Sub-Riemannian geometry is the geometry of a world with nonholonomic constraints. In such a world, one can move, send and receive information only in certain admissible directions but eventually can reach every position from any other. In the last two decades sub-Riemannian geometry has emerged as an independent research domain impacting on several areas of pure and applied mathematics, with applications to many areas such as quantum control, Hamiltonian dynamics, robotics and Lie theory. This comprehensive introduction proceeds from classical topics to cutting-edge theory and applications, assuming only standard knowledge of calculus, linear algebra and differential equations. The book may serve as a basis for an introductory course in Riemannian geometry or an advanced course in sub-Riemannian geometry, covering elements of Hamiltonian dynamics, integrable systems and Lie theory. It will also be a valuable reference source for researchers in various disciplines.

https://hal.archives-ouvertes.fr/hal-02019181/document

A Comprehensive Introduction to Sub-Riemannian Geometry

This article surveys the usual techniques of nonlinear optimal control such as the Pontryagin Maximum Principle and the conjugate point theory, and how they can be implemented numerically, with a special focus on applications to aerospace problems. In practice the knowledge resulting from the maximum principle is often insufficient for solving the problem, in particular because of the well-known problem of initializing adequately the shooting method. In this survey article it is explained how the usual tools of optimal control can be combined with other mathematical techniques to improve significantly their performances and widen their domain of application. The focus is put onto three important issues. The first is geometric optimal control, which is a theory that has emerged in the 1980s and is combining optimal control with various concepts of differential geometry, the ultimate objective being to derive optimal synthesis results for general classes of control systems. Its applicability and relevance is demonstrated on the problem of atmospheric reentry of a space shuttle. The second is the powerful continuation or homotopy method, consisting of deforming continuously a problem toward a simpler one and then of solving a series of parameterized problems to end up with the solution of the initial problem. After having recalled its mathematical foundations, it is shown how to combine successfully this method with the shooting method on several aerospace problems such as the orbit transfer problem. The third one consists of concepts of dynamical system theory, providing evidence of nice properties of the celestial dynamics that are of great interest for future mission design such as low-cost interplanetary space missions. The article ends with open problems and perspectives.

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Optimal Control and Applications to Aerospace: Some Results and Challenges

We give a complete classification of left-invariant sub-Riemannian structures on three-dimensional Lie groups in terms of the basic differential invariants. As a consequence, we explicitly find a sub-Riemannian isometry between the nonisomorphic Lie groups SL(2) and A +( $ \mathbb{R} $ )?×?S 1, where A +( $ \mathbb{R} $ ) denotes the group of orientation preserving affine maps on the real line.

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Sub-Riemannian structures on 3D lie groups

Neural network approximation: Three hidden layers are enough.

Deep learning has arguably achieved tremendous success in recent years. In simple words, deep learning uses the composition of many nonlinear functions to model the complex dependency between input features and labels. While neural networks have a long history, recent advances have greatly improved their performance in computer vision, natural language processing, etc. From the statistical and scientific perspective, it is natural to ask: What is deep learning? What are the new characteristics of deep learning, compared with classical methods? What are the theoretical foundations of deep learning? To answer these questions, we introduce common neural network models (e.g., convolutional neural nets, recurrent neural nets, generative adversarial nets) and training techniques (e.g., stochastic gradient descent, dropout, batch normalization) from a statistical point of view. Along the way, we highlight new characteristics of deep learning (including depth and over-parametrization) and explain their practical and theoretical benefits. We also sample recent results on theories of deep learning, many of which are only suggestive. While a complete understanding of deep learning remains elusive, we hope that our perspectives and discussions serve as a stimulus for new statistical research.

A Selective Overview of Deep Learning

In a number of previous papers of the first and third authors, caustics, cut-loci, spheres, and wave fronts of a system of sub-Riemannian geodesics emanating from a point q_{0} were studied. It turns out that only certain special arrangements of classical Lagrangian and Legendrian singularities occur outside q_0. As a consequence of this, for instance, the generic caustic is a globally stable object outside the origin q_{0}. Here we solve two remaining stability problems. The first part of the paper shows that in fact generic caustics have moduli at the origin, and the first module that occurs has a simple geometric interpretation. On the contrary, the second part of the paper shows a stability result at q_{0}. We define the “big wave front”: it is the graph of the multivalued function arclength \rightarrow wave-front reparametrized in a certain way. This object is a three-dimensional surface that also has a natural structure of the wave front. The projection of the singular set of this “big wave front” on the 3-dimensional space is nothing else but the caustic. We show that in fact this big wave front is Legendre-stable at the origin.

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On Sub-Riemannian Caustics and Wave Fronts for Contact Distributions in the Three-Space

On the representation of functions of several variables as a superposition of functions of a smaller number of variables

Geometry and topology of caustics - Caustics '06

On the representation of functions of two variables in the form ?[?(x) + ?(y)].- On functions of three variables.- The mathematics workshop for schools at Moscow State University.- The school mathematics circle at Moscow State University: harmonic functions.- On the representation of functions of several variables as a superposition of functions of a smaller number of variables.- Representation of continuous functions of three variables by the superposition of continuous functions of two variables.- Some questions of approximation and representation of functions.- Kolmogorov seminar on selected questions of analysis.- On analytic maps of the circle onto itself.- Small denominators. I. Mapping of the circumference onto itself.- The stability of the equilibrium position of a Hamiltonian system of ordinary differential equations in the general elliptic case.- Generation of almost periodic motion from a family of periodic motions.- Some remarks on flows of line elements and frames.- A test for nomographic representability using Decartes' rectilinear abacus.- Remarks on winding numbers.- On the behavior of an adiabatic invariant under slow periodic variation of the Hamiltonian.- Small perturbations of the automorphisms of the torus.- The classical theory of perturbations and the problem of stability of planetary systems.- Letter to the editor.- Dynamical systems and group representations at the Stockholm Mathematics Congress.- Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian.- Small denominators and stability problems in classical and celestial mechanics.- Small denominators and problems of stability of motion in classical and celestial mechanics.- Uniform distribution of points on a sphere and some ergodic properties of solutions of linear ordinary differential equations in a complex region.- On a theorem of Liouville concerning integrable problems of dynamics.- Instability of dynamical systems with several degrees of freedom.- On the instability of dynamical systems with several degrees of freedom.- Errata to V.I. Arnol'd's paper: "Small denominators. I.".- Small denominators and the problem of stability in classical and celestial mechanics.- Stability and instability in classical mechanics.- Conditions for the applicability, and estimate of the error, of an averaging method for systems which pass through states of resonance in the course of their evolution.- On a topological property of globally canonical maps in classical mechanics.

Representations of functions, celestial mechanics and KAM theory, 1957-1965

We study the existence of a decomposition of a given germ of 1-form on an affine space as a linear combination with positive coefficients of the differentials of concave functions. The number of entries should be minimal. Necessary and sufficient conditions upon the form germ with regular degeneracy to have such a disaggregation are found. This problem originates from recent papers of P.A. Chiappori and I. Ekeland on the mathematical economy.

Vladimir M. Zakalyukin

Papers

On Sub-Riemannian Caustics and Wave Fronts for Contact Distributions in the Three-Space

On the representation of functions of several variables as a superposition of functions of a smaller number of variables

Geometry and topology of caustics - Caustics '06

Representations of functions, celestial mechanics and KAM theory, 1957-1965

Concave Darboux Theorem