SRB Measures of Anosov Actions - Weich, Tobias; Guedes Bonthonneau, Yannick; Guillarmou, Colin (Journal of Differential Geometry (to appear) -- arXiv:2103.12127)

Non-Fock ground states in the translation-invariant Nelson model revisited non-perturbatively - Hasler, David; Hinrichs, Benjamin; Siebert, Oliver (Journal of Functional Analysis)
Keywords: Analysis

Ruelle-Taylor resonaces of Anosov actions - Weich, Tobias; Guedes Bonthonneau, Yannick; Guillarmou, Colin; Hilgert, Joachim (J. Europ. Math. Soc.)

Quantum-Classical Correspondences for Locally Symmetric Spaces - Hilgert, Joachim

Feynman-Kac formulas for semigroups generated by multi-polaron Hamiltonians in magnetic fields and on general domains - Hinrichs, Benjamin; Matte, Oliver (arXiv:2403.12147)
We prove Feynman-Kac formulas for the semigroups generated by selfadjoint operators in a class containing Fr\"ohlich Hamiltonians known from solid state physics. The latter model multi-polarons, i.e., a fixed number of quantum mechanical electrons moving in a polarizable crystal and interacting with the quantized phonon field generated by the crystal's vibrational modes. Both the electrons and phonons can be confined to suitable open subsets of Euclidean space. We also include possibly very singular magnetic vector potentials and electrostatic potentials. Our Feynman-Kac formulas comprise Fock space operator-valued multiplicative functionals and can be applied to every vector in the underlying Hilbert space. In comparison to the renormalized Nelson model, for which analogous Feynman-Kac formulas are known, the analysis of the creation and annihilation terms in the multiplicative functionals requires novel ideas to overcome difficulties caused by the phonon dispersion relation being constant. Getting these terms under control and generalizing other construction steps so as to cover confined systems are the main achievements of this article.

Locally homogeneous Axiom A flows I: projective Anosov subgroups and exponential mixing - Delarue, Benjamin; Monclair, Daniel; Sanders, Andrew (arXiv:2403.14257)
By constructing a non-empty domain of discontinuity in a suitable homogeneous space, we prove that every torsion-free projective Anosov subgroup is the monodromy group of a locally homogeneous contact Axiom A dynamical system with a unique basic hyperbolic set on which the flow is conjugate to the refraction flow of Sambarino. Under the assumption of irreducibility, we utilize the work of Stoyanov to establish spectral estimates for the associated complex Ruelle transfer operators, and by way of corollary: exponential mixing, exponentially decaying error term in the prime orbit theorem, and a spectral gap for the Ruelle zeta function. With no irreducibility assumption, results of Dyatlov-Guillarmou imply the global meromorphic continuation of zeta functions with smooth weights, as well as the existence of a discrete spectrum of Ruelle-Pollicott resonances and (co)-resonant states. We apply our results to space-like geodesic flows for the convex cocompact pseudo-Riemannian manifolds of Danciger-Gu\'eritaud-Kassel, and the Benoist-Hilbert geodesic flow for strictly convex real projective manifolds.

Quantum resonances and scattering poles of classical rank one locally symmetric spaces - Delarue, Benjamin; Hilgert, Joachim (arXiv:2403.14426)
For negatively curved symmetric spaces it is known that the poles of the scattering matrices defined via the standard intertwining operators for the spherical principal representations of the isometry group are either given as poles of the intertwining operators or as quantum resonances, i.e. poles of the meromorphically continued resolvents of the Laplace-Beltrami operator. We extend this result to classical locally symmetric spaces of negative curvature with convex-cocompact fundamental group using results of Bunke and Olbrich. The method of proof forces us to exclude the spectral parameters corresponding to singular Poisson transforms.

Hankel transform, K-Bessel functions and zeta distributions in the Dunkl setting - Brennecken, Dominik (Journal of Mathematical Analysis and Applications)
Keywords: Applied Mathematics; Analysis

Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces - Papageorgiou, Efthymia (Journal of Evolution Equations)
Abstract This work deals with the extension problem for the fractional Laplacian on Riemannian symmetric spaces G / K of noncompact type and of general rank, which gives rise to a family of convolution operators, including the Poisson operator. More precisely, motivated by Euclidean results for the Poisson semigroup, we study the long-time asymptotic behavior of solutions to the extension problem for $$L^1$$ L 1 initial data. In the case of the Laplace–Beltrami operator, we show that if the initial data are bi- K -invariant, then the solution to the extension problem behaves asymptotically as the mass times the fundamental solution, but this convergence may break down in the non-bi- K -invariant case. In the second part, we investigate the long-time asymptotic behavior of the extension problem associated with the so-called distinguished Laplacian on G / K . In this case, we observe phenomena which are similar to the Euclidean setting for the Poisson semigroup, such as $$L^1$$ L 1 asymptotic convergence without the assumption of bi- K -invariance.
Keywords: Mathematics (miscellaneous)

Temperedness of locally symmetric spaces: The product case - Weich, Tobias; Wolf, Lasse Lennart (Geom Dedicata)
Let $X=X_1× X_2$ be a product of two rank one symmetric spaces of non-compact type and $Γ$ a torsion-free discrete subgroup in $G_1× G_2$. We show that the spectrum of $Γ \backslash X$ is related to the asymptotic growth of $Γ$ in the two direction defined by the two factors. We obtain that $L^2(Γ \backslash G)$ is tempered for large class of $Γ$.

