FUNDAMENTALS OF ADVANCED MATHEMATICAL PHYSICS
measure theory, ordinary differential equations
The exams consists of a practical part and of an oral examination. The first involves a take-home project on a topic chosen by the teacher and the student. An oral examination consisting in the discussion of the project and the verification of the understanding of the therory
Aim of the class is to introduce the tools of this theory in a constructive way, so to facilitate its usage in practical applications, as well as in the teaching experiments in high school.
Singular (fractal) measures and the associated orthogonal polynomials. Relevance of the theory in other branches of mathematics and in Physics. Historical development of the theory. Links with the theory of dynamical systems. Numerical experiments.
SIngular measures. Hausdorff dimension, multifractal dimensions. Orthogonal polynomials, definition and basic properties. Recurrence relation. Jacobi matrix and its eigenvalues - eigenvectors. Christoffel-Darboux relation and applications. Gaussian integration. Iterated function systems and applications. Logarithmic potential. Equilibrium measure. Capacity. Asymptotic behavior of orthogonal polynomials. Regular measures according to Ullman-Stahl-Totik. Examples are provided in all chapters.
classroom teaching
Borrowed from
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