Manifold Calculus Research Articles

Koszul self‐duality of manifolds

AbstractWe show that Koszul duality for operads in can be expressed via generalized Thom complexes. As an application, we prove the Koszul self‐duality of the right module associated to a framed manifold . We discuss implications for factorization homology, embedding calculus, and confirm an old conjecture of Ching on the relation of Goodwillie calculus to manifold calculus.

Modeling, Simulation and Control of a Spacecraft: Automated Rendezvous under Positional Constraints

The aim of this research paper is to propose a framework to model, simulate and control the motion of a small spacecraft in the proximity of a space station. In particular, rendezvous in the presence of physical obstacles is tackled by a virtual potential theory within a modern manifold calculus setting and simulated numerically. The roto-translational motion of a spacecraft as well as the control fields are entirely formulated through a coordinate-free Lie group-type formalism. Likewise, the proposed control strategies are expressed in a coordinate-free setting through structured control fields. Several numerical simulations guide the reader through an evaluation of the most convenient control strategy among those devised in the present work.

Manifold Calculus in System Theory and Control—Second Order Structures and Systems

The present tutorial paper constitutes the second of a series of tutorials on manifold calculus with applications in system theory and control. The aim of the present tutorial, in particular, is to explain and illustrate some key concepts in manifold calculus such as covariant derivation and manifold curvature. Such key concepts are then applied to the formulation, to the control, and to the analysis of non-linear dynamical systems whose state-space are smooth (Riemannian) manifolds. The main flow of exposition is enriched by a number of examples whose aim is to clarify the notation used and the main theoretical findings through practical calculations.

Delooping the functor calculus tower

We study a connection between mapping spaces of bimodules and of infinitesimal bimodules over an operad. As main application and motivation of our work, we produce an explicit delooping of the manifold calculus tower associated to the space of smooth maps D m → D n $D^m\rightarrow D^n$ of discs, n ⩾ m $n\geqslant m$ , avoiding any given multisingularity and coinciding with the standard inclusion near the boundary ∂ D m $\partial D^m$ . In particular, we give a new proof of the delooping of the space of disc embeddings in terms of little discs operads maps with the advantage that it can be applied to more general mapping spaces. As a spin-off result, we discover a homotopy recurrence relation on the components of the little discs operads.

Non‐geometric rough paths on manifolds

We provide a theory of manifold-valued rough paths of bounded 3 > p $3 > p$ -variation, which we do not assume to be geometric. Rough paths are defined in charts, relying on the vector space-valued theory of Friz and Hairer (A course on rough paths, 2014), and coordinate-free (but connection-dependent) definitions of the rough integral of cotangent bundle-valued controlled paths, and of rough differential equations driven by a rough path valued in another manifold, are given. When the path is the realisation of semimartingale, we recover the theory of Itô integration and stochastic differential equations on manifolds (Émery, Stochastic calculus in manifolds, 1989). We proceed to present the extrinsic counterparts to our local formulae, and show how these extend the work in Cass et al. (Proc. Lond. Math. Soc. (3) 111 (2015) 1471–1518) to the setting of non-geometric rough paths and controlled integrands more general than 1-forms. In the last section, we turn to parallel transport and Cartan development: the lack of geometricity leads us to make the choice of a connection on the tangent bundle of the manifold T M $TM$ , which figures in an Itô correction term in the parallelism rough differential equation; such connection, which is not needed in the geometric/Stratonovich setting, is required to satisfy properties which guarantee well-definedness, linearity, and optionally isometricity of parallel transport. We conclude by providing a few examples that explore the additional subtleties introduced by our change in perspective.

Virtual Attractive-Repulsive Potentials Control Theory: A Review and an Extension to Riemannian Manifolds

The present paper is concerned with an instance of automatic control for autonomous vehicles based on the theory of virtual attractive-repulsive potentials (VARP). The first part of this paper presents a review of the VARP control theory as developed specifically by B. Nguyen, Y.-L. Chuang, D. Tung, C. Hsieh, Z. Jin, L. Shi, D. Marthaler, A. Bertozzi and R. Murray, in the paper ‘Virtual attractive-repulsive potentials for cooperative control of second order dynamic vehicles on the Caltech MVWT’, which appeared in the Proceedings of the 2005 American Control Conference, (Portland, OR, USA) held in June 2005 (pp. 1084–1089). The aim of the first part of the present paper is to recall the mathematical and logical steps that lead to controlling an autonomous robot by a VARP-based control theory. The concepts recalled in the first part of the present paper, with special reference to the physical interpretation of the terms in the developed control field, serve as the starting point to develop a more convoluted control theory for (second-order) dynamical systems whose state spaces are (possibly high-dimensional) curved manifolds. The second part of this paper is, in fact, devoted to extending the classical VARP control theory to regulate dynamical systems whose state spaces possess the mathematical structure of smooth manifolds through manifold calculus. Manifold-type state spaces present a high degree of symmetry, due to mutual non-linear constraints between single physical variables. A comprehensive set of numerical experiments complements the review of the VARP theory and the theoretical developments towards its extension to smooth manifolds.

