arXiv research

The paper extends Cartan development to infinite dimensional Lie groups.

Paper investigates optimal transport map estimation in infinite-dimensional spaces.

This paper proves neural networks can approximate any infinite-dimensional map with uniform guarantees.

Introduces infinite-dimensional differential geometry using Bastiani calculus.

Universal approximation theorem for differentiable maps on infinite-dimensional manifolds

Diffeologies unify infinite-dimensional geometry and PDEs, enhancing classical function spaces.

In this paper, we study the Lévy-Milman concentration phenomenon of 1-Lipschitz maps into infinite dimensional metric spaces. Our main theorem asserts that the concentration to an infinite dimensional $\ell^p$-ball with the $\ell^q$-distance function for $1\leq p<q\leq +\infty$ is equivalent to the concentration to the…

Research covers geometry, analysis, and integration on infinite-dimensional spaces.

Generalized Stacey-Roberts lemma for Banach manifolds.

Theory of symplectic reduction in infinite dimensions developed.

The paper proves Sard's theorem for polynomial maps in infinite dimensions.

We define a Toledo number for actions of surface groups and complex hyperbolic lattices on infinite dimensional Hermitian symmetric spaces, which allows us to define maximal representations. When the target is not of tube type we show that there cannot be Zariski-dense maximal representations, and whenever the existenc…

Generalizes neural network approximation to infinite-dimensional manifolds and derivatives.

The abstract discusses transversality for infinite dimensional manifolds.

This paper explores topological aspects of index theory for infinite-dimensional manifolds.

We study mapping class groups of infinite type surfaces with isolated punctures and their actions on the loop graphs introduced by Bavard-Walker. We classify all of the mapping classes in these actions which are loxodromic with a WWPD action on the corresponding loop graph. The WWPD property is a weakening of Bestvina-…

Generalizes neural networks for infinite-dimensional mappings, including PDE solutions.

The paper shows how to use fine shape to understand infinite-dimensional spaces.

Smooth structures on infinite dimensional Grassmannians and non-commutative cross-ratios.

Using symplectic techniques and spectral analysis of smooth paths of self-adjoint operators, we characterize the set of conjugate instants along a geodesic in an infinite dimensional Riemannian Hilbert manifold.

Book on infinite-dimensional Lie groups, covering basics and various classes.

Study shows how Poisson brackets factor on infinite dimensional manifolds.

New infinite-type loxodromic elements found in surface mapping classes.

We describe the exponential map from an infinite-dimensional Lie algebra to an infinite-dimensional group of operators on a Hilbert space. Notions of differential geometry are introduced for these groups. In particular, the Ricci curvature, which is understood as the limit of the Ricci curvature of finite-dimensional g…

We define submersions f between manifolds M and N modelled on locally convex spaces. If the range N is finite-dimensional or a Banach manifold, then these coincide with the naive notion of a submersion. We study pre-images of submanifolds under submersions and pre-images under mappings whose differentials have dense im…

Chern-Weil and Chern-Simons theory extend to certain infinite-rank bundles that appear in mathematical physics. We discuss what is known of the invariant theory of the corresponding infinite-dimensional Lie groups. We use these techniques to detect cohomology classes for spaces of maps between manifolds and for diffeom…

We analyze the signature type of a cascade of periodic orbits associated to period doubling renormalizable maps of the two dimensional disk. The signature is a sequence of rational numbers which describes how periodic orbits turn each other and is invariant by topological conjugacies that preserve orientation. We prove…

This paper concerns a study of three families of non-compact type symmetric spaces of infinite dimension. Although they have infinite dimension they have finite rank. More precisely, we show they have finite telescopic dimension. We also show the existence of Furstenberg maps for some group actions on these spaces. Suc…

Develops theory for conditional optimal transport in infinite-dimensional spaces.

For a compact manifold M and a differentiable stack \cX presented by a Lie groupoid X, we show the Hom-stack Hom(M,\cX) is presented by a Fréchet-Lie groupoid Map(M,X) and so is an infinite-dimensional differentiable stack. We further show that if \cX is an orbifold, presented by a proper étale Lie groupoid, then Map(M…

Enhances conformal geometry in higher dimensions with infinite-dimensional algebra.

Develops neural network approximations for infinite-dimensional input-output maps.

We prove geometric superrigidity for actions of cocompact lattices in semisimple Lie groups of higher rank on infinite dimensional Riemannian manifolds of nonpositive curvature and finite telescopic dimension.

Unified method for CNNs to approximate equivariant maps across various groups.

Symplectic method solves infinite-dimensional Schrödinger equations.

We show that there are homotopy equivalences $h:N\to M$ between closed manifolds which are induced by cell-like maps $p:N\to X$ and $q:M\to X$ but which are not homotopic to homeomorphisms. The phenomenon is based on construction of cell-like maps that kill certain $\mathbb L$-classes. The image space in these construc…

Normal forms for equivariant maps in infinite dimensions established.

Study shows infinite order in mapping class groups for certain 3D shapes.

Local gluing connects flow lines in finite time intervals.

We prove a Frobenius theorem for Banach distributions on manifolds that are modelled over locally convex spaces. Moreover, we recall how Frobenius theorems can be applied to infinite-dimensional Lie groups and obtain, that given a Lie subalgebra of the Lie algebra of a Lie group that is modelled over a locally convex s…

Mapping spaces of supermanifolds are usually thought as exclusively in functorial terms (i.e. trough the Grothendieck functor of points). In this work we provide a geometric description of such mapping spaces in terms of infinite-dimensional super-vector bundles.

Optimal transport for functional data using Hilbert-Schmidt operators.

We have been studying the index theory for some special infinite-dimensional manifolds with a "proper cocompact" actions of the loop group LT of the circle T, from the viewpoint of the noncommutative geometry. In this paper, we will introduce the LT-equivariant KK-theory and we will construct three KK-elements: the ind…

Geometric approach solves Euler equations with random forces.

We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic theory of manifolds and Lie groups is developed. Special attention is paid to the case of mappings between topological vector spaces ov…

It is shown that every abelian regular Lie group is a quotient of its Lie algebra via the exponential mapping.

Study the Lax equation in infinite-dimensional Lie algebras and Lie groups.

This is an overview article. In his Habilitationsvortrag, Riemann described infinite dimensional manifolds parameterizing functions and shapes of solids. This is taken as an excuse to describe convenient calculus in infinite dimensions which allows for short and transparent proofs of the main facts of the theory of man…