The program of the school includes series of main lectures, workshop and poster session. The following programme is tentative and is updated regularly.

The timetable of the Scientific Program: PDF

### Courses

A) Foundations of Finsler Geometry and its Generalizations

A1) Finsler and pseudo-Finsler geometry - a fresh look at a century-old problem (Nicoleta Voicu):

The lectures present a completely coordinate-free treatment of variational problems in Finsler and pseudo-Finsler spaces, based on differential forms and Lie derivatives.

Contents:

- A brief history of Finsler geometry. Tangent bundle, pulled-back bundle, projectivised sphere bundle approaches. Basic Finslerian geometric objects.

- Coordinate-free variational calculus on Finsler spaces. Lepage forms, first and second variation formulas, Noether currents.

- Mathematical aspects of Finslerian relativity. Pseudo-Finsler spaces and their specific problems: singularities of metric tensor, non-compactness of indicatrix. Field-theoretical functionals on pseudo-Finsler spaces and their extremals.

A2) Variational foundations of Finsler geometry: Projective spaces, Grassmann bundles and the Hilbert form (Demeter Krupka)

Contents: PDF

B) Geometric control theory, sub-Riemannian geometry, and their applications in robotics and vision (Yuri Sachkov)

Lecture 1: Introduction to geometric control (Smooth manifolds, vector fields and flows, Lie brackets. Dynamical and control systems. Statement of controllability and optimal control problems. Examples of control problems: stopping a train, linear oscillator, car with trailers, Dubins car, rotations of rigid body, rolling sphere, curve reconstruction, quantum systems.)

Lecture 2: Controllability and attainability (Controllability of linear systems. Local controllability of nonlinear systems. Orbit theorem. Frobenius theorem. Attainable sets of full-rank systems.)

Lecture 3: Optimal control problems (Existence of optimal controls (Filippov's theorem). Elements of symplectic geometry. Necessary optimality conditions (Pontryagin maximum principle). Solution to optimal control problems.)

Lecture 4: Controllability and optimal control on Lie groups (Lie groups, Lie algebras, homogeneous spaces. Left-invariant control systems on Lie groups. Controllability of left-invariant and bilinear systems. Elements of sub-Riemannian geometry. Sub-Riemannian geometry on Lie groups (Heisenberg group, SO(3), SL(2), Engel group, Cartan group).)

Lecture 5: Applications. (The plate-ball problem. Mobile robots with trailers. Euler elasticae. Image inpainting. Vessel tracking and diabetic retinopathy diagnostics.)

Literature

- A.A. Agrachev, Yu.L. Sachkov, Control Theory from the Geometric Viewpoint, Springer-Verlag, 2004.

- A.A. Agrachev, D. Barilari, and U. Boscain, Introduction to Riemannian and sub-Riemannian geometry, https://webusers.imj-prg.fr/ davide.barilari/Notes.php.

- Yu.L. Sachkov, Controllability and symmetries of invariant systems on Lie groups and homogeneous spaces (in Russian), Moscow, Fizmatlit, 2007.

- Yu.L. Sachkov, Control Theory on Lie Groups, Journal of Mathematical Sciences, Vol. 156, No. 3, 2009, 381-439.

### Workshop

The Workshop will have two sections:
A. 100 years after Finsler - Foundations and advances in Finsler geometry (organizer: N. Voicu)
B. Geometric control theory (organizer: Yu. Sachkov).

The goal is to present recent results in differential geometry, geometric control theory, and applications. Presentations of posters are also possible. The concrete workshop program of oral contributions will be scheduled with respect to the number of registered talks.

Note that the lecture series and the workshop program will be conducted independetly, so that all participants may attend both.

Workshop talks

 Tadashi Aikou Negativity of holomorphic vector bundles and complex Finsler structures Ludovit Balko Some numerical homotopy invariants of certain classes of Flag manifolds (poster PDF) Chayan Kumar Mishra Curvature Inheritance symmetry on Finsler space Praveen Agarwal Note on Multple Gamma functions and Multiple Hurwitz zeta functions Xinyue Cheng TBA Dipankar Debnath On N(k) mixed quasi Einstein manifold and some global properties CTJ Dodson Information geometry as a possible control tool Manuel Hohmann Cosmological observations in Finsler spacetimes Pavel Chalmoviansky Envelopes of planar curves in applications (poster PDF) Miguel Angel Javaloyes On the definition and examples of cones and Finsler spacetimes Young Ho Kim Application of Laplace operator on manifolds Chang-Wan Kim Rigidity theorems in Finsler geometry (poster) Ville Kivioja Classification of Lie groups by metric geometry Demeter Krupka Variational forces Enrico Le Donne Tangents of sub-Riemannian geodesics Huageng Liu Discrete optimal control of a car on the sphere Colin MacLaurin Time slicings of black holes Kishore Marathe What is Physical Mathematics? Mohamed Menad Subdivision schemes in modelling geometry (poster) Terhi Moisala Infinite-dimensional Carnot groups and Rademacher's theorem Gilbert Nibaruta Conformal change of Chern connection Alvaro Pampano Criticality of Sub-Riemannian Geodesics Projections and Applications Dhriti Sundar Patra The Fisher-Marsden conjecture on contact manifold Ioan Radu Peter Hardy type inequaliities in Finsler geometry Christian Pfeifer Finsler spacetimes Liviu Popescu Lie algebroids framework for drift-less control affine systems Paul Popescu Constructions and examples related to skew-symmetric algebroids Ghanashyam Kumar Prajapati On Semi C-Reducibility of general (α, β)-Finsler metrics Matthew Romney Quasiconformal mappings on sub-Riemannian manifolds Narasimhamurthy Senajji Kampalappa Einstein theory of Finsler-Randers Cosmological Model Hemangi Shah Geometry  of  Asymptotically  Harmonic  Manifolds with Minimal  Horospheres Gao Shan Hamel's field variational integrators for geometrically exact beam and Chaplygin-Timoshenko sleigh Gauree Shanker TBA Sarita Singh Parametric Experimental & Numerical Study of Human Bifurcated Artey with their topological behaviour Bankteshwar Tiwari TBA Francesca Tripaldi $\\ell^{q,1}$-cohomology of Heisenberg groups Zbynek Urban Global variational principle Paweł Walczak Integral formulae for codimension-one foliated Randers spaces (O)

### Social program

Excursions will be organized on Wednesday, August 22. We offer three options: either a trip to castles and Saxon fortresses around Brasov, or a hiking trip in Piatra Craiului National Park (hiking boots are necessary for the hiking trip), or a tour of the city of Brasov.

On Thursday, August 23 in the evening, a conference dinner will take place in Brasov.

Partners & Cooperations