The seminar covers both classical results (e.g., the Rashevsky-Chow theorem and the Pontryagin maximum principle) and new directions (Gromov's theorem on nilpotent approximation and the Nagano-Sussmann orbit theorem). Participants will gain a comprehensive understanding of the geometric and analytical tools underlying sub-Riemannian geometry and their applications.

Program
Below is an extended version of the program, which is unlikely to fit into a single semester. Therefore, topics 1–3 will be covered in the first semester. If there is interest among the participants, the seminar is planned to continue into the second semester with a program that will be a modification of items 4–6, tailored to the audience's interests.

1. Basics and Motivation
1.1 Nonholonomicity and Frobenius theorem [2].
1.2 Definition of a sub-Riemannian manifold [1].
1.3 Examples: almost Riemannian manifold, isoperimetric problems, and the Heisenberg group [1], [3].
1.4 Three equivalent definitions of distance (separate text).
1.5 ResNet neural networks [17].
1.6 The Nagano–Sussmann orbit theorem and the Rashevsky-Chow theorem [2].
1.7 Examples. Control of an N-level quantum system using 1- and 2-qubit transformations (separate text).
1.8 Sub-Finsler and sub-Lorentzian manifolds [11].

2. Geometric Control Theory Machinery

2.1 Necessary condition for optimality — the Pontryagin maximum principle [2], [3].
2.2 Hamiltonian formalism. Lagrangian formalism [2], [3].
2.3 Equations of shortest paths: normal and abnormal geodesics. Extremals on Heisenberg and Engel groups, Reeds-Shepp car [2].
2.4 Smoothness of normal geodesics. Exponential map [1]

3. Analysis of Extremals

3.1 Existence of shortest paths (Filippov's theorem) [8], [2].
3.2 Local optimality of strictly normal extremals and conjugate points. Example: Heisenberg group [2].
3.3 Liu-Sussmann example of a strictly abnormal shortest path [12].
3.4 Example of an abnormal extremal that is not optimal on any time subinterval [14].
3.5 The problem of smoothness of abnormal geodesics [14], [1], [18], [5], [10].
3.6 Hsu's differential inclusion [7].

4. Local Geometry and Asymptotic Analysis

4.1 Privileged coordinates [1], [4].
4.2 The Ball-Box theorem [1], [4].
4.3 Hausdorff dimension of a sub-Riemannian manifold. Hausdorff dimension of the Heisenberg group [1], [4].
4.4 Carnot groups and Finsler structures on them. Their Hausdorff dimension [11].
4.5 Tangent cone and Gromov's theorem on nilpotent approximation [4], [1].

5. Algebraic Structure and Integrability

5.1 Left-invariant problems on Lie groups. Trivialization of tangent and cotangent bundles [1].
5.2 Hamiltonian reduction and vertical subsystem. Example: Heisenberg group [1].
5.3 Example: control of nonholonomic mechanical systems, rolling a ball on a plane, Reeds-Shepp car [2].
5.4 Lie-Poisson bracket on the Lie coalgebra, coadjoint representation orbits, Casimirs [9], [19].
5.5 Left-invariant equations on Heisenberg, Engel, and Cartan groups, and on the group SO(3) [16].
5.6 Integrability of equations on all three-dimensional Lie groups, on Engel and Cartan groups. Integrability on a specific Carnot group with growth vector (2,3,5,6) [16], [13].
5.7 Examples: classification of equations on three-dimensional Lie groups, geodesic equations on all these groups, and their integration [1].

6. Miscellaneous

6.1 Pansu's theorem on the differentiability of mappings of Carnot groups [15].
6.2 Popp's volume [1].
6.3 Sub-Laplacians as hypoelliptic operators, Hörmander's condition [6].
6.4 Curvature in sub-Riemannian geometry [14], [1].

References
[1] Agrachev A. A., Barillari D., Boscain U. A Comprehensive Introduction to Sub-Riemannian Geometry. Cambridge University Press, 2019. (Cambridge Studies in Advanced Mathematics, Vol. 181).
[2] Аграчев А. А., Сачков Ю. Л. Геометрическая теория управления. М.: Физматлит, 2004.
[3] Алексеев В. М., Тихомиров В. М., Фомин С. В. Оптимальное управление. М.: Наука, 1979.
[4] Gromov M. Carnot-Carathéodory spaces seen from within. In: Sub-Riemannian geometry, Birkh¨auser, 1996. P. 79–323. (Progress in Mathematics, Vol. 144).
[5] Hakavuori E., Le Donne E. Non-minimality of corners in sub-riemannian geometry // Inventiones mathematicae, 206(3). 2016.
[6] Hörmander L. Hypoelliptic second order differential equations // Acta Mathematica. 1967. Vol. 119. P. 147–171.
[7] Hsu L. Calculus of Variations via the Griffiths formalism, Differential Geometry, 1992, 36, p. 551-589.
[8] Филиппов А. Ф. Дифференциальные уравнения с разрывной правой частью. М.: Наука, 1985. (English translation: Filippov A. F. Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Publishers, 1988).
[9] Кириллов А. А. Лекции по методу орбит. Н.: ИДМИ 2002.
[10] Lokutsievskiy L. V., Zelikin M.I., Derivatives of Sub-Riemannian Geodesics are Lp-Hölder Continuous, ESAIM: COCV, Volume 29, 2023.
[11] Le Donne E. Metric Lie Groups. Carnot-Carath´eodory Spaces from the Homogeneous Viewpoint.
[12] Liu W., Sussmann H. J. Shortest paths for sub-Riemannian metrics on rank-two distributions. AMS, v. 118, n. 564, 1995.
[13] Локуциевский Л. В., Сачков Ю. Л. Об интегрируемости по Лиувиллю субримановых задач на группах Карно глубины 4 и больше // Математический сборник. 2018. Т. 209, № 5. С. 74–119.
[14] Montgomery R. A Tour of Subriemannian Geometries, their Geodesics and Applications. Mathematical Surveys and Monographs, Vol. 91, 2002.
[15] Pansu P. Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un // Annals of Mathematics. Second Series. 1989. Vol. 129, No. 1. P. 1–60.
[16] Сачков Ю.Л. Введение в геометрическую теорию управления. – М.: URSS, 2021, 160 C.
[17] Scagliotti A. Deep learning approximation of diffeomorphisms via linear-control systems // Mathematical Control and Related Fields, 2023, 13(3): 1226-1257. arXiv:2110.12393
[18] Chitour Y., Jean F., Monti R., Rifford L., Sacchelli L., Sigalotti M., Socionovo A. Not all sub-Riemannian minimizing geodesics are smooth, 2025, arXiv:2501.18920
[19] Больсинов А. В., Фоменко А. Т. Интегрируемые гамильтоновы системы. Геометрия, топология, классификация. Издательский дом «Удмуртский университет», 1999.

Seminar organizer
Lokutsievskiy Lev Vyacheslavovich

Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center