Manifold Calculus in System Theory and Control—Fundamentals and First-Order Systems

The aim of the present tutorial paper is to recall notions from manifold calculus and to illustrate how these tools prove useful in describing system-theoretic properties. Special emphasis is put on embedded manifold calculus (which is coordinate-free and relies on the embedding of a manifold into a larger ambient space). In addition, we also consider the control of non-linear systems whose states belong to curved manifolds. As a case study, synchronization of non-linear systems by feedback control on smooth manifolds (including Lie groups) is surveyed. Special emphasis is also put on numerical methods to simulate non-linear control systems on curved manifolds. The present tutorial is meant to cover a portion of the mentioned topics, such as first-order systems, but it does not cover topics such as covariant derivation and second-order dynamical systems, which will be covered in a subsequent tutorial paper.

Improvement and Assessment of a Blind Image Deblurring Algorithm Based on Independent Component Analysis

The aim of the present paper is to improve an existing blind image deblurring algorithm, based on an independent component learning paradigm, by manifold calculus. The original technique is based on an independent component analysis algorithm applied to a set of pseudo-images obtained by Gabor-filtering a blurred image and is based on an adapt-and-project paradigm. A comparison between the original technique and the improved method shows that independent component learning on the unit hypersphere by a Riemannian-gradient algorithm outperforms the adapt-and-project strategy. A comprehensive set of numerical tests evidenced the strengths and weaknesses of the discussed deblurring technique.

Gradient-based Learning Methods Extended to Smooth Manifolds Applied to Automated Clustering

Grassmann manifold based sparse spectral clustering is a classification technique that consists in learning a latent representation of data, formed by a subspace basis, which is sparse. In order to learn a latent representation, spectral clustering is formulated in terms of a loss minimization problem over a smooth manifold known as Grassmannian. Such minimization problem cannot be tackled by one of traditional gradient-based learning algorithms, which are only suitable to perform optimization in absence of constraints among parameters. It is, therefore, necessary to develop specific optimization/learning algorithms that are able to look for a local minimum of a loss function under smooth constraints in an efficient way. Such need calls for manifold optimization methods. In this paper, we extend classical gradient-based learning algorithms on at parameter spaces (from classical gradient descent to adaptive momentum) to curved spaces (smooth manifolds) by means of tools from manifold calculus. We compare clustering performances of these methods and known methods from the scientific literature. The obtained results confirm that the proposed learning algorithms prove lighter in computational complexity than existing ones without detriment in clustering efficacy.

An application of the h-principle to manifold calculus

Manifold calculus is a form of functor calculus that analyzes contravariant functors from some categories of manifolds to topological spaces by providing analytic approximations to them. In this paper, using the technique of the h-principle, we show that for a symplectic manifold N, the analytic approximation to the Lagrangian embeddings functor $$\mathrm {Emb}_{\mathrm {Lag}}(-,N)$$ is the totally real embeddings functor $$\mathrm {Emb}_{\mathrm {TR}}(-,N)$$. More generally, for subsets $${\mathcal {A}}$$ of the m-plane Grassmannian bundle $${{\,\mathrm{{Gr}}\,}}(m,TN)$$ for which the h-principle holds for $${\mathcal {A}}$$-directed embeddings, we prove the analyticity of the $${\mathcal {A}}$$-directed embeddings functor $${{\,\mathrm{Emb}\,}}_{{\mathcal {A}}}(-,N)$$.

Low stages of the Taylor tower for r-immersions

We study the beginning of the Taylor tower, supplied by manifold calculus of functors, for the space of r-immersions, which are immersions without r-fold self-intersections. We describe the first r layers of the tower and discuss the connectivities of the associated maps. We also prove several results about r-immersions that are of independent interest.

Occupants in simplicial complexes

Let $M$ be a smooth manifold and $K\subset M$ be a simplicial complex of codimension at least 3. Functor calculus methods lead to a homotopical formula of $M\setminus K$ in terms of spaces $M\setminus T$ where $T$ is a finite subset of $K$. This is a generalization of the author's previous work with Michael Weiss where the subset $K$ is assumed to be a smooth submanifold of $M$ and uses his generalization of manifold calculus adapted for simplicial complexes.

