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Strongly continuous fields of operators over varying Hilbert spaces Open
After introducing a natural notion of continuous fields of locally convex spaces, we establish a new theory of strongly continuous families of possibly unbounded self-adjoint operators over varying Hilbert spaces. This setting allows to tr…
Mini-Workshop: Recent Results on Loop Spaces Open
This Oberwolfach mini-workshop gathered mathematicians with an expertise in loop spaces, in order to discuss recent breakthrough results in this field, and the implications of these. A particular focus was given on recent results that rigo…
Lower bounds on the normal injectivity radius of hypersurfaces and bounded geometries on manifolds with boundary Open
We prove for the first time a pointwise lower estimate of the normal injectivity radius of an embedded hypersurface in an arbitrary Riemannian manifold. Main applications include: (i) a pointwise lower estimate of the graphing radius of a …
Neumann Cut-Offs and Essential Self-adjointness on Complete Riemannian Manifolds with Boundary Open
We generalize some fundamental results for noncompact Riemannian manfolds without boundary, that only require completeness and no curvature assumptions, to manifolds with boundary: let M be a smooth Riemannian manifold with boundary $$\par…
Asymptotic equivalence of identification operators in geometric scattering theory Open
Given two measures \mu_{1} and \mu_{2} on a measurable space X such that d\mu_{2}=\rho_{1,2}\,d\mu_{1} for some bounded measurable function \rho_{1,2}:X\rightarrow (0,\infty) , there exist two natural identification operators J_{1,2},\tild…
A note on the scattering theory of Kato-Ricci manifolds Open
In this note we prove a new $L^1$ criterion for the existence and completeness of the wave operators corresponding to the Laplace-Beltrami operators corresponding to two Riemannian metrics on a fixed noncompact manifold. Our result relies …
Fermionic Dyson expansions and stochastic Duistermaat-Heckman localization on loop spaces Open
Given a self-adjoint operator $H\geq 0$ and (appropriate) densely defined and closed operators $P_{1},\dots, P_{n}$ in a Hilbert space $\mathscr{H}$, we provide a systematic study of bounded operators given by iterated integrals \begin{ali…
Neumann cut-offs and essential self-adjointness on complete Riemannian manifolds with boundary Open
We generalize some fundamental results for noncompact Riemannian manfolds without boundary, that only require completeness and no curvature assumptions, to manifolds with boundary: let $M$ be a smooth Riemannian manifold with boundary $\pa…
Asymptotic Equivalence of Identification Operators in Geometric Scattering Theory Open
Given two measures $μ_1$ and $μ_2$ on a measurable space $X$ such that $dμ_2=ρ_{1,2} \, dμ_1$ for some bounded measurable function $ρ_{1,2}:X\to (0,\infty)$, there exist two natural identification operators $J_{1,2},\tilde{J}_{1,2}:L^2(X,μ…
Locally convex aspects of the Kato and the Dynkin class on manifolds Open
We consider the Kato and the Dynkin class and their local counterparts on a smooth Riemannian manifold as Fréchet spaces. Based on recent results by Carron, Mondello and Tewodrose we show that for a Riemannian manifold $(X,g)$ of dimension…
Essential Spectrum and Feller Type Properties Open
We give necessary and sufficient conditions for a regular semi-Dirichlet form to enjoy a new Feller type property, which we call weak Feller property . Our characterization involves potential theoretic as well as probabilistic aspects and …
A new notion of subharmonicity on locally smoothing spaces, and a conjecture by Braverman, Milatovic, Shubin Open
Given a strongly local Dirichlet space and $λ\geq 0$, we introduce a new notion of $λ$--subharmonicity for $L^1_\loc$--functions, which we call \emph{local $λ$--shift defectivity}, and which turns out to be equivalent to distributional $λ$…
Feynman–Kac formula for perturbations of order $$\le 1$$, and noncommutative geometry Open
Let Q be a differential operator of order $$\le 1$$ on a complex metric vector bundle $$\mathscr {E}\rightarrow \mathscr {M}$$ with metric connection $$\nabla $$ over a possibly noncompact Riemannian manifold $$\mathscr {M}$$ . Under very …
Estimates for the covariant derivative of the heat semigroup on differential forms, and covariant Riesz transforms Open