Polyhedral bounds on the joint spectrum and temperedness of locally symmetric spaces - Lutsko, Christopher; Weich, Tobias; Wolf, Lasse Lennart (arXiv:2402.02530)
Given a real semisimple connected Lie group $G$ and a discrete torsion-free subgroup $Γ < G$ we prove a precise connection between growth rates of the group $Γ$, polyhedral bounds on the joint spectrum of the ring of invariant differential operators, and the decay of matrix coefficients. In particular, this allows us to completely characterize temperedness of $L^2(Γ\backslash G)$ in this general setting.

A $C^k$-seeley-extension-theorem for Bastiani’s differential calculus - Hanusch, Maximilian (Canadian Journal of Mathematics)
Keywords: extension of differentiable maps

Possible points of blow-up in chemotaxis systems with spatially heterogeneous logistic source - Black, Tobias; Fuest, Mario; Lankeit, Johannes; Mizukami, Masaaki (Nonlinear Analysis: Real World Applications)
Keywords: Applied Mathematics; Computational Mathematics; General Economics; Econometrics and Finance; General Engineering; General Medicine; Analysis

The Lax Equation and Weak Regularity of Asymptotic Estimate Lie Groups - Hanusch, Maximilian (Annals of Global Analysis and Geometry)
Keywords: Lax equation; generalized Baker-Campbell-Dynkin-Hausdorff formula; regularity of Lie groups

Decompositions of Analytic 1-Manifolds - Hanusch, Maximilian (Indagationes Mathematicae.)
Keywords: Lie group actions and analytic 1-submanifolds

Super-Gaussian decay of exponentials: A sufficient condition - Hinrichs, Benjamin; Janssen, Daan W.; Ziebell, Jobst (Journal of Mathematical Analysis and Applications)
Keywords: Applied Mathematics; Analysis

Feynman-Kac formula for fiber Hamiltonians in the relativistic Nelson model in two spatial dimensions - Hinrichs, Benjamin; Matte, Oliver (arXiv:2309.09005)
In this proceeding we consider a translation invariant Nelson type model in two spatial dimensions modeling a scalar relativistic particle in interaction with a massive radiation field. As is well-known, the corresponding Hamiltonian can be defined with the help of an energy renormalization. First, we review a Feynman-Kac formula for the semigroup generated by this Hamiltonian proven by the authors in a recent preprint (where several matter particles and exterior potentials are treated as well). After that, we employ a few technical key relations and estimates obtained in our preprint to present an otherwise self-contained derivation of new Feynman-Kac formulas for the fiber Hamiltonians attached to fixed total momenta of the translation invariant system. We conclude by inferring an alternative derivation of the Feynman-Kac formula for the full translation invariant Hamiltonian.

Absence of principal eigenvalues for higher rank locally symmetric spaces - Weich, Tobias; Wolf, Lasse Lennart (Communications in Mathematical Physics)
Given a geometrically finite hyperbolic surface of infinite volume it is a classical result of Patterson that the positive Laplace-Beltrami operator has no $L^2$-eigenvalues $≥ 1/4$. In this article we prove a generalization of this result for the joint $L^2$-eigenvalues of the algebra of commuting differential operators on Riemannian locally symmetric spaces $Γ\backslash G/K$ of higher rank. We derive dynamical assumptions on the $Γ$-action on the geodesic and the Satake compactifications which imply the absence of the corresponding principal eigenvalues. A large class of examples fulfilling these assumptions are the non-compact quotients by Anosov subgroups.

Meromorphic Continuation of Weighted Zeta Functions on Open Hyperbolic Systems - Schütte, Philipp; Weich, Tobias; Barkhofen, Sonja (Communications in Mathematical Physics)
In this article we prove meromorphic continuation of weighted zeta functions in the framework of open hyperbolic systems by using the meromorphically continued restricted resolvent of Dyatlov and Guillarmou (2016). We obtain a residue formula proving equality between residues of weighted zetas and invariant Ruelle distributions. We combine this equality with results of Guillarmou, Hilgert and Weich (2021) in order to relate the residues to Patterson-Sullivan distributions. Finally we provide proof-of-principle results concerning the numerical calculation of invariant Ruelle distributions for 3-disc scattering systems.

Invariant Ruelle Distributions on Convex-Cocompact Hyperbolic Surfaces -- A Numerical Algorithm via Weighted Zeta Functions - Schütte, Philipp; Weich, Tobias (arXiv:2308.13463)
We present a numerical algorithm for the computation of invariant Ruelle distributions on convex co-compact hyperbolic surfaces. This is achieved by exploiting the connection between invariant Ruelle distributions and residues of meromorphically continued weighted zeta functions established by the authors together with Barkhofen (2021). To make this applicable for numerics we express the weighted zeta as the logarithmic derivative of a suitable parameter dependent Fredholm determinant similar to Borthwick (2014). As an additional difficulty our transfer operator has to include a contracting direction which we account for with techniques developed by Rugh (1992). We achieve a further improvement in convergence speed for our algorithm in the case of surfaces with additional symmetries by proving and applying a symmetry reduction of weighted zeta functions.

