arXiv research

The paper extends von Neumann's theory to normed modules and shows how they can be represented.

The paper studies symplectic forms on projective limits of Banach bundles and their Darboux Theorem.

Constructs Poisson structure on Banach Lie algebroid predual.

First, we extend the notion of second order differential equations (SODE) on a smooth manifold to anchored Banach vector bundles. Then we define the Banach Lie algebroids as Lie algebroids structures modeled on anchored Banach vector bundles and prove that they form a category.

Study proves existence of precotangent bundles for Grassmannians.

Integrability criterion for projective limits of Banach distributions on Fréchet manifolds.

A {1}-structure on a Banach manifold M (with model space E) is an E-valued 1-form on M that induces on each tangent space an isomorphism onto E. Given a Banach principal bundle P with connected base space and a {1}-structure on P, we show that its automorphism group can be turned into a Banach-Lie group acting smoothly…

This paper is devoted to the framework of direct limit of anchored Banach bundles over a convenient manifold which is a direct limit of Banach manifold. In particular we give a criterion of integrability for distributions on such convenient manifolds which are locally direct limits of particular sequences of Banach anc…

Let $X$ be an $n$-dimensional manifold and $V_1,\ldots,V_n\subset C^\infty(X,\mathbb R)$ finite-dimensional vector spaces. For systems of equations $\{f_i = a_i\colon\: f_i\in V_i,\:a_i \in\mathbb R,\:i=1,\ldots,n\}$ we discover a relationship between the average number of their solutions and mixed volumes of convex bo…

Under appropriate assumptions, we generalize the concept of linear almost Poisson struc- tures, almost Lie algebroids, almost differentials in the framework of Banach anchored bundles and the relation between these objects. We then obtain an adapted formalism for mechanical systems which is illustrated by the evolution…

Regulated curves on Banach manifolds with continuous projections and regulated derivatives are studied.

We investigate infinitesimal properties of sets of ordered $n$-uples of idempotents in a symmetric Banach $*$-algebra. These sets are called flag manifolds and carry several interesting bundles that hold an important role in some areas of operator theory. In this direction, we introduce and study Stiefel bundles on fla…

Study Nijenhuis operators on Banach homogeneous spaces, extending previous work.

Extends Gaussian process theory to Banach spaces.

Let $X$ be a (real or complex) Banach space, and $\mathcal{I}(X)$ be the set of all (non-zero and non-identity) idempotents; i.e., bounded linear operators on $X$ whose squares equal themselves. We show that the Banach submanifold $\mathcal{I}(X)$ of $\mathcal{L}(X)$ is a locally trivial analytic affine-Banach bundle o…

New kernels defined for various spaces, including measures.

Study Nijenhuis operators on homogeneous spaces related to C*-algebras.

The aim of this note is to analyse the structure of the $L^0$-normed $L^0$-modules over a metric measure space. These are a tool that has been introduced by N. Gigli to develop a differential calculus on spaces verifying the Riemannian Curvature Dimension condition. More precisely, we discuss under which conditions an …

Some recent work in Frechet geometry is briefly reviewed. In particular an earlier result on the structure of second tangent bundles in the finite dimensional case was extended to infinite dimensional Banach manifolds and Frechet manifolds that could be represented as projective limits of Banach manifolds. This led to …

Study on measure-valued CARMA processes in Banach spaces.

The second order tangent bundle $T^{2}M$ of a smooth manifold $M$ consists of the equivalent classes of curves on $M$ that agree up to their acceleration. It is known that in the case of a finite $n$-dimensional manifold $M$, $T^{2}M$ becomes a vector bundle over $M$ if and only if $M$ is endowed with a linear connecti…

The paper uses Banach spaces to analyze neural networks.

The tangent bundle $T^kM$ of order $k$, of a smooth Banach manifold $M$ consists of all equivalent classes of curves that agree up to their accelerations of order $k$. For a Banach manifold $M$ and a natural number $k$ first we determine a smooth manifold structure on $T^kM$ which also offers a fiber bundle structure f…

We show how to construct measures on Banach manifolds associated to supersymmetric quantum field theories. These measures are mathematically well-defined objects inspired by the formal path integrals appearing in the physics literature on quantum field theory. We give three concrete examples of our construction. The fi…

The aim of this article is to study effective Reifenberg theorems for measures in a Hilbert or Banach space. For Hilbert spaces, we see all the results from $\mathbb{R}^n$ continue to hold with no additional restrictions. For a general Banach spaces we will see that the classical Reifenberg theorem holds, and that a we…

The study defines divergence for multivector fields on infinite-dimensional manifolds.

We prove the differentiability of Lipschitz maps X-->V, where X is a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V is a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure …

Develops an L^p theory for Dolbeault-Dirac operators on compact Kähler manifolds.

Targeting at sparse learning, we construct Banach spaces B of functions on an input space X with the properties that (1) B possesses an l1 norm in the sense that it is isometrically isomorphic to the Banach space of integrable functions on X with respect to the counting measure; (2) point evaluations are continuous lin…

The abstract extends deformation and tangent groupoid constructions to infinite-dimensional manifolds.

We define the notion of strong projective limit of Banach Lie algebroids. We study the associated structures of Fréchet bundles and the compatibility with the different morphisms. This kind of structure seems to be a convenient framework for various situations.

Develops a new method for robust risk measurement by averaging nearby payoffs.

New algorithm for estimating multivariate quantiles using stochastic optimal transport.

Study extends JB-algebra structure group results to infinite dimensions.

Sharp bounds for max-sliced Wasserstein distances derived for empirical distributions.

We consider the construction of the basic bundle gerbe on SU(n) introduced by Meinrenken and show that it extends to a range of groups with unitary actions on a Hilbert space including U(n), diagonal tori and the Banach Lie group of unitary operators differing from the identity by an element of a Schatten ideal. In all…

Study first-order locally convex Lie algebroids in Bastiani calculus.

Develops thermodynamic formalism for quasimorphisms on negatively curved spaces.

Study variance-reduced method for estimating fixed points in Banach spaces.

Local connection forms provide a very useful tool for handling connections on principal bundles, because they ignore any complexities of the total space and, essentially, involve only two fundamental features of the structure group, namely the adjoint representation and the left (logarithmic) differential. The main res…

Newton's method tackles nonlinear mappings into vector bundles with connections and retractions.

Extends Mahalanobis distance to Banach spaces for anomaly detection.

The paper defines projective structures for Lie bialgebras and Poisson-Lie groups.

Formalizes integral curves on Banach manifolds in Lean.

In this paper, using the structures of cone and bicone fields on vector bundles, the author introduces a ILB (inverse limit of Banach)- manifold structure on $\mathcal M$ the space of Riemannian metrics on a noncompact manifold $M$. In the last section, it is proven that, this way, on the open submanifold $\mathcal{M}_…

Researchers develop neural networks for approximating functions in Banach spaces.

Study Riemannian metric bundles and their connections to K-theory.

Examines different approaches to Poisson structures in Banach spaces.