Smooth Manifold Research Articles

A groupoid approach to the Wodzicki residue and the Kontsevich–Vishik trace

Building on the work of Debord and Skandalis, van Erp and Yuncken introduced a groupoid approach to pseudo-differential operators which has various advantages over the classical approach using Hörmander’s symbolic calculus. In a recent work by Couchet and Yuncken, they showed that for pseudo-differential operators of order -\dim(M) acting on a smooth manifold M the Wodzicki residue can be naturally obtained from van Erp and Yuncken’s approach. In this short note, we extend their work to pseudo-differential operators of any order. We also describe of the Kontsevich–Vishik trace.

Tautological rings of fake quaternionic spaces

AbstractThe tautological ring $$R^*(M)$$ R ∗ ( M ) of a smooth manifold M is the ring of characteristic classes generated by the Miller–Morita–Mumford classes, and is often more accessible than the ring of all characteristic classes of smooth M-bundles. In this paper, we introduce a new method to obtain upper bounds on the Krull dimension of $$R^*(M)$$ R ∗ ( M ) for manifolds homotopy equivalent to a fixed, simply connected Poincaré duality space by using recent progress in rational homotopy theory and the family signature theorem. In particular, we show that the Krull dimension of the tautological ring vanishes for almost all manifolds homotopy equivalent to $$\mathbb {H}P^2$$ H P 2 .

Hermite interpolation with retractions on manifolds

AbstractInterpolation of data on non-Euclidean spaces is an active research area fostered by its numerous applications. This work considers the Hermite interpolation problem: finding a sufficiently smooth manifold curve that interpolates a collection of data points on a Riemannian manifold while matching a prescribed derivative at each point. A novel procedure relying on the general concept of retractions is proposed to solve this problem on a large class of manifolds, including those for which computing the Riemannian exponential or logarithmic maps is not straightforward, such as the manifold of fixed-rank matrices. The well-posedness of the method is analyzed by introducing and showing the existence of retraction-convex sets, a generalization of geodesically convex sets. A classical result on the asymptotic interpolation error of Hermite interpolation is extended to the manifold setting. Finally numerical experiments on the manifold of fixed-rank matrices and the Stiefel manifold of matrices with orthonormal columns illustrate these results and the effectiveness of the method.

Approximated harmonic maps with tension fields in Zygmund class

Suppose that u is a map from D8 to a compact smooth Riemannian manifold N with bounded energy. We show that there exists a constant λ>0 which depends only on N and E(u,D8) such that if the tension field τ belongs to Zygmund class Llnλ⁡L(D8), then the Hopf differential of u belongs to the Zygmund class Lln3⁡L(D1) and the norm ‖h‖Lln3⁡L(D1) depends only on N,E(u,D8) and ‖τ‖Llnλ⁡L(D8). As a direct corollary, we obtain the energy identity and necklessness of a blow-up sequence un with bounded energy E(un) and bounded τ(un) in Llnλ⁡L(D8).

Diffeomorphisms of Foliated Manifolds I

The set $Diff(M)$ of all diffeomorphisms of manifold $M$ onto itself is the group related to composition and inverse mapping. The group of diffeomorphisms of smooth manifolds is of great importance in differential geometry and analysis. It is known that the group $Diff(M)$ is topological group in compact open topology.In this paper we investigate the group $Diff_{F}(M)$ of diffeomorphisms foliated manifold $(M,F)$ with foliated compact open topology. In this paper we prove that if all leaves of the the foliation $F$ are closed subsets of $M$ then the foliated compact open topology of the group $Diff_{F}(M)$ coincides with compact open topology. In addition it is studied the question on the dimension of the group of isometries of foliated manifold is studied when foliation generated by riemannian submersion.