On Lieb-Robinson bounds for a class of continuum fermions - Hinrichs, Benjamin; Lemm, Marius; Siebert, Oliver (arXiv:2310.17736)
We consider the quantum dynamics of a many-fermion system in $\mathbb R^d$ with an ultraviolet regularized pair interaction as previously studied in [M. Gebert, B. Nachtergaele, J. Reschke, and R. Sims, Ann. Henri Poincar\'e 21.11 (2020)]. We provide a Lieb-Robinson bound under substantially relaxed assumptions on the potentials. We also improve the associated one-body Lieb-Robinson bound on $L^2$-overlaps to an almost ballistic one (i.e., an almost linear light cone) under the same relaxed assumptions. Applications include the existence of the infinite-volume dynamics and clustering of ground states in the presence of a spectral gap. We also develop a fermionic continuum notion of conditional expectation and use it to approximate time-evolved fermionic observables by local ones, which opens the door to other applications of the Lieb-Robinson bounds.

A Lower Bound on the Critical Momentum of an Impurity in a Bose-Einstein Condensate - Hinrichs, Benjamin; Lampart, Jonas (arXiv:2311.05361)
In the Bogoliubov-Fr\"ohlich model, we prove that an impurity immersed in a Bose-Einstein condensate forms a stable quasi-particle when the total momentum is less than its mass times the speed of sound. The system thus exhibits superfluid behavior, as this quasi-particle does not experience friction. We do not assume any infrared or ultraviolet regularization of the model, which contains massless excitations and point-like interactions.

Higher rank quantum-classical correspondence - Hilgert, Joachim; Weich, Tobias; Wolf, Lasse Lennart (Analysis & PDE)
For a compact Riemannian locally symmetric space $Γ\backslash G/K$ of arbitrary rank we determine the location of certain Ruelle-Taylor resonances for the Weyl chamber action. We provide a Weyl-lower bound on an appropriate counting function for the Ruelle-Taylor resonances and establish a spectral gap which is uniform in $Γ$ if $G/K$ is irreducible of higher rank. This is achieved by proving a quantum-classical correspondence, i.e. a 1:1-correspondence between horocyclically invariant Ruelle-Taylor resonant states and joint eigenfunctions of the algebra of invariant differential operators on $G/K$.

Spectral correspondences for finite graphs without dead ends - Bux, Kai-Uwe; Hilgert, Joachim; Weich, Tobias (arXiv:2307.10876)
We compare the spectral properties of two kinds of linear operators characterizing the (classical) geodesic flow and its quantization on connected locally finite graphs without dead ends. The first kind are transfer operators acting on vector spaces associated with the set of non backtracking paths in the graphs. The second kind of operators are averaging operators acting on vector spaces associated with the space of vertices of the graph. The choice of vector spaces reflects regularity properties. Our main results are correspondences between classical and quantum spectral objects as well as some automatic regularity properties for eigenfunctions of transfer operators.

Spectral correspondences for rank one locally symmetric spaces: the case of exceptional parameters - Arends, Christian; Hilgert, Joachim (Journal de l’École polytechnique — Mathématiques)
In this paper we complete the program of relating the Laplace spectrum for rank one compact locally symmetric spaces with the first band Ruelle-Pollicott resonances of the geodesic flow on its sphere bundle. This program was started by Flaminio and Forni for hyperbolic surfaces, continued by Dyatlov, Faure and Guillarmou for real hyperbolic spaces and by Guillarmou, Hilgert and Weich for general rank one spaces. Except for the case of hyperbolic surfaces a countable set of exceptional spectral parameters always left untreated since the corresponding Poisson transforms are neither injective nor surjective. We use vector valued Poisson transforms to treat also the exceptional spectral parameters. For surfaces the exceptional spectral parameters lead to discrete series representations of $SL(2,\mathbb R)$. In higher dimensions the situation is more complicated, but can be described completely.
Keywords: Ruelle resonances; Poisson transforms; locally symmetric spaces; principal series representations

Spectral correspondences for rank one locally symmetric spaces - The case of exceptional parameters - Hilgert, Joachim; Arends, C. (J. de l'École polytechnique — Mathématiques)

Aspects of control theory on infinite-dimensional Lie groups and G-manifolds - Hilgert, Joachim; Glöckner, H. (J. Diff. Equations)

A pairing formula for resonant states on finite regular graphs - Hilgert, Joachim; Arends, C.; Frahm, J.

Edge Laplacians and vector valued Poisson transforms for graphs - Hilgert, Joachim; Arends, C.; Frahm, J.

Quantization in fibering polarizations, Mabuchi rays and geometric Peter--Weyl theorem - Hilgert, Joachim; Baier, T.; Kaya, O.; Mourao, J.; Nunes, J.

Ruelle-Taylor resonances of Anosov actions - Hilgert, Joachim; Guedes Bonthonneau, Y.; Guillarmou, C.; Weich, Tobias

Deep neural networks on diffeomorphism groups for optimal shape reparametrization - Celledoni, Elena; Glöckner, Helge; Riseth, Jørgen; Schmeding, Alexander (BIT Numerical Mathematics)

Aspects of control theory on infinite-dimensional Lie groups and G-manifolds - Glöckner, Helge; Hilgert, Joachim (Journal of Differential Equations)
Keywords: 22E65; 28B05; 34A12; 34H05; 46E30; 46E40

Diffeomorphism groups of convex polytopes - Glöckner, Helge (Journal of Convex Analysis)
Let $E$ be a finite-dimensional real vector space and $M\subseteq E$ be a convex polytope with non-empty interior. We turn the group of all $C^∞$-diffeomorphisms of $M$ into a regular Lie group.

