# Mathematics courses in Jyväskylä Summer School 2019

## MA1: Large-scale geometry of Lie groups

Lecturer: Prof. Yves de Cornulier (CNRS and University Lyon 1 Claude Bernard, France)

Time: 12.-16.8.2019, 10x45min

Coordinator: Enrico Le Donne

Modes of study: Obligatory attendance on lectures, and completing exercises.

Credits: 2 ECTS

Evaluation: Pass/fail

Contents: Introduction to some notions of large-scale geometry that turn out to be relevant in the geometric study of Lie groups: One is the notion of quasi-isometry, another related one is the notion of sublinearly bilipschitz equivalence. Introduction to the notion of asymptotic cone of a metric space. Presentation of Pansu's result that one can identify the asymptotic cone of a simply connected nilpotent Lie group to its Carnot-graded group, which is another nilpotent Lie group, with a special metric. Descriptions of the asymptotic cone for various other Lie groups. Introduction to Gromov-hyperbolicity, and characterization, among connected Lie groups, of those that are Gromov-hyperbolic.

Learning outcomes: An overview of basic concepts of large-scale geometry, which are useful much beyond the realm of Lie groups; a acquaintance with the geometry of Lie groups.

Prerequisites: basic knowledge of Lie groups and general topology.

## MA2: Introduction to geometric control theory

Lecturer: Prof. Mario Sigalotti (Inria & Sorbonne Université, France)

Time: 12.-16.8.2019, 10x45min

Coordinator: Enrico Le Donne

Modes of study: Obligatory attendance on lectures, and completing exercises.

Credits: 2 ECTS

Evaluation: Pass/fail

Contents: After an introduction to the control theory framework and its terminology, three main topics will be discussed: controllability, optimal control, and stabilization. The goal is to present the standard tools of geometric control (Lie brackets, accessibility, Pontryagin Maximum Principle, ...) and also to illustrate them by means of some recent results. For controllability, applications to the bilinear Schrödinger equation will be presented. For stability, we shall discuss maximal Lyapunov exponents for bilinear control systems. For optimal control, some recent results on the regularity of optimal trajectories will be presented.

Learning outcomes: An overview of the questions considered in geometric control theory and of the tools which are used to tackle them. A subjective view of some of the active research areas in the field.

Prerequisites: Basic theory of ODEs (existence, uniqueness, regular dependence with respect to parameters), elementary differential geometry (manifolds, vector fields, tangent and cotangent manifolds).