@article{EdekoGerlachKuehner2019,
  author    = {Edeko, Nikolai and Gerlach, Moritz Reinhardt and K{\"u}hner, Viktoria},
  title     = {Measure-preserving semiflows and one-parameter Koopman semigroups},
  series = {Semigroup forum},
  volume    = {98},
  journal   = {Semigroup forum},
  number    = {1},
  publisher = {Springer},
  address   = {New York},
  issn      = {0037-1912},
  doi       = {10.1007/s00233-018-9960-3},
  pages     = {48 -- 63},
  year      = {2019},
  abstract  = {For a finite measure space X, we characterize strongly continuous Markov lattice semigroups on Lp(X) by showing that their generator A acts as a derivation on the dense subspace D(A)L(X). We then use this to characterize Koopman semigroups on Lp(X) if X is a standard probability space. In addition, we show that every measurable and measure-preserving flow on a standard probability space is isomorphic to a continuous flow on a compact Borel probability space.},
  language  = {en}
}
@article{GerlachGlueck2018,
  author    = {Gerlach, Moritz Reinhardt and Gl{\"u}ck, Jochen},
  title     = {Lower bounds and the asymptotic behaviour of positive operator semigroups},
  series = {Ergodic theory and dynamical systems},
  volume    = {38},
  journal   = {Ergodic theory and dynamical systems},
  publisher = {Cambridge Univ. Press},
  address   = {New York},
  issn      = {0143-3857},
  doi       = {10.1017/etds.2017.9},
  pages     = {3012 -- 3041},
  year      = {2018},
  abstract  = {If (T-t) is a semigroup of Markov operators on an L-1-space that admits a nontrivial lower bound, then a well-known theorem of Lasota and Yorke asserts that the semigroup is strongly convergent as t -> infinity. In this article we generalize and improve this result in several respects. First, we give a new and very simple proof for the fact that the same conclusion also holds if the semigroup is merely assumed to be bounded instead of Markov. As a main result, we then prove a version of this theorem for semigroups which only admit certain individual lower bounds. Moreover, we generalize a theorem of Ding on semigroups of Frobenius-Perron operators. We also demonstrate how our results can be adapted to the setting of general Banach lattices and we give some counterexamples to show optimality of our results. Our methods combine some rather concrete estimates and approximation arguments with abstract functional analytical tools. One of these tools is a theorem which relates the convergence of a time-continuous operator semigroup to the convergence of embedded discrete semigroups.},
  language  = {en}
}
@article{GerlachGlueck2019,
  author    = {Gerlach, Moritz Reinhardt and Gl{\"u}ck, Jochen},
  title     = {Mean ergodicity vs weak almost periodicity},
  series = {Studia mathematica},
  volume    = {248},
  journal   = {Studia mathematica},
  number    = {1},
  publisher = {Polska Akademia Nauk, Instytut Matematyczny},
  address   = {Warszawa},
  issn      = {0039-3223},
  doi       = {10.4064/sm170918-20-3},
  pages     = {45 -- 56},
  year      = {2019},
  abstract  = {We provide explicit examples of positive and power-bounded operators on c(0) and l(infinity) which are mean ergodic but not weakly almost periodic. As a consequence we prove that a countably order complete Banach lattice on which every positive and power-bounded mean ergodic operator is weakly almost periodic is necessarily a KB-space. This answers several open questions from the literature. Finally, we prove that if T is a positive mean ergodic operator with zero fixed space on an arbitrary Banach lattice, then so is every power of T .},
  language  = {en}
}
@article{HanischLudewig2022,
  author    = {Hanisch, Florian and Ludewig, Matthias},
  title     = {The fermionic integral on loop space and the Pfaffian line bundle},
  series = {Journal of mathematical physics},
  volume    = {63},
  journal   = {Journal of mathematical physics},
  number    = {12},
  publisher = {American Inst. of Physics},
  address   = {College Park, Md.},
  issn      = {0022-2488},
  doi       = {10.1063/5.0060355},
  pages     = {26},
  year      = {2022},
  abstract  = {As the loop space of a Riemannian manifold is infinite-dimensional, it is a non-trivial problem to make sense of the "top degree component " of a differential form on it. In this paper, we show that a formula from finite dimensions generalizes to assign a sensible "top degree component " to certain composite forms, obtained by wedging with the exponential (in the exterior algebra) of the canonical presymplectic 2-form on the loop space. This construction is a crucial ingredient for the definition of the supersymmetric path integral on the loop space.},