Mapping groups associated with real-valued function spaces and direct limits of Sobolev-Lie groups - Glöckner, Helge; Tárrega, Luis (Journal of Lie Theory)

$L^2$-spectrum, growth indicator function and critical exponent on locally symmetric spaces - Wolf, Lasse L.; Zhang, Hong-Wei (arXiv:2311.11770)
In this short note we observe, on locally symmetric spaces of higher rank, a connection between the growth indicator function introduced by Quint and the modified critical exponent of the Poincar\'e series equipped with the polyhedral distance. As a consequence, we provide a different characterization of the bottom of the $L^2$-spectrum of the Laplace-Beltrami operator in terms of the growth indicator function. Moreover, we explore the relationship between these three objects and the temperedness.

Resonances and Weighted Zeta Functions for Obstacle Scattering via Smooth Models - Delarue, Benjamin; Schütte, Philipp; Weich, Tobias (Annales Henri Poincaré)
Abstract We consider a geodesic billiard system consisting of a complete Riemannian manifold and an obstacle submanifold with boundary at which the trajectories of the geodesic flow experience specular reflections. We show that if the geodesic billiard system is hyperbolic on its trapped set and the latter is compact and non-grazing, the techniques for open hyperbolic systems developed by Dyatlov and Guillarmou (Ann Henri Poincaré 17(11):3089–3146, 2016) can be applied to a smooth model for the discontinuous flow defined by the non-grazing billiard trajectories. This allows us to obtain a meromorphic resolvent for the generator of the billiard flow. As an application we prove a meromorphic continuation of weighted zeta functions together with explicit residue formulae. In particular, our results apply to scattering by convex obstacles in the Euclidean plane.
Keywords: Mathematical Physics; Nuclear and High Energy Physics; Statistical and Nonlinear Physics

Singular cohomology of symplectic quotients by circle actions and Kirwan surjectivity - Delarue, Benjamin; Ramacher, Pablo; Schmitt, Maximilian (arXiv:2312.03634)
Let $M$ be a symplectic manifold carrying a Hamiltonian $S^1$-action with momentum map $J:M → \mathbb{R}$ and consider the corresponding symplectic quotient $\mathcal{M}_0:=J^{-1}(0)/S^1$. We extend Sjamaar's complex of differential forms on $\mathcal{M}_0$, whose cohomology is isomorphic to the singular cohomology $H(\mathcal{M}_0;\mathbb{R})$ of $\mathcal{M}_0$ with real coefficients, to a complex of differential forms on $\mathcal{M}_0$ associated with a partial desingularization $\widetilde{\mathcal{M}}_0$, which we call resolution differential forms. The cohomology of that complex turns out to be isomorphic to the de Rham cohomology $H(\widetilde{ \mathcal{M}}_0)$ of $\widetilde{\mathcal{M}}_0$. Based on this, we derive a long exact sequence involving both $H(\mathcal{M}_0;\mathbb{R})$ and $H(\widetilde{ \mathcal{M}}_0)$ and give conditions for its splitting. We then define a Kirwan map $\mathcal{K}:H_{S^1}(M) → H(\widetilde{\mathcal{M}}_0)$ from the equivariant cohomology $H_{S^1}(M)$ of $M$ to $H(\widetilde{\mathcal{M}}_0)$ and show that its image contains the image of $H(\mathcal{M}_0;\mathbb{R})$ in $H(\widetilde{\mathcal{M}}_0)$ under the natural inclusion. Combining both results in the case that all fixed point components of $M$ have vanishing odd cohomology we obtain a surjection $\check κ:H^ev_{S^1}(M) → H^ev(\mathcal{M}_0;\mathbb{R})$ in even degrees, while already simple examples show that a similar surjection in odd degrees does not exist in general. As an interesting class of examples we study abelian polygon spaces.

A Riemann-Roch formula for singular reductions by circle actions - Delarue, Benjamin; Ioos, Louis; Ramacher, Pablo (arXiv:2302.09894)
We compute a Riemann-Roch formula for the invariant Riemann-Roch number of a quantizable Hamiltonian $S^1$-manifold $(M,ω,\mathcal{J})$ in terms of the geometry of its symplectic quotient, allowing $0$ to be a singular value of the moment map $\mathcal{J}:M→\mathbb{R}$. The formula involves a new explicit local invariant of the singularities. Our approach relies on a complete singular stationary phase expansion of the associated Witten integral.

