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

The preliminary 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

*100 years after Finsler*- Foundations and advances in Finsler geometry

*(organizer: N. Voicu)*

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.

**Workshop talks**

### Social program

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