Modified Laplacian connects to Yang-Mills instantons on manifolds.
problem Understanding instantons on 4D manifolds.
method Infinite dimensional Lévy Laplacian defined on manifolds, parameterized by curves in orthogonal rotations.
result Instantons on 4D manifolds are related to the modified Lévy Laplacian under specific curve conditions.
Develops trace class operators and inverse Laplacian theory for infinite dimensions.
problem Understanding trace class operators and inverse Laplacian on infinite dimensional spaces.
method Presentation of trace class operators and construction of inverse Laplacian on closed manifolds.
result Original trace computations involving the inverse Laplacian on the torus.
We consider the Harnack inequality for harmonic functions with respect to three types of infinite dimensional operators. For the infinite dimensional Laplacian, we show no Harnack inequality is possible. We also show that the Harnack inequality fails for a large class of Ornstein-Uhlenbeck processes, although functions…
The Levy Laplacian is studied on manifolds, with heat flow solutions tending to constant functionals over time.
problem Understanding the Levy Laplacian on manifolds and its heat flow behavior.
method Various definitions of the Levy Laplacian are proven equivalent. Heat flows of differential forms are used to construct solutions.
result Solutions of the heat equation with the Levy Laplacian tend to locally constant functionals over time.
Study of Yang-Mills fields on 4-manifolds using modified Lévy Laplacians.
problem Connection between Yang-Mills fields and modified Lévy Laplacians on 4-manifolds.
method Analysis of modified Lévy Laplacians and their relation to Yang-Mills equations under nontrivial holonomy groups.
result Existence of a modified Lévy Laplacian related to Yang-Mills self-duality equations.
We show that, on any asymptotically hyperbolic surface, the essential spectrum of the Lichnerowicz Laplacian ΔL contains the ray [1/4,+∞[. If moreover the scalar curvature is constant then -2 and 0 are infinite dimensional eigenvalues. If, in addition, the inequality <Δu,u>L2≥41∣∣u∣∣L22…
We show that the resolvent of the Laplacian on SL(3,R)/SO(3) can be lifted to a meromorphic function on a Riemann surface which is a branched covering of C. The poles of this function are called the resonances of the Laplacian. We determine all resonances and show that the corresponding residue op…
The paper proposes a Gaussian mixture model for Hilbert-space-valued data.
problem Challenges in characterizing probability measures for infinite-dimensional random objects.
method Gaussian mixture framework based on kernel mean embeddings.
result The proposed algorithm yields a dense class of approximations in infinite-dimensional spaces.
We consider the Hodge Laplacian on manifolds with incomplete edge singularities, with infinite dimensional von Neumann spaces and intricate elliptic boundary value theory. We single out a class of its algebraic self-adjoint extensions. Our microlocal heat kernel construction for algebraic boundary conditions is guided …
Koopman operator theory simplifies complex systems analysis.
problem Analyzing nonlinear dynamical systems and complex networks.
method Estimating Koopman operator from data to reveal system properties.
result Koopman operators provide insights into system characteristics.
Consider the sum of the first N eigenspaces for the Laplacian on a Riemannian manifold. A basis for this space determines a map to Euclidean space and for N sufficiently large the map is an embedding. In analogy with a fruitful idea of Kähler geometry, we define (Riemannian) Bergman metrics of degree N to be thos…
Generalizes Li-Yau Harnack inequality to path space of manifolds.
problem Extending classical Harnack inequalities to infinite-dimensional path space.
method Defines finite-dimensional gradients and Laplacians on path space, proving a generalized Harnack inequality.
result Established a new Harnack inequality on path space of manifolds.
Geometric framework for Milnor classifying spaces in diffeological spaces.
problem Milnor classifying spaces in diffeological spaces.
method Developed spherical and projective models with natural diffeological structures, constructed Riemannian metrics, defined differential forms, and introduced Clifford structures.
result Established a coherent geometric setting combining classifying spaces, diffeology, and higher geometric structures.
We consider a rigidity problem for the spectral gap of the Laplacian on an RCD(K,∞)-space (a metric measure space satisfying the Riemannian curvature-dimension condition) for positive K. For a weighted Riemannian manifold, Cheng--Zhou showed that the sharp spectral gap is achieved only when a 1-dimensional G…
Most Finsler metrics have infinite-dimensional holonomy groups.
problem Understanding the holonomy groups of Finsler metrics.
method Analyzing the set of Finsler metrics on a manifold.
result An open dense subset of Finsler metrics have infinite-dimensional holonomy groups.