Large-Time Behavior of Two Families of Operators Related to the Fractional Laplacian on Certain Riemannian Manifolds - Papageorgiou, Efthymia (Potential Analysis)
Abstract This note is concerned with two families of operators related to the fractional Laplacian, the first arising from the Caffarelli-Silvestre extension problem and the second from the fractional heat equation. They both include the Poisson semigroup. We show that on a complete, connected, and non-compact Riemannian manifold of non-negative Ricci curvature, in both cases, the solution with $$L^1$$ L 1 initial data behaves asymptotically as the mass times the fundamental solution. Similar long-time convergence results remain valid on more general manifolds satisfying the Li-Yau two-sided estimate of the heat kernel. The situation changes drastically on hyperbolic space, and more generally on rank one non-compact symmetric spaces: we show that for the Poisson semigroup, the convergence to the Poisson kernel fails -but remains true under the additional assumption of radial initial data.
Keywords: Analysis

Asymptotics for the infinite Brownian loop on noncompact symmetric spaces - Papageorgiou, Efthymia (Journal of Elliptic and Parabolic Equations)
Abstract The infinite Brownian loop on a Riemannian manifold is the limit in distribution of the Brownian bridge of length T around a fixed origin when $$T → +∞ $$ T → + ∞ . The aim of this note is to study its long-time asymptotics on Riemannian symmetric spaces G / K of noncompact type and of general rank. This amounts to the behavior of solutions to the heat equation subject to the Doob transform induced by the ground spherical function. Unlike the standard Brownian motion, we observe in this case phenomena which are similar to the Euclidean setting, namely $$L^1$$ L 1 asymptotic convergence without requiring bi- K -invariance for initial data, and strong $$L^{∞ }$$ L ∞ convergence.
Keywords: Applied Mathematics; Numerical Analysis; Analysis

New examples of harmonic maps to the hyperbolic plane via Bäcklund transformation - Polychrou, G.; Papageorgiou, Efthymia; Fotiadis, A.; Daskaloyannis, C. (Revista Matemática Complutense)
Abstract We study harmonic maps from a subset of the complex plane to a subset of the hyperbolic plane. In Fotiadis and Daskaloyannis (Nonlinear Anal 214, 112546, 2022), harmonic maps are related to the sinh-Gordon equation and a Bäcklund transformation is introduced, which connects solutions of the sinh-Gordon and sine-Gordon equation. We develop this machinery in order to construct new harmonic maps to the hyperbolic plane.
Keywords: General Mathematics

The Dunkl-Laplace transform and Macdonald’s hypergeometric series - Brennecken, Dominik; Rösler, Margit (Transactions of the American Mathematical Society)

The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds - Cekić, Mihajlo; Delarue, Benjamin; Dyatlov, Semyon; Paternain, Gabriel P. (Inventiones mathematicae)
Abstract We show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold $$Σ $$ Σ with Betti number $$b_1$$ b 1 , the order of vanishing of the Ruelle zeta function at zero equals $$4-b_1$$ 4 - b 1 , while in the hyperbolic case it is equal to $$4-2b_1$$ 4 - 2 b 1 . This is in contrast to the 2-dimensional case where the order of vanishing is a topological invariant. The proof uses the microlocal approach to dynamical zeta functions, giving a geometric description of generalized Pollicott–Ruelle resonant differential forms at 0 in the hyperbolic case and using first variation for the perturbation. To show that the first variation is generically nonzero we introduce a new identity relating pushforwards of products of resonant and coresonant 2-forms on the sphere bundle $$SΣ $$ S Σ with harmonic 1-forms on $$Σ $$ Σ .
Keywords: General Mathematics

Non-Lie subgroups in Lie groups over local fields of positive characteristic - Glöckner, Helge (p-Adic Numbers, Ultrametric Analysis, and Applications)
Keywords: 20Exx; 22Exx; 32Cxx

Manifolds of mappings on Cartesian products - Glöckner, Helge; Schmeding, Alexander (Annals of Global Analysis and Geometry)
Keywords: 58D15; 22E65; 26E15; 26E20; 46E40; 46T20; 58A05

Aspects of differential calculus related to infinite-dimensional vector bundles and Poisson vector spaces - Glöckner, Helge (Axioms)
We prove various results in infinite-dimensional differential calculus that relate the differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: (1) in the theory of infinite-dimensional vector bundles, to construct new bundles from given ones, such as dual bundles, topological tensor products, infinite direct sums, and completions (under suitable hypotheses); (2) in the theory of locally convex Poisson vector spaces, to prove continuity of the Poisson bracket and continuity of passage from a function to the associated Hamiltonian vector field. Topological properties of topological vector spaces are essential for the studies, which allow the hypocontinuity of bilinear mappings to be exploited. Notably, we encounter kR-spaces and locally convex spaces E such that E×E is a kR-space.

Birkhoff decompositions for loop groups with coefficient algebras - Glöckner, Helge (arXiv:2206.11711)
Starting with a finite-dimensional complex Lie algebra, we extend scalars using suitable commutative topological algebras. We study Birkhoff decompositions for the corresponding loop groups. Some results remain valid for loop groups with valued in complex Banach-Lie groups.

