filter based global geomagnetic model for the holocene}, series = {Journal of geophysical research : Solid earth}, volume = {127}, journal = {Journal of geophysical research : Solid earth}, number = {2}, publisher = {American Geophysical Union}, address = {Washington}, issn = {2169-9313}, doi = {10.1029/2021JB023166}, pages = {17}, year = {2022}, abstract = {We propose a global geomagnetic field model for the last 14 thousand years, based on thermoremanent records. We call the model ArchKalmag14k. ArchKalmag14k is constructed by modifying recently proposed algorithms, based on space-time correlations. Due to the amount of data and complexity of the model, the full Bayesian posterior is numerically intractable. To tackle this, we sequentialize the inversion by implementing a Kalman-filter with a fixed time step. Every step consists of a prediction, based on a degree dependent temporal covariance, and a correction via Gaussian process regression. Dating errors are treated via a noisy input formulation. Cross correlations are reintroduced by a smoothing algorithm and model parameters are inferred from the data. Due to the specific statistical nature of the proposed algorithms, the model comes with space and time-dependent uncertainty estimates. The new model ArchKalmag14k shows less variation in the large-scale degrees than comparable models. Local predictions represent the underlying data and agree with comparable models, if the location is sampled well. Uncertainties are bigger for earlier times and in regions of sparse data coverage. We also use ArchKalmag14k to analyze the appearance and evolution of the South Atlantic anomaly together with reverse flux patches at the core-mantle boundary, considering the model uncertainties. While we find good agreement with earlier models for recent times, our model suggests a different evolution of intensity minima prior to 1650 CE. In general, our results suggest that prior to 6000 BCE the data is not sufficient to support global models.}, language = {en} } @article{BellingeriFrizPaychaetal.2022, author = {Bellingeri, Carlo and Friz, Peter and Paycha, Sylvie and Preiß, Rosa Lili Dora}, title = {Smooth rough paths, their geometry and algebraic renormalization}, series = {Vietnam journal of mathematics}, volume = {50}, journal = {Vietnam journal of mathematics}, number = {3}, publisher = {Springer}, address = {Singapore}, issn = {2305-221X}, doi = {10.1007/s10013-022-00570-7}, pages = {719 -- 761}, year = {2022}, abstract = {We introduce the class of "smooth rough paths" and study their main properties. Working in a smooth setting allows us to discard sewing arguments and focus on algebraic and geometric aspects. Specifically, a Maurer-Cartan perspective is the key to a purely algebraic form of Lyons' extension theorem, the renormalization of rough paths following up on [Bruned et al.: A rough path perspective on renormalization, J. Funct. Anal. 277(11), 2019], as well as a related notion of "sum of rough paths". We first develop our ideas in a geometric rough path setting, as this best resonates with recent works on signature varieties, as well as with the renormalization of geometric rough paths. We then explore extensions to the quasi-geometric and the more general Hopf algebraic setting.}, language = {en} } @article{LieStahnSullivan2022, author = {Lie, Han Cheng and Stahn, Martin and Sullivan, Tim J.}, title = {Randomised one-step time integration methods for deterministic operator differential equations}, series = {Calcolo}, volume = {59}, journal = {Calcolo}, number = {1}, publisher = {Springer}, address = {Milano}, issn = {0008-0624}, doi = {10.1007/s10092-022-00457-6}, pages = {33}, year = {2022}, abstract = {Uncertainty quantification plays an important role in problems that involve inferring a parameter of an initial value problem from observations of the solution. Conrad et al. (Stat Comput 27(4):1065-1082, 2017) proposed randomisation of deterministic time integration methods as a strategy for quantifying uncertainty due to the unknown time discretisation error. We consider this strategy for systems that are described by deterministic, possibly time-dependent operator differential equations defined on a Banach space or a Gelfand triple. Our main results are strong error bounds on the random trajectories measured in Orlicz norms, proven under a weaker assumption on the local truncation error of the underlying deterministic time integration method. Our analysis establishes the theoretical validity of randomised time integration for differential equations in infinite-dimensional settings.}, language = {en} }