  language  = {en}
}
@article{Metzger2023,
  author    = {Metzger, Jan},
  title     = {Refined position estimates for surfaces of Willmore type in Riemannian manifolds},
  series = {Communications in analysis and geometry},
  volume    = {30},
  journal   = {Communications in analysis and geometry},
  number    = {10},
  publisher = {International Press of Boston},
  address   = {Somerville, Mass.},
  issn      = {1019-8385},
  doi       = {10.4310/CAG.2022.v30.n10.a5},
  pages     = {2315 -- 2346},
  year      = {2023},
  abstract  = {In this paper we consider surfaces which are critical points of the Willmore functional subject to constrained area. In the case of small area we calculate the corrections to the intrinsic geometry induced by the ambient curvature. These estimates together with the choice of an adapted geometric center of mass lead to refined position estimates in relation to the scalar curvature of the ambient manifold.},
  language  = {en}
}
@article{RœllyZass2020,
  author    = {Rœlly, Sylvie and Zass, Alexander},
  title     = {Marked Gibbs point processes with unbounded interaction},
  series = {Journal of statistical physics},
  volume    = {179},
  journal   = {Journal of statistical physics},
  number    = {4},
  publisher = {Springer},
  address   = {New York},
  issn      = {0022-4715},
  doi       = {10.1007/s10955-020-02559-3},
  pages     = {972 -- 996},
  year      = {2020},
  abstract  = {We construct marked Gibbs point processes in R-d under quite general assumptions. Firstly, we allow for interaction functionals that may be unbounded and whose range is not assumed to be uniformly bounded. Indeed, our typical interaction admits an a.s. finite but random range. Secondly, the random marks-attached to the locations in R-d-belong to a general normed space G. They are not bounded, but their law should admit a super-exponential moment. The approach used here relies on the so-called entropy method and large-deviation tools in order to prove tightness of a family of finite-volume Gibbs point processes. An application to infinite-dimensional interacting diffusions is also presented.},
  language  = {en}
}
@article{Seyedhosseini2022,
  author    = {Seyedhosseini, Mehran},
  title     = {A variant of Roe algebras for spaces with cylindrical ends with applications in relative higher index theory},
  series = {Journal of noncommutative geometry},
  volume    = {16},
  journal   = {Journal of noncommutative geometry},
  number    = {2},
  publisher = {European Mathematical Society},
  address   = {Zurich},
  issn      = {1661-6952},
  doi       = {10.4171/JNCG/457},
  pages     = {595 -- 624},
  year      = {2022},
  abstract  = {In this paper, we define a variant of Roe algebras for spaces with cylindrical ends and use this to study questions regarding existence and classification of metrics of positive scalar curvature on such manifolds which are collared on the cylindrical end. We discuss how our constructions are related to relative higher index theory as developed by Chang, Weinberger, and Yu and use this relationship to define higher rho-invariants for positive scalar curvature metrics on manifolds with boundary. This paves the way for the classification of these metrics. Finally, we use the machinery developed here to give a concise proof of a result of Schick and the author, which relates the relative higher index with indices defined in the presence of positive scalar curvature on the boundary.},
  language  = {en}
}
@article{Zass2021,
  author    = {Zass, Alexander},
  title     = {Gibbs point processes on path space},
  series = {Markov processes and related fields},
  volume    = {28},
  journal   = {Markov processes and related fields},
  number    = {3},
  publisher = {Polymat},
  address   = {Moscow},
  issn      = {1024-2953},
  pages     = {329 -- 364},
  year      = {2021},
