### 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.

- 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…
- 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.