# CIMPA School

# CONTEMPORARY GEOMETRY

# University of Namibia, Windhoek, Namibia, January 16-27, 2023

**Abstracts of courses**

**Aissa Wade (Pennsylvania State University)****
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**INTRODUCTION TO POISSON COHOMOLOGY****
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**The aim of this mini course is to introduce fundamental topics in Poisson geometry. We will first cover basic Poisson algebras and Lie algebroids.
Then we will discuss the linearization problem for Poisson structures. But our focus will be on the theory of Poisson cohomology and their applications.****
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**Youngju Kim (Konkuk University)****
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**AN INTRODUCTION TO HYPERBOLIC GEOMETRY****
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**In this course we will mainly discuss the geometry of hyperbolic plane. We will begin with basic notion such as hyperbolic plane, the boundary at infinity, Mobius transformations, Poincare metric, hyperbolic trigonometry and hyperbolic area etc.
Then we will discuss discrete groups of isometries and hyperbolic surfaces.****
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**Jeongseok Oh (Imperial College)****
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**ALGEBRAIC GEOMETRY IN QUASIMAP THEORY****
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**The goal of this course is to introduce quasimap invariants defined by Professor Bumsig Kim et al. and their use in some comparison results including mirror symmetry. We plan to cover some necessary concepts such as varieties vs schemes, schemes vs stacks, moduli spaces, Chow (co)homology groups, Euler classes, Chern characters and localisations briefly.
****
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**Ludovic Riffort (Laboratoire A.J. Dieudonné Université Cote d'Azur)****
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**AN INTRODUCTION TO SUB-RIEMANNIAN GEOMETRY****
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**Sub-Riemannian geometry provides a mathematical model for many problems involving non-holonomic constraints.
The paradigmatic example is the dynamics of parking a car. We can describe the configuration of a car by its
position (x,y)∈ ℝ**^{2} and the orientation of the steering wheel θ ∈ S^{1}. Indeed,
not all movements are allowed: for example, the car cannot move sideways (a practical example of nonholonomic constraint). Nevertheless, through a combination of admissible movements, the car can attain the equivalent of a sideways displacement, making it possible to maneuver it into the parking spot.

The total distance traveled is clearly longer than the straight line from the initial to the final position, and a non-trivial problem is to determine how to do it in an optimal way, i.e. minimizing the total distance. This gives rise to a metric space structure on the car's configuration manifold ℝ^{2}× S^{1}, and constitute an example of sub-Riemannian structure.

The aim of this course is to provide an introduction to geometry and analysis on sub-Riemannian manifolds, and to illustrate open problems and research directions of this very active research domain.
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**Morgan Kamga-Pene (University of Namibia)****
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**INTRODUCTION TO DIFFERENTIAL GEOMETRY****
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**The course will introduce the basic concepts of differential geometry, like tensors, connections, curvatures, Riemannian metrics, geodesics, and will serve as a preparation to the other courses of the school.
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