Higher Cheeger ratios of features in Laplace-Beltrami eigenfunctions

This paper investigates links between the eigenvalues and eigenfunctions of the Laplace-Beltrami operator, and the higher Cheeger constants of smooth Riemannian manifolds, possibly weighted and/or with boundary. The higher Cheeger constants give a loose description of the major geometric features of a manifold. We give a constructive upper bound on the higher Cheeger constants, in terms of the eigenvalue of any eigenfunction with the corresponding number of nodal domains. Specifically, we show that for each such eigenfunction, a positive-measure collection of its superlevel sets have their Cheeger ratios bounded above in terms of the corresponding eigenvalue.Some manifolds have their major features entwined across several eigenfunctions, and no single eigenfunction contains all the major features. In this case, there may exist carefully chosen linear combinations of the eigenfunctions, each with large values on a single feature, and small values elsewhere. We can then apply a soft-thresholding operator to these linear combinations to obtain new functions, each supported on a single feature. We show that the Cheeger ratios of the level sets of these functions also give an upper bound on the Laplace-Beltrami eigenvalues. We extend these level set results to nonautonomous dynamical systems, and show that the dynamic Laplacian eigenfunctions reveal sets with small dynamic Cheeger ratios.

Smooth connectivity in real algebraic varieties

AbstractA standard question in real algebraic geometry is to compute the number of connected components of a real algebraic variety in affine space. This manuscript provides algorithms for computing the number of connected components, the Euler characteristic, and deciding the connectivity between two points for a smooth manifold arising as the complement of a real hypersurface of a real algebraic variety. When considering the complement of the set of singular points of a real algebraic variety, this yields an approach for determining smooth connectivity in a real algebraic variety. The method is based upon gradient ascent/descent paths on the real algebraic variety inspired by a method proposed by Hong, Rohal, Safey El Din, and Schost for complements of real hypersurfaces. Several examples are included to demonstrate the approach.

Computing Euclidean distance and maximum likelihood retraction maps for constrained optimization

Riemannian optimization uses local methods to solve optimization problems whose constraint set is a smooth manifold. A linear step along some descent direction usually leaves the constraints, and hence retraction maps are used to approximate the exponential map and return to the manifold. For many common matrix manifolds, retraction maps are available, with more or less explicit formulas. For implicitly-defined manifolds, suitable retraction maps are difficult to compute. We therefore develop an algorithm which uses homotopy continuation to compute the Euclidean distance retraction for any implicitly-defined submanifold of Rn, and prove convergence results.We also consider statistical models as Riemannian submanifolds of the probability simplex with the Fisher metric. Replacing Euclidean distance with maximum likelihood results in a map which we prove is a retraction. In fact, we prove the retraction is second-order; with the Levi-Civita connection associated to the Fisher metric, it approximates geodesics to second-order accuracy.

Level sets of nonsmooth functions, Part 2: Lipschitz and piecewise-differentiable manifolds

This paper introduces piecewise-differentiable (PCr) manifolds according to a unified general framework that also applies to nonsmooth Lipschitz manifolds and smooth manifolds. We present definitions of nonsmooth manifolds and embedded submanifolds for abstract sets as well as for subsets of Rn, and explore the relationships between them. The PCr and Lipschitz Rank Theorems from Part 1 of this series, in terms of the Clarke Jacobian and B-subdifferential generalized derivative sets, are used to characterize level sets of functions between nonsmooth manifolds as embedded submanifolds. We illustrate how the Level Set Theorems developed in this paper can be applied to functions on Euclidean space, including a piecewise-differentiable process model for distillation columns.

Volume and Euler classes in bounded cohomology of transformation groups

Abstract Let $M$ be an oriented smooth manifold and $\operatorname{Homeo}\!(M,\omega )$ the group of measure preserving homeomorphisms of $M$ , where $\omega$ is a finite measure induced by a volume form. In this paper, we define volume and Euler classes in bounded cohomology of an infinite dimensional transformation group $\operatorname{Homeo}_0\!(M,\omega )$ and $\operatorname{Homeo}_+\!(M,\omega )$ , respectively, and in several cases prove their non-triviality. More precisely, we define: • Volume classes in $\operatorname{H}_b^n(\operatorname{Homeo}_0\!(M,\omega ))$ , where $M$ is a hyperbolic manifold of dimension $n$ . • Euler classes in $\operatorname{H}_b^2(\operatorname{Homeo}_+(S,\omega ))$ , where $S$ is an oriented closed hyperbolic surface. We show that Euler classes have positive norms for any closed hyperbolic surface and volume classes have positive norms for all hyperbolic surfaces and certain hyperbolic $3$ -manifolds; hence, they are non-trivial.

