Krerley Oliveira
Summary: I covered the section of the course related to methods for computing and estimating eigenvalues and eigenvectors. My focus was on the foundational aspects of this theory, including classic theorems and methodologies. Several examples were thoroughly discussed with the students. I placed particular emphasis on class discussions and practical exercises, which included hands-on activities and a set of homework assignments.
Etienne Pardoux
Summary: I will treat conditional expectation, uniform integrability tightness, and convergence in law. The exact content will depend upon how reactive the students are. I will give them exercises along the way.
Frank Neumann
Samuël Borza
Summary: Theory of distributions, sub-Riemannian structures, and admissible trajectory, controllability and Chow-Rachevsky theorem, Cauchy-Carathéodory theorem and the endpoint map, necessary conditions for minimality (Pontryagin’s Maximum Principle, normal and abnormal extremals), the Heisenberg group, the Grushin plane, the Martinet flat structure, contact structures, and Carnot groups, metric tangent for sub-Riemannian structures.
Wilhelm Klingenberg
Summary: Necessary background in analysis (density functions and its push forward by a map of domains, convexity and the second derivative condition, Legendre transform, convex dual of a functional, and Jensen’s inequality), Monge minimization problem of transport for a continuous cost function c(x, y) with an example in one space dimension, the dual maximization problem due to Leonid Kantorovich, Brenier Theorem.
Mark Rudelson
Summary: Wigner Semicircle Law for normalized eigenvalues of large random symmetric matrices was proved, which can be viewed as a non-commutative version of the Central Limit Theorem. For this purpose the following technical tools were introduced and developed: Stieltjes transform, Hanson-Wright inequality, self-consistent equation for Stieltjes transform.
Tony EZOME
Summary: I started by recalling the needed background from commutative algebra (noetherian modules and rings, finitely generated algebras, Hilbert basis theorem, integral/algebraic elements in a ring/field extension, transcendence degree of a field extension, the algebraic closure of a field) with useful exercice sessions.
Christian Maire
Summary: In this course one gave the proof of the following result. Theorem. Let p be a prime number, and let G be a p-group. Then there exists a Galois extension K/Q such that G=Gal(K/Q). The course aimed to give the key arguments, and explained the difficulty for p=2. During the last part of the course, I also explained some basic properties regarding class group, and the philosophy of Class Field Theory.
Florian Luca
Summary: Average orders of arithmetic functions, maximal orders, normal orders, the Turan-Kubilius Theorem Introduction to probabilistic number theory, density of sets of integers. Smooth numbers, Applications: there are fewer pseudoprimes than primes. Carmichael numbers. also explained some basic properties regarding class group, and the philosophy of Class Field Theory
JOAN-CARLES LARIO
Summary: Introduction to Cyclotomic Number Fields and Cyclotomic Function
Fields with some of their applications to Diophantine and Algebraic Geometric problems, respectively.