Real and complex analysis - MAT311
This series of lectures is an introduction to real and complex analysis.
The lectures are organized around 4 central themes :
- Lebesgue's theory of measure and integration ;
- Fourier analysis ;
- Hilbert spaces and variational methods ;
- the theory of holomorphic functions (functions of one complex variable which are complex differentiable).
This course aims to give to all students some common base of knowledge in functional analysis, opening up the way to many different fields : pure mathematics, applied mathematics, mechanics , theoretical physics, ... In particular, this course is an important tool box for the second and third year courses in pure and applied mathematics in particular, MAT431 and
MAT432, which are the natural extensions of this course.
The theory of measure and integration developed by H. Lebesgue is a key tool in many branches of mathematics and is commonly used in applications (e.g. in numerical analysis). This theory also offers a natural framework to probability theory which is presented in the second year's course in applied mathematics (MAP 432) and it is the foundation of geometric
measure theory. Illustration through applications in Fourier analysis and in the definition of Hilbert space will be given. Fourier analysis finds many applications in the solvability of partial differential equations but also in signal processing (see course MAP 555),… Hilbert spaces theory is at the crossroad of analysis and geometry, it constitutes a first step towards the theory of operators and spectral theory, this is also an essential tool for the solvability of variational problems (see the course
in optimization MAP 431) and partial differential equations (see the courses MAT431, MAT432 or MAP 431) which arise in physics as well as in mechanics (heat equation, wave equation Schoedinger's equation). The theory of holomorphic functions finds a variety of applications either in pure mathematics (number theory, minimal surfaces and geometry, ...) or in applied fields (fluid mechanics, …).
The mathematical concepts introduced in this course will be illustrated by applications : the use of Fourier analysis in the modeling of diffraction in optics, the use of Hilbert spaces in quantum mechanics and in variational problems, the use of holomorphic functions in fluid mechanics (aerodynamics) or in the study of minimal surfaces , ...
No particular prerequisite are needed to follow this course beside the background of "classes préparatoires". Nevertheless, one lecture will be devoted to some complements in topology (topology of normed vector spaces and metric spaces) since these are essential both in the development of the theory of Lebesgue's integration and the theory of Hilbert spaces.
The lectures are organized around 4 central themes :
- Lebesgue's theory of measure and integration ;
- Fourier analysis ;
- Hilbert spaces and variational methods ;
- the theory of holomorphic functions (functions of one complex variable which are complex differentiable).
This course aims to give to all students some common base of knowledge in functional analysis, opening up the way to many different fields : pure mathematics, applied mathematics, mechanics , theoretical physics, ... In particular, this course is an important tool box for the second and third year courses in pure and applied mathematics in particular, MAT431 and
MAT432, which are the natural extensions of this course.
The theory of measure and integration developed by H. Lebesgue is a key tool in many branches of mathematics and is commonly used in applications (e.g. in numerical analysis). This theory also offers a natural framework to probability theory which is presented in the second year's course in applied mathematics (MAP 432) and it is the foundation of geometric
measure theory. Illustration through applications in Fourier analysis and in the definition of Hilbert space will be given. Fourier analysis finds many applications in the solvability of partial differential equations but also in signal processing (see course MAP 555),… Hilbert spaces theory is at the crossroad of analysis and geometry, it constitutes a first step towards the theory of operators and spectral theory, this is also an essential tool for the solvability of variational problems (see the course
in optimization MAP 431) and partial differential equations (see the courses MAT431, MAT432 or MAP 431) which arise in physics as well as in mechanics (heat equation, wave equation Schoedinger's equation). The theory of holomorphic functions finds a variety of applications either in pure mathematics (number theory, minimal surfaces and geometry, ...) or in applied fields (fluid mechanics, …).
The mathematical concepts introduced in this course will be illustrated by applications : the use of Fourier analysis in the modeling of diffraction in optics, the use of Hilbert spaces in quantum mechanics and in variational problems, the use of holomorphic functions in fluid mechanics (aerodynamics) or in the study of minimal surfaces , ...
No particular prerequisite are needed to follow this course beside the background of "classes préparatoires". Nevertheless, one lecture will be devoted to some complements in topology (topology of normed vector spaces and metric spaces) since these are essential both in the development of the theory of Lebesgue's integration and the theory of Hilbert spaces.