Some Aspects of Differential Topology of Subcartesian Spaces

In this paper, we investigate the differential topological properties of a large class of singular spaces: subcarteisan space. First, a minor further result on the partition of unity for differential spaces is derived. Second, the tubular neighborhood theorem for subcartesian spaces with constant structural dimensions is established. Third, the concept of Morse functions on smooth manifolds is generalized to differential spaces. For subcartesian space with constant structural dimension, a class of examples of Morse functions is provided. With the assumption that the subcartesian space can be embedded as a bounded subset of an Euclidean space, it is proved that any smooth bounded function on this space can be approximated by Morse functions. The infinitesimal stability of Morse functions on subcartesian spaces is studied. Classical results on Morse functions on smooth manifolds can be treated directly as corollaries of our results here.

Arc-smooth functions and cuspidality of sets

AbstractA function f is arc-smooth if the composite f ◦ c with every smooth curve c in its domain of definition is smooth. On open sets in smooth manifolds the arc-smooth functions are precisely the smooth functions by a classical theorem of Boman. Recently, we extended this result to certain tame closed sets (namely, Hölder sets and simple fat subanalytic sets). In this paper we link, in a precise way, the cuspidality of the (boundary of the) set to the loss of regularity, i.e., how many derivatives of f ◦ c are needed in order to determine the derivatives of f. We also discuss how flatness of f ◦ c affects flatness of f. Besides Hölder sets and subanalytic sets we treat sets that are definable in certain polynomially bounded o-minimal expansions of the real field.

Morph-SSL: Self-Supervision With Longitudinal Morphing for Forecasting AMD Progression From OCT Volumes.

The lack of reliable biomarkers makes predicting the conversion from intermediate to neovascular age-related macular degeneration (iAMD, nAMD) a challenging task. We develop a Deep Learning (DL) model to predict the future risk of conversion of an eye from iAMD to nAMD from its current OCT scan. Although eye clinics generate vast amounts of longitudinal OCT scans to monitor AMD progression, only a small subset can be manually labeled for supervised DL. To address this issue, we propose Morph-SSL, a novel Self-supervised Learning (SSL) method for longitudinal data. It uses pairs of unlabelled OCT scans from different visits and involves morphing the scan from the previous visit to the next. The Decoder predicts the transformation for morphing and ensures a smooth feature manifold that can generate intermediate scans between visits through linear interpolation. Next, the Morph-SSL trained features are input to a Classifier which is trained in a supervised manner to model the cumulative probability distribution of the time to conversion with a sigmoidal function. Morph-SSL was trained on unlabelled scans of 399 eyes (3570 visits). The Classifier was evaluated with a five-fold cross-validation on 2418 scans from 343 eyes with clinical labels of the conversion date. The Morph-SSL features achieved an AUC of 0.779 in predicting the conversion to nAMD within the next 6 months, outperforming the same network when trained end-to-end from scratch or pre-trained with popular SSL methods. Automated prediction of the future risk of nAMD onset can enable timely treatment and individualized AMD management.

Cohesive self-dual Yang-Mills modules over four manifolds

This research is motivated by Atiyah, Hitchin, and Singers paper Self-duality in Four-Dimensional Riemannian Geometry , which introduced a relationship between self-dual Yang-Mills fields on smooth manifolds and holomorphic vector bundles on their twistor spaces. Here, self-duality is a specific structure in 4-dimensional manifolds and Yang-Mills fields are gauge fields that satisfy Yang-Mills equations in 4-dimensions and are corresponded to the holomorphic bundles on twistor spaces. In this paper, we extend the relationship from vector bundles to a generalization of the Yang-Mills fields. To achieve this purpose, we apply Atiyah, Hitchin, and Singers theorem to cohesive modules, which was originally introduced by Block in in studying coherent sheaves over complex manifolds and the relations between homomorphic torus and its dual non-commutative torus. We introduce the notion of cohesive self-dual Yang-Mills modules and show that the twistor correspondence actually induces the equivalence between the dg category of cohesive self-dual Yang Mills modules P_(A_SD ) and the dg category of holomorphic cohesive modules P_(A_Hol ) on the twistor spaces.

