6 short introductory courses.

Each course will take around eight hours.

The courses are of postgraduate level with tutorials sessions. 

Curves and Surfaces in Minkowski spaces: 08 hours (Athoumane Niang: UCAD, Senegal)

The differential geometry of curves and surfaces can be recommended to first-year graduate students, strong senior students, and students specializing in geometry. The material is given in two parts. 

  1. Theory of Curves in the plane and the three-dimensional Minkowski space. The first part contains the standard theoretical material on differential geometry of curves. It contains a small number of exercises and simple problems of a local nature. We will focus on the following concepts: Definition and methods of presentation of curves; Tangent line and Osculating plane; Length of a Curve; Curvature of a Curve; Torsion of a Curve; The Frenet Formulas and the Natural Equation of a Curve; etc… 
  2. Extrinsic Geometry of Surfaces in Three-dimensional Minkowski spaces. The second part contains the standard theoretical material on differential geometry of surfaces. We will focus on the following concepts: Definition and Methods of Generating Surfaces; the Tangent Plane; First Fundamental Form of a Surface; Second Fundamental Form of a Surface; Some Classes of Curves on a Surface; The Main Equations of Surface Theory.

Differential Geometry: 08 hours (Aissa Wade, USA)

In this course, we will discuss the basic theory of smooth manifolds. In lecture 1, we define what it means for M to be a smooth manifold or for M to be a submanifold of another manifold. We discuss immersions, fiber bundles, and vector bundles. In lecture 2, we treat the tangent bundle, submersions, and the cotangent bundle. Lecture 3 deals with the exterior algebra and Stokes’ Theorem. In lecture 4, we present some applications of Stokes Theorem. 

Riemannian Geometry: 08 hours (Cyriaque Atindogbé or Joel Tossa: IMSP, Benin)

In this course, we will introduce the important notion of a Riemannian manifold. More precisely: 

  • Riemannian Manifolds
  • The Levi-Civita Connection
  • Geodesics
  • The Riemann Curvature Tensor 
  • Curvature and Local Geometry 

Optimization on Banach spaces: 08 hours (Ngalla Djitte, Senegal)

In this course, we will discuss the basic theory of shape optimization. 

1.     Introduction, Examples;

2.     Continuity with respect to the domain;

3.     Existence of optimal shapes; 

4.     Shape derivatives;

5.     Relaxation, homogenization. 

Partial Differential Equations (PDE): 08 hours (Diaraf Seck: Senegal) 

1. In this course, we will discuss the basic theory of PDE.

2. Classic examples, well-posed problems in the sense of Hadamard; 

3. Existence, Uniqueness and Regularity of Solutions;

4. Comparison and Maximum Principles:

  • Comparison principles;
  • The weak maximum principle;
  • The strong maximum principle;
  • A priori estimates.

5. Fixed Point Theorems and Their Applications:

  • The Schauder fixed point theorem;
  • Applications of the Schauder theorem. 

Calculus of Variations: 08 hours, (Ibrahima Faye, Senegal)

1.  Introduction, Examples;

2.  Classical methods, Direct methods – Euler-Lagrange equation; 

3. Hamiltonian formulation, Hamilton-Jacobi equation;

  • Field theory, Hilbert theorem;
  • The Dirichlet integral;
  • Relaxation theorem.

4.  Regularity

  • Weyl lemma;
  • De Giogi-Nash-Moser theory;

5.  Minimum surfaces;

6.  The isoperimetric inequality. 

In addition to these courses, there will be presentations by researchers on their works in progress to initiate and develop collaborations between African researchers. These presentations will be scheduled in the afternoon of the school.