Spectral Asymptotics for Kinetic Brownian Motion on Surfaces of Constant Curvature - Kolb, Martin; Weich, Tobias; Wolf, Lasse Lennart (Annales Henri Poincaré)
Abstract The kinetic Brownian motion on the sphere bundle of a Riemannian manifold $$\mathbb {M}$$ M is a stochastic process that models a random perturbation of the geodesic flow. If $$\mathbb {M}$$ M is an orientable compact constantly curved surface, we show that in the limit of infinitely large perturbation the $$L^2$$ L 2 -spectrum of the infinitesimal generator of a time-rescaled version of the process converges to the Laplace spectrum of the base manifold.
Keywords: Mathematical Physics; Nuclear and High Energy Physics; Statistical and Nonlinear Physics

Ruelle–Pollicott resonances for manifolds with hyperbolic cusps - Guedes Bonthonneau, Yannick; Weich, Tobias (Journal of the European Mathematical Society)
Keywords: Applied Mathematics; General Mathematics

Regularity of Lie groups - Hanusch, Maximilian (Communications in Analysis and Geometry)
Keywords: regularity of Lie groups

Analysis 1 und 2 Skript/Buch - Hanusch, Maximilian

Stabilization of arbitrary structures in a doubly degenerate reaction-diffusion system modeling bacterial motion on a nutrient-poor agar. - Winkler, Michael (Calculus of Variations and Partial Differential Equations)

Chemotaxis-Stokes interaction with very weak diffusion enhancement: Blow-up exclusion via detection of absorption-induced entropy structures involving multiplicative couplings. - Winkler, Michael (Advanced Nonlinear Studies)

Small-signal solutions of a two-dimensional doubly degenerate taxis system modeling bacterial motion in a nutrient-poor environments. - Winkler, Michael (Nonlinear Analysis: Real World Applications)

A unifying approach toward boundedness in Keller-Segel type cross-diffusion systems via conditional $L^∞$ estimates for taxis gradients. - Winkler, Michael (Mathematische Nachrichten)

Oscillatory decay in a degenerate parabolic equation. - Winkler, Michael (Partial Differential Equations and Applications)

A critical blow-up exponent for flux limitation in a Keller-Segel system - Winkler, Michael (Indiana University Mathematics Journal)

Unlimited growth in logarithmic Keller-Segel systems - Winkler, Michael (Journal of Differential Equations)

Radial solutions to a chemotaxis-consumption model involving prescribed signal concentrations on the boundary - Lankeit, Johannes; Winkler, Michael (Nonlinearity)

Reaction-driven relaxation in threee-dimensional Keller-Segel-Navier-Stokes interaction. - Winkler, Michael (Communications in Mathematical Physics)

A family of mass-critical Keller-Segel systems. - Winkler, Michael (Proceedings of the London Mathematical Society)

Global weak solutions and absorbing sets in a chemotaxis-Navier–Stokes system with prescribed signal concentration on the boundary - Black, Tobias; Winkler, Michael (Mathematical Models and Methods in Applied Sciences)
An initial-boundary value problem for a coupled chemotaxis-Navier–Stokes model with porous medium type diffusion is considered. Previous related literature has provided profound knowledge in cases when the system is augmented with no-flux/no-flux/no-slip boundary conditions for the density of cells, the chemical concentration and the fluid velocity field, respectively; in particular, available qualitative results strongly indicate that only trivial solution behavior can be expected on large time scales. In line with refined modeling approaches to oxygen evolution near fluid-air interfaces, this study now focuses on situations involving a fixed chemoattractant concentration on the boundary. Despite an apparent loss of mathematically favorable energy structures thereby induced, by means of an alternative variational approach a basic theory of global existence is developed in a natural framework of weak solvability. Beyond this, some additional qualitative information on the large time behavior of these solutions is derived by identifying a certain global relaxation property. Specifically, a second result asserts, within a suitable topological setting, the existence of a bounded set which eventually absorbs each individual of the obtained trajectories, and the diameter of which is bounded only by the physically relevant quantities of total population size and prescribed boundary concentration of the chemical signal.
Keywords: Applied Mathematics; Modeling and Simulation

Global weak solutions to a chemotaxis-Navier-Stokes system in $R^3$. - Kang, Kyungkeun; Lee, Jihoon; Winkler, Michael (Discrete and Continuous Dynamical Systems)

Elementary symmetric polynomials and martingales for Heckman-Opdam processes - Rösler, Margit; Voit, Michael (Contemporary Mathematics)
We consider the generators $L_k$ of Heckman-Opdam diffusion processes in the compact and non-compact case in $N$ dimensions for root systems of type $A$ and $B$, with a multiplicity function of the form $k=κk_0$ with some fixed value $k_0$ and a varying constant $κ\in\,[0,\infty[$. Using elementary symmetric functions, we present polynomials which are simultaneous eigenfunctions of the $L_k$ for all $κ\in\,]0,\infty[$. This leads to martingales associated with the Heckman-Opdam diffusions $ (X_{t,1},\ldots,X_{t,N})_{t\ge0}$. As our results extend to the freezing case $κ=\infty$ with a deterministic limit after some renormalization, we find formulas for the expectations $\mathbb E(\prod_{j=1}^N(y-X_{t,j})),$ $y\in\mathbb C$.