Derives PF-ODE for infinite-dimensional functions, improving function generation tasks.
problem Efficient inference in infinite-dimensional diffusion models.
method Derives PF-ODE in infinite-dimensional function spaces.
result Reduces function evaluations while maintaining sample quality.
Study on infinite-dimensional Heisenberg groups using hypoelliptic heat kernels.
problem Properties of hypoelliptic heat kernels on infinite-dimensional reduced Heisenberg groups.
method Construction and study of hypoelliptic heat kernels on infinite-dimensional reduced Heisenberg groups.
result Hypoelliptic logarithmic Sobolev inequalities on the space.
Complete, conformally flat metrics of constant positive scalar curvature on the complement of k points in the n-sphere, k≥2, n≥3, were constructed by R\. Schoen [S2]. We consider the problem of determining the moduli space of all such metrics. All such metrics are asymptotically periodic, and we develop…
The Leibniz rule for derivations is invariant under cyclic permutations of co-multiples within the arguments of derivations. We explore the implications of this principle: in effect, we construct a class of noncommutative bundles in which the sheaves of algebras of walks along a tesselated affine manifold form the base…
Smooth structures on infinite dimensional Grassmannians and non-commutative cross-ratios.
problem Smooth structures on infinite dimensional Grassmannians and non-commutative cross-ratios.
method Analyzing and expanding the notion of non-commutative cross-ratios, proving their smoothness.
result Smoothness of non-commutative cross-ratios.
Proves uniqueness of embedding complex manifold into infinite-dimensional space.
problem Balanced embedding of non-compact complex manifold into infinite-dimensional projective space.
method Fine estimates of asymptotics of a balanced embedding.
result Uniqueness of embedding proven.
The paper extends Cartan development to infinite dimensional Lie groups.
problem Generalizing Cartan development to infinite dimensional Lie groups.
method Generalization of Cartan development to infinite dimensional manifolds and Lie groups.
result The tangent mapping of a Cartan development is another Cartan development.
The paper explores infinite-dimensional nonholonomic and vakonomic systems.
problem Understanding dynamics of infinite-dimensional systems with constraints.
method Visualizing and revisiting classical and new examples of nonholonomic and vakonomic systems.
result Infinite-dimensional systems exhibit both nonholonomic and vakonomic dynamics.
We construct a infinite-dimensional manifold structure adapted to analytic Lie pseudogroups of infinite type. More precisely, we prove that any isotropy subgroup of an analytic Lie pseudogroup of infinite type is a regular infinite-dimensional Lie group, modelled on a locally convex strict inductive limit of Banach spa…
Paper investigates optimal transport map estimation in infinite-dimensional spaces.
problem Estimating optimal transport maps in infinite-dimensional spaces is challenging.
method Characterizes γ-smoothness for optimal transport maps and develops a polynomial-rate estimator. result Shows polynomial-order minimax risk for optimal transport map estimation.
Introduces infinite-dimensional differential geometry using Bastiani calculus.
problem Calculus breakdown in infinite-dimensional settings.
method Uses Bastiani calculus for directional derivatives.
result Develops and connects infinite-dimensional Lie groups and weak Riemannian geometry.
We extend rectified flow to infinite-dimensional Hilbert space.
problem Extending rectified flow to infinite-dimensional spaces.
method Established a rigorous functional formulation using the superposition principle for continuity equations.
result Demonstrated superior performance compared to existing models.
Constructs infinite-dimensional Siegel disc as symplectic and Kaehler quotient.
problem No specific problem stated; focuses on mathematical construction.
method Symplectic and Kaehler quotient construction.
result Infinite-dimensional Siegel disc constructed as symplectic and Kaehler quotient.
New non-trivial Kaehler-Ricci solitons found in infinite dimensional complex space forms.
problem Non-trivial Kaehler-Ricci solitons in infinite dimensional complex space forms.
method Exhibited families of non trivial radial Kaehler-Ricci solitons in infinite dimensional complex space forms.
result Non-trivial Kaehler-Ricci solitons exist in infinite dimensional complex space forms, contradicting previous finite dimensional results.
This research develops approximation theory for OOMs of infinite-dimensional processes.
problem Developing an approximation theory for OOMs of infinite-dimensional processes.
method Establishing an inner product structure and proving continuity of observable operators.
result A fundamental obstacle in making an infinite-dimensional space of future distributions into a Hilbert space is described.
