Advanced Analysis and Computational Mathematics - 3MMACIA

Goal(s)

To provide students with an advanced background in mathematics with the fundamental analytical tools for engineering, which will be useful throughout their studies in applied mathematics.
Students who have completed a mathematics-oriented undergraduate program often have a limited view of function integration, primarily based on the Riemann integral. Our aim is to rigorously introduce the Lebesgue integral and measure theory.
We will then present differential calculus in a very general framework, and conclude the lecture with an introduction to Hilbert spaces, including an application to Fourier series.
Basic numerical techniques for solving linear systems and nonlinear equations will also be introduced at the end of the semester.

Content(s)

1. Measure Theory and the Lebesgue Integral
- Measurable spaces and measurable functions. Positive measures. Lebesgue integral.
- Computational tools (antiderivatives, dominated convergence, Fubini’s theorem, parameter-dependent integrals, change of variables).
- L^p spaces on R. Convolution. Density results.

2. Banach Spaces and Fixed Point Theorem

3. Differential Calculus in Banach Spaces. Implicit Function Theorem

4. Hilbert Spaces
- Hilbert bases, Riesz-Fischer theorem, projection onto closed convex sets.
- Periodic functions, Fourier series.

5. Basic Numerical Methods
- Iterative methods for solving linear systems.
- Nonlinear equations (Newton and quasi-Newton methods).

Mathematics curriculum from MP/MPI preparatory classes or second-year undergraduate level with a focus on mathematics.