arXiv research

Introduces infinite-dimensional differential geometry using Bastiani calculus.

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

Develops a new algebraic framework for differential geometry of infinite dimensional spaces.

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

Shape analysis and compuational anatomy both make use of sophisticated tools from infinite-dimensional differential manifolds and Riemannian geometry on spaces of functions. While comprehensive references for the mathematical foundations exist, it is sometimes difficult to gain an overview how differential geometry and…

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…

Diffeology extends differential geometry to complex spaces.

Develops derived differential geometry for supermanifolds.

Introduces a new geometric framework for non-perturbative BV-theory.

Differential equations are derived for a continuous limit of iterated Schwarzian reflection of analytic curves, and solutions are interpreted as geodesics in an infinite-dimensional symmetric space geometry.

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

Study fundamental groups of geometric transformation groups using loop spaces.

Generalized Stacey-Roberts lemma for Banach manifolds.

The geometry of an admissible Bäcklund transformation for an exterior differential system is described by an admissible Cartan connection for a geometric structure on a tower with infinite--dimensional skeleton which is the universal prolongation of a $|1|$--graded semi-simple Lie algebra.

This thesis introduces the notion of "relative gerbes" for smooth maps of manifolds, and discusses their differential geometry. The equivalence classes of relative gerbes are classified by the relative integral cohomology in degree three. Furthermore, by using the concept of relative gerbes, the pre-quantization of Lie…

Symplectic method solves infinite-dimensional Schrödinger equations.

Introduces a new geometric framework for probability distributions.

Using standard analysis only, we present an extension ${^\bullet\R}$ of the real field containing nilpotent infinitesimals. On the one hand we want to present a very simple setting to formalize infinitesimal methods in Differential Geometry, Analysis and Physics. On the other hand we want to show that these infinitesim…

Geometric framework for Milnor classifying spaces in diffeological spaces.

Introduces new geometric framework for probability densities on manifolds.

The central object of synthetic differential geometry is microlinear spaces. In our previous paper [Microlinearity in Frolicher spaces -beyond the regnant philosophy of manifolds-, International Journal of Pure and Applied Mathematics, 60 (2010), 15-24] we have emancipated microlinearity from within well-adapted models…

The paper studies geometric properties of Grassman manifolds within Euclidean spaces.

Study of generalized vector bundles and their geometric tools.

The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed in recent work (collaboration with H. Gloeckner and K.-H. Neeb), without any restriction on the dimension or on the cha…

The geometry of symmetric spaces, polar actions, isoparametric submanifolds and spherical buildings is governed by spherical Weyl groups and simple Lie groups. A natural generalization of semisimple Lie groups are affine Kac-Moody groups as they mirror their structure theory and have good explicitely known representati…

The relationship between minimal algebraic Kac-Moody groups and twin buildings is well known as is the relationship between formal completions in one direction and affine buildings. Nevertheless, as the completion of a Kac-Moody group in one direction destroys the opposite BN-pair, there exists no longer a twin buildin…

Infinite-dimensional contact geometry explored.

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 …

Develops derived differential geometry theory.

Universal approximation theorem for differentiable maps on infinite-dimensional manifolds

Newton's method solves variational problems on manifolds.

Paper tackles infinite-dimensional optimization and Bayesian learning for stochastic differential equations.

Generalization of the cross ratio to polarizations of linear finite and infinite-dimensional spaces (in particular to Sato Grassmannian) is given and explored. This cross ratio appears to be a cocycle of the canonical (tautalogical) bundle over the Grassmannian with coefficients in the sheaf of its endomorphisms. Opera…

This study describes the Fisher-Rao metric on Gaussian measures in infinite-dimensional spaces.

New mechanism for pure differential privacy on functional summaries using Laplace-like process.

If $X$ is a manifold then the set $C^\infty(X)$ of smooth functions $f:X\to\mathbb R$ is a $C^\infty$-ring, a rich algebraic structure with many operations. $C^\infty$-schemes are schemes over $C^\infty$-rings, a way of using Algebro-Geometric techniques in Differential Geometry. They include smooth manifolds, but also…

Riemannian stochastic gradient descent approximates a diffusion process called Riemannian stochastic modified flow.

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

This paper opens the series of articles supplemental to the series (hep-th/9405050,q-alg/9610026,q-alg/9611003,q-alg/9611019,funct-an/9611003), which also lies in lines of general ideology exposed in the review (mp_arc/96-477). The main purpose of the activity, which has its origin and motivation presumably in the auth…

Develops a mathematical framework for causal fermion systems in infinite dimensions.

Spray-invariant sets maintain geodesics on infinite-dimensional manifolds.

Endowing differentiable functions from a compact manifold to a Lie group with the pointwise group operations one obtains the so-called current groups and, as a special case, loop groups. These are prime examples of infinite-dimensional Lie groups modelled on locally convex spaces. In the present paper, we generalise th…

Some natural hidden symmetries in the Verma modules over the Virasoro algebra are constructed in terms of geometric quantization. Their differential geometric meaning is established and their expression via $q_R$-conformal symmetries in the Verma modules over the Lie algebra $sl(2,C)$ is found. The analysis and the unr…

Riemannian symmetric spaces are fundamental objects in finite dimensional differential geometry. An important problem is the construction of symmetric spaces for generalizations of simple Lie groups, especially their closest infinite dimensional analogues known as Kac-Moody groups. We solve this problem and construct a…

We show that the standard picture regarding the notion of stability of constant scalar curvature metrics in Kähler geometry described by S.K. Donaldson, which involves the geometry of infinite-dimensional groups and spaces, can be applied to the constant scalar curvature metrics in Sasaki geometry with only few modific…

The paper extends Cartan development to infinite dimensional Lie groups.

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…

We give a unified method for the general equivalence problem of extrinsic geometry, on the basis of our formulation of a general extrinsic geometry as that of an osculating map $\varphi\colon (M,\mathfrak f) \to L/L^0 \subset \operatorname{Flag}(V,φ)$ from a filtered manifold $(M,\mathfrak f)$ to a homogeneous space $L…