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…
STARK improves denoising of low-depth spatial transcriptomics images.
problem Denoising spatial transcriptomics images at ultra-low sequencing depths.
method Adaptive regularization with kernel ridge regression and graph Laplacian.
result STARK optimizes denoising performance over competing methods.
This paper explores topological aspects of index theory for infinite-dimensional manifolds.
problem Formulating index theory for infinite-dimensional manifolds with LT-actions.
method Introducing RKK-theory and constructing assembly maps for proper LT-spaces.
result Formulation of infinite-dimensional Poincaré duality and assembly maps.
Infinite-dimensional SBDMs improve image generation across multiple resolutions.
problem Efficient image generation at high resolutions and across different levels.
method Developed SBDMs in infinite-dimensional setting, using trace class operators and operator networks.
result Improved efficiency and generalization across resolution levels.
Adjoint sampler targets infinite-dimensional function spaces for efficient sampling.
problem Limited theory and algorithms for sampling infinite-dimensional function spaces.
method Adjoint Sampler for infinite-dimensional function spaces based on stochastic maximum principle.
result FAS achieves superior performance in synthetic and real systems.
New calculus solves boundary value problems for elliptic operators.
problem Boundary value problems for 0-elliptic operators.
method Developed a new calculus called symbolic 0-calculus to handle boundary value problems.
result Construct left and right parametrices for 0-elliptic operators with boundary conditions.
Study shows how Poisson brackets factor on infinite dimensional manifolds.
problem Understanding Poisson brackets on infinite dimensional manifolds.
method Analyzes Poisson brackets on smoothly paracompact manifolds with specific properties.
result Dual map of a Poisson bracket factors as a smooth section of a vector bundle.
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 control and observer methods for complex systems.
problem Controlling and observing infinite-dimensional systems with boundary actuation.
method Energy-Casimir method and port-Hamiltonian system representation.
result Control law and observer designed for Kirchhoff-Love plate example.
The study defines divergence for multivector fields on infinite-dimensional manifolds.
problem Defining divergence for multivector fields on infinite-dimensional manifolds.
method Definition of divergence consistent with finite-dimensional geometry, properties transferred from finite to infinite dimensions.
result Natural properties of divergence are preserved in infinite dimensions.
Symplectic method solves infinite-dimensional Schrödinger equations.
problem Solving Schrödinger equations on infinite-dimensional Hilbert spaces with unbounded Hamiltonians.
method Analytic vectors, manifolds modelled on normed spaces, symplectic differential geometry, Marsden--Weinstein reduction.
result Mapped t-dependent Schrödinger equations onto projective spaces. Research covers geometry, analysis, and integration on infinite-dimensional spaces.
problem Exploring geometric and analytical structures in infinite-dimensional settings.
method Analyzes numerical schemes, Lie groups, connections, and integration theory.
result Developed new methods for integration and analysis on infinite-dimensional manifolds.
New infinite-dimensional representations with bounded multiplicity found for Lie groups.
problem Finding representations with bounded multiplicity for Lie groups.
method Proving existence of infinite-dimensional irreducible representations with bounded multiplicity property.
result Infinite-dimensional irreducible representations with bounded multiplicity found for non-compact semisimple Lie groups.
The theory of product preserving functors and Weil functors is partly extended to infinite dimensional manifolds, using the theory of C∞-algebras.
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 ℓp-ball with the ℓq-distance function for 1≤p<q≤+∞ is equivalent to the concentration to the…
We formulate the Riemannian calculus of the probability set embedded with L2-Wasserstein metric. This is an initial work of transport information geometry. Our investigation starts with the probability simplex (probability manifold) supported on vertices of a finite graph. The main idea is to embed the probability m…
Infinite-dimensional diffusion models tackle generative tasks for complex data.
problem Challenges in applying diffusion models to infinite-dimensional data.
method Directly formulate diffusion-based generative models in infinite dimensions.
result Developed guidelines for designing infinite-dimensional diffusion models.
Diffeologies unify infinite-dimensional geometry and PDEs, enhancing classical function spaces.
problem Combining infinite-dimensional geometry and PDEs for optimization problems.
method Review and extension of classical function spaces and mapping spaces.
result Diffeologies provide a unified framework for evolution equations and optimization problems.