Implications of data topology for deep generative models

Many deep generative models, such as variational autoencoders (VAEs) and generative adversarial networks (GANs), learn an immersion mapping from a standard normal distribution in a low-dimensional latent space into a higher-dimensional data space. As such, these mappings are only capable of producing simple data topologies, i.e., those equivalent to an immersion of Euclidean space. In this work, we demonstrate the limitations of such latent space generative models when trained on data distributions with non-trivial topologies. We do this by training these models on synthetic image datasets with known topologies (spheres, torii, etc.). We then show how this results in failures of both data generation as well as data interpolation. Next, we compare this behavior to two classes of deep generative models that in principle allow for more complex data topologies. First, we look at chart autoencoders (CAEs), which construct a smooth data manifold from multiple latent space chart mappings. Second, we explore score-based models, e.g., denoising diffusion probabilistic models, which estimate gradients of the data distribution without resorting to an explicit mapping to a latent space. Our results show that these models do demonstrate improved ability over latent space models in modeling data distributions with complex topologies, however, challenges still remain.

Canonical torsor bundles of prescribed rational functions on complex curves

The prescribed rational functions constitute a subset of rational functions satisfying certain symmetry and analyticity conditions. Over a smooth complex curve [Formula: see text], we construct explicitly a bundle [Formula: see text] with values in the prescribed rational functions. An intrinsic coordinate-independent formulation (the main result of the paper, Proposition 1) for such bundles is given. The construction presented in this paper is useful for studies of the canonical cosimplicial cohomology of infinite-dimensional Lie algebras on smooth manifolds, as well as for the purposes of conformal field theory and the theory of foliations.

Riemannian geometry for efficient analysis of protein dynamics data

An increasingly common viewpoint is that protein dynamics datasets reside in a nonlinear subspace of low conformational energy. Ideal data analysis tools should therefore account for such nonlinear geometry. The Riemannian geometry setting can be suitable for a variety of reasons. First, it comes with a rich mathematical structure to account for a wide range of geometries that can be modeled after an energy landscape. Second, many standard data analysis tools developed for data in Euclidean space can be generalized to Riemannian manifolds. In the context of protein dynamics, a conceptual challenge comes from the lack of guidelines for constructing a smooth Riemannian structure based on an energy landscape. In addition, computational feasibility in computing geodesics and related mappings poses a major challenge. This work considers these challenges. The first part of the paper develops a local approximation technique for computing geodesics and related mappings on Riemannian manifolds in a computationally feasible manner. The second part constructs a smooth manifold and a Riemannian structure that is based on an energy landscape for protein conformations. The resulting Riemannian geometry is tested on several data analysis tasks relevant for protein dynamics data. In particular, the geodesics with given start- and end-points approximately recover corresponding molecular dynamics trajectories for proteins that undergo relatively ordered transitions with medium-sized deformations. The Riemannian protein geometry also gives physically realistic summary statistics and retrieves the underlying dimension even for large-sized deformations within seconds on a laptop.

Representations of complex tori and GL(2, C)

Groups and their representations have been studied for a long time. One can extend the notion of a group by asking the group axioms to hold in other categories. A group in the category of smooth manifolds is a Lie group, and a group in the category of algebraic varieties is an algebraic group. In this paper, we discuss the representation theory of algebraic groups, in particular complex tori and GL(2,C).

High Excursion Probabilities for Gaussian Fields on Smooth Manifolds

High Excursion Probabilities for Gaussian Fields on Smooth Manifolds

Differential operators on C⁎-algebras and applications to smooth functional calculus and Schwartz functions on the tangent groupoid

We introduce the notion of a differential operator on C⁎-algebras. This is a noncommutative analogue of a differential operator on a smooth manifold. We show that the common closed domain of all differential operators is closed under smooth functional calculus. As a corollary, we show that Schwartz functions on Connes tangent groupoid are closed under smooth functional calculus.