Chemotaxis and cross-diffusion models in complex environments: Models and analytic problems toward a multiscale vision. - Bellomo, Nicolas; Outada, Nisrine; Soler, Juan; Tao, Youshan; Winkler, Michael (Mathematical Models & Methods in Applied Sciences)

Global existence in reaction-diffusion systems with mass control under relaxed assumptions merely referring to cross-absorptiva effects. - Lankeit, Johannes; Winkler, Michael (Journal of Evolution Equations)

A double critical mass phenomenon in a no-flux-Dirichlet Keller-Segel system. - Fuhrmann, Jan; Lankeit, Johannes; Winkler, Michael (Journal de Mathématiques Pures et Appliquées)

A smallness condition ensuring boundedness in a two-dimensional chemotaxis-Navier-Stokes system involving Dirichlet boundary conditions for the signal. - Wang, Yulan; Winkler, Michael; Xiang, Zhaoyin (Acta Mathematica Sinica (English Series))

Global mass-preserving solutions to a chemotaxis-fluid model involving Dirichlet boundary conditions for the signal. - Wang, Yulan; Winkler, Michael; Xiang, Zhaoyin (Analysis and Applications)

Potential kernels for radial Dunkl Laplacians - Graczyk, P.; Luks, Tomasz; Sawyer, P. (Canadian Journal of Mathematics)

Asymptotic stability of spatial homogeneity in a haptotaxis model for oncolytic virotherapy. - Tao, Youshan; Winkler, Michael (Proceedings of the Royal Society of Edinburgh Section A: Mathematics)

Existence theory and qualitative analysis for a fully cross-diffusive predator-prey system - Tao, Youshan; Winkler, Michael (SIAM Journal on Mathematical Analysis)

Nonnegative solutions to a doubly degenerate nutrient taxis system - Li, Genglin; Winkler, Michael (Communications on Pure and Applied Analysis)

Prescribed signal concentration on the boundary: eventual smoothness in a chemotaxis-Navier–Stokes system with logistic proliferation - Black, Tobias; Wu, Chunyan (Calculus of Variations and Partial Differential Equations)
Keywords: Applied Mathematics; Analysis

Semiclassical formulae For Wigner distributions - Barkhofen, Sonja; Schütte, Philipp; Weich, Tobias (Journal of Physics A: Mathematical and Theoretical)
In this paper we give an overview over some aspects of the modern mathematical theory of Ruelle resonances for chaotic, i.e. uniformly hyperbolic, dynamical systems and their implications in physics. First we recall recent developments in the mathematical theory of resonances, in particular how invariant Ruelle distributions arise as residues of weighted zeta functions. Then we derive a correspondence between weighted and semiclassical zeta functions in the setting of negatively curved surfaces. Combining this with results of Hilgert, Guillarmou and Weich yields a high frequency interpretation of invariant Ruelle distributions as quantum mechanical matrix coefficients in constant negative curvature. We finish by presenting numerical calculations of phase space distributions in the more physical setting of 3-disk scattering systems.

Poisson transforms for trees of bounded degree - Bux, Kai-Uwe; Hilgert, Joachim; Weich, Tobias (Journal of Spectral Theory)
Keywords: Geometry and Topology; Mathematical Physics; Statistical and Nonlinear Physics

Poisson transforms for trees of bounded degree - Hilgert, Joachim; Weich, Tobias; Bux, K.-U. (J. of Spectral Theory)

Ethan D. Bolker und Maura B. Mast: Common Sense Mathematics, Second Edition. AMS/MAA Press 2021 - Hilgert, Joachim (Mathematische Semesterberichte)

Asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions - Delarue, Benjamin; Ramacher, Pablo (Journal of Symplectic Geometry)

Decompositions of locally compact contraction groups, series and extensions - Glöckner, Helge; Willis, George A. (Journal of Algebra)
A locally compact contraction group is a pair (G,α), where G is a locally compact group and α:G→G an automorphism such that αn(x)→e pointwise as n→∞. We show that every surjective, continuous, equivariant homomorphism between locally compact contraction groups admits an equivariant continuous global section. As a consequence, extensions of locally compact contraction groups with abelian kernel can be described by continuous equivariant cohomology. For each prime number p, we use 2-cocycles to construct uncountably many pairwise non-isomorphic totally disconnected, locally compact contraction groups (G,α) which are central extensions0→Fp((t))→G→Fp((t))→0 of the additive group of the field of formal Laurent series over Fp=Z/pZ by itself. By contrast, there are only countably many locally compact contraction groups (up to isomorphism) which are torsion groups and abelian, as follows from a classification of the abelian locally compact contraction groups.
Keywords: Contraction group; Torsion group; Extension; Cocycle; Section; Equivariant cohomology; Abelian group; Nilpotent group; Isomorphism types

Locally pro-p contraction groups are nilpotent - Glöckner, Helge; Willis, George A. (J. Reine Angew. Math.)
Keywords: 22D05; 22A05; 20E18

Direct limits of regular Lie groups - Glöckner, Helge (Mathematische Nachrichten)

Contraction groups and the big cell for endomorphisms of Lie groups over local fields - Glöckner, Helge (arXiv:2101.02981)
Let $G$ be a Lie group over a totally disconnected local field and $α$ be an analytic endomorphism of $G$. The contraction group of $α$ ist the set of all $x\in G$ such that $α^n(x)→ e$ as $n→∞$. Call sequence $(x_{-n})_{n≥ 0}$ in $G$ an $α$-regressive trajectory for $x\in G$ if $α(x_{-n})=x_{-n+1}$ for all $n≥ 1$ and $x_0=x$. The anti-contraction group of $α$ is the set of all $x\in G$ admitting an $α$-regressive trajectory $(x_{-n})_{n≥ 0}$ such that $x_{-n}→ e$ as $n→∞$. The Levi subgroup is the set of all $x\in G$ whose $α$-orbit is relatively compact, and such that $x$ admits an $α$-regressive trajectory $(x_{-n})_{n≥ 0}$ such that $\{x_{-n}\colon n≥ 0\}$ is relatively compact. The big cell associated to $α$ is the set $Ω$ of all all products $xyz$ with $x$ in the contraction group, $y$ in the Levi subgroup and $z$ in the anti-contraction group. Let $π$ be the mapping from the cartesian product of the contraction group, Levi subgroup and anti-contraction group to $Ω$ which maps $(x,y,z)$ to $xyz$. We show: $Ω$ is open in $G$ and $π$ is étale for suitable immersed Lie subgroup structures on the three subgroups just mentioned. Moreover, we study group-theoretic properties of contraction groups and anti-contraction groups.