  abstract  = {We present general existence and uniqueness results for marked models with pair interactions, exemplified through Gibbs point processes on path space. More precisely, we study a class of infinite-dimensional diffusions under Gibbsian interactions, in the context of marked point configurations: the starting points belong to R-d, and the marks are the paths of Langevin diffusions. We use the entropy method to prove existence of an infinite-volume Gibbs point process and use cluster expansion tools to provide an explicit activity domain in which uniqueness holds.},
  language  = {en}
}
@article{HuangHuangReichetal.2022,
  author    = {Huang, Daniel Zhengyu and Huang, Jiaoyang and Reich, Sebastian and Stuart, Andrew M.},
  title     = {Efficient derivative-free Bayesian inference for large-scale inverse problems},
  series = {Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data},
  volume    = {38},
  journal   = {Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data},
  number    = {12},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {0266-5611},
  doi       = {10.1088/1361-6420/ac99fa},
  pages     = {40},
  year      = {2022},
  abstract  = {We consider Bayesian inference for large-scale inverse problems, where computational challenges arise from the need for repeated evaluations of an expensive forward model. This renders most Markov chain Monte Carlo approaches infeasible, since they typically require O(10(4)) model runs, or more. Moreover, the forward model is often given as a black box or is impractical to differentiate. Therefore derivative-free algorithms are highly desirable. We propose a framework, which is built on Kalman methodology, to efficiently perform Bayesian inference in such inverse problems. The basic method is based on an approximation of the filtering distribution of a novel mean-field dynamical system, into which the inverse problem is embedded as an observation operator. Theoretical properties are established for linear inverse problems, demonstrating that the desired Bayesian posterior is given by the steady state of the law of the filtering distribution of the mean-field dynamical system, and proving exponential convergence to it. This suggests that, for nonlinear problems which are close to Gaussian, sequentially computing this law provides the basis for efficient iterative methods to approximate the Bayesian posterior. Ensemble methods are applied to obtain interacting particle system approximations of the filtering distribution of the mean-field model; and practical strategies to further reduce the computational and memory cost of the methodology are presented, including low-rank approximation and a bi-fidelity approach. The effectiveness of the framework is demonstrated in several numerical experiments, including proof-of-concept linear/nonlinear examples and two large-scale applications: learning of permeability parameters in subsurface flow; and learning subgrid-scale parameters in a global climate model. Moreover, the stochastic ensemble Kalman filter and various ensemble square-root Kalman filters are all employed and are compared numerically. The results demonstrate that the proposed method, based on exponential convergence to the filtering distribution of a mean-field dynamical system, is competitive with pre-existing Kalman-based methods for inverse problems.},
  language  = {en}
}
@article{ShlapunovTarchanov2022,
  author    = {Shlapunov, Alexander A. and Tarchanov, Nikolaj Nikolaevič},
  title     = {Inverse image of precompact sets and regular solutions to the Navier-Stokes equations},
  series = {Vestnik Udmurtskogo Universiteta. Matematika, mechanika, kompʹjuternye nauki},
  volume    = {32},
  journal   = {Vestnik Udmurtskogo Universiteta. Matematika, mechanika, kompʹjuternye nauki},
  number    = {2},
  publisher = {Udmurtskij gosudarstvennyj universitet},
  address   = {Iževsk},
  issn      = {1994-9197},
  doi       = {10.35634/vm220208},
  pages     = {278 -- 297},
  year      = {2022},
  abstract  = {We consider the initial value problem for the Navier-Stokes equations over R-3 x [0, T] with time T > 0 in the spatially periodic setting. We prove that it induces open injective mappings A(s): B-1(s) -> B-2(s-1) where B-1(s), B-2(s-1) are elements from scales of specially constructed function spaces of Bochner-Sobolev typeparametrized with the smoothness index s is an element of N. Finally, we prove that a map Asis surjective if and only if the inverse image A(s)(- 1) (K) of any pre compact set K from the range of the map Asis bounded in the Bochner space L-s([0, T], L-r(T-3))with the Ladyzhenskaya-Prodi-Serrin numbers s, r.},
  language  = {en}
}