Very good homogeneous functors in manifold calculus

Let $M$ be a smooth manifold, and let $\mathcal {O}(M)$ be the poset of open subsets of $M$. Let $\mathcal{C} $ be a category that has a zero object and all small limits. A homogeneous functor (in the sense of manifold calculus) of degree $k$ from $\mathc

Manifold calculus adapted for simplicial complexes

We develop a generalization of manifold calculus in the sense of Goodwillie-Weiss where the manifold is replaced by a simplicial complex. We consider functors from the category of open subsets of a fixed simplical complex into the category of topological spaces and prove an analogue of the approximation theorem. Namely, under certain conditions such a functor can be approximated by a tower of (appropriately adapted) polynomial functors.

A streamlined proof of the convergence of the Taylor tower for embeddings in ${\mathbb R}^n$

Manifold calculus of functors has in recent years been successfully used in the study of the topology of various spaces of embeddings of one manifold in another. Given a space of embeddings, the theory produces a Taylor tower whose purpose is to approxima

Polynomial functors in manifold calculus

Let M be a smooth manifold, and let O(M) be the poset of open subsets of M. Manifold calculus, due to Goodwillie and Weiss, is a calculus of functors suitable for studying contravariant functors (cofunctors) F:O(M)⟶Spaces from O(M) to the category of spaces. Weiss showed that polynomial cofunctors of degree ≤k are determined by their values on Ok(M), where Ok(M) is the full subposet of O(M) whose objects are open subsets diffeomorphic to the disjoint union of at most k balls. Afterwards Pryor showed that one can replace Ok(M) by more general subposets and still recover the same notion of polynomial cofunctor. In this paper, we generalize these results to cofunctors from O(M) to any simplicial model category M. If Fk(M) stands for the unordered configuration space of k points in M, we also show that the category of homogeneous cofunctors O(M)⟶M of degree k is weakly equivalent to the category of linear cofunctors O(Fk(M))⟶M provided that M has a zero object. Using a new approach, we also show that if M is a general model category and F:Ok(M)⟶M is an isotopy cofunctor, then the homotopy right Kan extension of F along the inclusion Ok(M)↪O(M) is also an isotopy cofunctor.

Spaces of smooth embeddings and configuration categories

In the homotopical study of spaces of smooth embeddings, the functor calculus method (Goodwillie-Klein-Weiss manifold calculus) has opened up important connections to operad theory. Using this and a few simplifying observations, we arrive at an operadic description of the obstructions to deforming smooth immersions into smooth embeddings. We give an application which in some respects improves on recent results of Arone-Turchin and Dwyer-Hess concerning high-dimensional variants of spaces of long knots.

From maps between coloured operads to Swiss-Cheese algebras

In the present work, we extract pairs of topological spaces from maps between coloured operads. We prove that those pairs are weakly equivalent to explicit algebras over the one dimensional Swiss-Cheese operad 𝒮𝒞 1 . Thereafter, we show that the pair formed by the space of long embeddings and the manifold calculus limit of (l)-immersions from ℝ d to ℝ n is an 𝒮𝒞 d+1 -algebra.

EMBEDDINGS, NORMAL INVARIANTS AND FUNCTOR CALCULUS

This paper investigates the space of codimension zero embeddings of a Poincaré duality space in a disk. One of our main results exhibits a tower that interpolates from the space of Poincaré immersions to a certain space of “unlinked” Poincaré embeddings. The layers of this tower are described in terms of the coefficient spectra of the identity appearing in Goodwillie’s homotopy functor calculus. We also answer a question posed to us by Sylvain Cappell. The appendix proposes a conjectural relationship between our tower and the manifold calculus tower for the smooth embedding space.

Special Open Sets in Manifold Calculus

Embedding Calculus, as described by Weiss, is a calculus of functors, suitable for studying contravariant functors from the poset of open subsets of a smooth manifold $M$, denoted $\mathcal{O}(M)$, to a category of topological spaces (of which the functor $Emb(-,N)$ for some fixed manifold $N$ is a prime example). Polynomial functors of degree $k$ can be characterized by their restriction to $\mathcal{O}_k(M)$, the full subposet of $\mathcal{O}(M)$ consisting of open sets which are a disjoint union of at most $k$ components, each diffeomorphic to the open unit ball. In this work, we replace $\mathcal{O}_k(M)$ by more general subposets and see that we still recover the same notion of polynomial cofunctor.