With $$\vec {\Delta }_j\ge 0$$ is the uniquely determined self-adjoint realization of the Laplace operator acting on j -forms on a geodesically complete Riemannian manifold M and $$\nabla $$ the Levi-Civita covariant derivative, we prove a…
A differential topological invariant on spin manifolds from supersymmetric path integrals Open
We show that the N=1/2 supersymmetric path integral on a closed even dimensional Riemannian spin manifold, realized via Chen forms and recent results from noncommutative geometry, induces a differential topological invariant (which does no…
A Chern-Simons transgression formula for supersymmetric path integrals on spin manifolds Open
Earlier results show that the N = 1/2 supersymmetric path integral on a closed even dimensional Riemannian spin manifold (X,g) can be constructed in a mathematically rigorous way via Chen differential forms and techniques from non-commutat…
Odd characteristic classes in entire cyclic homology and equivariant loop space homology Open
Given a compact manifold M and a smooth map g\colon M\to U(l\times l;\mathbb{C}) from M to the Lie group of unitary l\times l matrices with entries in \mathbb{C} , we construct a Chern character \mathrm{Ch}^-(g) which lives in the odd part…
Estimates for the covariant derivative of the heat semigroup on differential forms, and covariant Riesz transforms Open
With $\vecΔ_j\geq 0$ is the uniquely determined self-adjoint realization of the Laplace operator acting on $j$-forms on a geodesically complete Riemannian manifold $M$ and $\nabla$ the Levi-Civita covariant derivative, we prove amongst oth…
Heat flow regularity, Bismut–Elworthy–Li’s derivative formula, and pathwise couplings on Riemannian manifolds with Kato bounded Ricci curvature Open
We prove that if the Ricci tensor $\mathrm{Ric}$ of a geodesically complete Riemannian manifold $M$, endowed with the Riemannian distance $\mathsf{d}$ and the Riemannian measure $\mathfrak{m}$, is bounded from below by a continuous functio…
Feynman-Kac formula for perturbations of order $\leq 1$ and noncommutative geometry Open
Let $Q$ be a differential operator of order $\leq 1$ on a complex metric vector bundle $\mathscr{E}\to \mathscr{M}$ with metric connection $\nabla$ over a possibly noncompact Riemannian manifold $\mathscr{M}$. Under very mild regularity as…
Molecules as metric measure spaces with Kato-bounded Ricci curvature Open
Set , with the ground state of an arbitrary molecule with electrons in the infinite mass limit (neglecting spin/statistics). Let be the set of singularities of the underlying Coulomb potential. We show that the metric measure space given b…
Scattering Theory and Spectral Stability under a Ricci Flow for Dirac Operators Open
Given a noncompact spin manifold $M$ with a fixed topological spin structure and two complete Riemannian metrics $g$ and $h$ on $M$ with bounded sectional curvatures, we prove a criterion for the existence and completeness of the wave oper…
Heat flow regularity, Bismut's derivative formula, and pathwise Brownian couplings on Riemannian manifolds with Dynkin bounded Ricci curvature Open
We prove that if the Ricci curvature of a geodesically complete Riemannian manifold $X$, endowed with the Riemannian distance $\rho$ and the Riemannian volume measure $\mathfrak{m}$, is bounded from below by a Dynkin decomposable function …
Heat flow regularity, Bismut-Elworthy-Li's derivative formula, and\n pathwise couplings on Riemannian manifolds with Kato bounded Ricci curvature Open
We prove that if the Ricci tensor $\\mathrm{Ric}$ of a geodesically complete\nRiemannian manifold $M$, endowed with the Riemannian distance $\\mathsf{d}$ and\nthe Riemannian measure $\\mathfrak{m}$, is bounded from below by a continuous\nf…
H\"older estimates for magnetic Schr\"odinger semigroups on open subsets of $\mathbb{R}^d$ from mirror coupling Open
We use the mirror coupling of Brownian motion to show that under a $\beta\in (0,1)$-dependent Kato type assumption\footnote{which is satisfied under a suitable $L^q$-assumption on the electro-magnetic potential, where $q$ depends on $\beta…
$\mathrm{RCD}^*(K,N)$ spaces and the geometry of multi-particle Schrödinger semigroups Open
With $(X,\mathfrak{d},\mathfrak{m})$ an $\mathrm{RCD}^*(K,N)$ space for some $K\in\mathbf{R}$, $N\in [1,\infty)$, let $H$ be the self-adjoint Laplacian induced by the underlying Cheeger form. Given $α\in [0,1]$ we introduce the $α$-Kato cl…