Resonances and weighted zeta functions for obstacle scattering via smooth models - Schütte, Philipp; Weich, Tobias; Delarue, Benjamin
We consider a geodesic billiard system consisting of a complete Riemannian manifold and an obstacle submanifold with boundary at which the trajectories of the geodesic flow experience specular reflections. We show that if the geodesic billiard system is hyperbolic on its trapped set and the latter is compact and non-grazing the techniques for open hyperbolic systems developed by Dyatlov and Guillarmou can be applied to a smooth model for the discontinuous flow defined by the non-grazing billiard trajectories. This allows us to obtain a meromorphic resolvent for the generator of the billiard flow. As an application we prove a meromorphic continuation of weighted zeta functions together with explicit residue formulae. In particular, our results apply to scattering by convex obstacles in the Euclidean plane.

Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces - Küster, Benjamin; Weich, Tobias (International Mathematics Research Notices)
Abstract For a compact Riemannian locally symmetric space $\mathcal M$ of rank 1 and an associated vector bundle $\mathbf V_{τ }$ over the unit cosphere bundle $S^{* }\mathcal M$, we give a precise description of those classical (Pollicott–Ruelle) resonant states on $\mathbf V_{τ }$ that vanish under covariant derivatives in the Anosov-unstable directions of the chaotic geodesic flow on $S^{* }\mathcal M$. In particular, we show that they are isomorphically mapped by natural pushforwards into generalized common eigenspaces of the algebra of invariant differential operators $D(G,σ )$ on compatible associated vector bundles $\mathbf W_{σ }$ over $\mathcal M$. As a consequence of this description, we obtain an exact band structure of the Pollicott–Ruelle spectrum. Further, under some mild assumptions on the representations $τ$ and $σ$ defining the bundles $\mathbf V_{τ }$ and $\mathbf W_{σ }$, we obtain a very explicit description of the generalized common eigenspaces. This allows us to relate classical Pollicott–Ruelle resonances to quantum eigenvalues of a Laplacian in a suitable Hilbert space of sections of $\mathbf W_{σ }$. Our methods of proof are based on representation theory and Lie theory.
Keywords: General Mathematics

Wave Front Sets of Nilpotent Lie Group Representations - Weich, Tobias; Budde, Julia (arXiv:2103.02968v1)

Symmetries of analytic curves - Hanusch, Maximilian (Differential Geometry and its Applications)
Keywords: Geometry and Topology; Analysis

Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with gradient-dependent flux limitation. - Winkler, Michael (Nonlinear Analysis: Real World Applications)

Suppressing blow-up by gradient-dependent flux limitation in a planar Keller-Segel-Navier-Stokes system. - Winkler, Michael (Zeitschrift für Angewandte Mathematik und Physik)

Can rotational fluxes impede the tendency toward spatial homogeneity in nutrient taxis (-Stokes) systems? - Winkler, Michael (International Mathematics Research Notices)

Boundedness in a three-dimensional Keller-Segel-Stokes system with subcritical sensitivity. - Winkler, Michael (Applied Mathematics Letters)

Does spatial homogeneity ultimately prevail in nutrient taxis systems? A paradigm for structure support by rapid diffusion decay in an autonomous parabolic flow. - Winkler, Michael (Transactions of the American Mathematical Society)

Positive intertwiners for Bessel functions of type B - Rösler, Margit; Voit, Michael (Proceedings of the American Mathematical Society)
Keywords: Applied Mathematics; General Mathematics

A fully cross-diffusive two-component evolution system: Existence and qualitative analysis via entropy-consistent thin-film-type approximation. - Tao, Youshan; Winkler, Michael (Journal of Functional Analysis)

The dampening role of large repulsive convection in a chemotaxis system modeling tumor angiogenesis. - Tao, Youshan; Winkler, Michael (Nonlinear Analysis, Theory, Methods and Applications)

Immediate regularization of measure-type population densities in a two-dimensional chemotaxis system with signal consumption. - Wang, Yulan; Winkler, Michael; Xiang, Zhaoyin (Science China Mathematics)

Small-density solutiions in Keller-Segel systems involving rapidly decaying diffusivities. - Ding, Mengyao; Winkler, Michael (Nonlinear Differential Equatiions and Applicatiions)

Global smooth solutions in a two-dimensional cross-diffusion system modeling propagation of urban crime. - Tao, Youshan; Winkler, Michael (Communications in Mathematical Sciences)

Taxis-driven formation of singular hotspots in a May-Nowak type model for virus infection. - Tao, Youshan; Winkler, Michael (SIAM Journal on Mathematical Analysis)