Cécile Armana
Summary: We will present global aspects of algebraic number theory:
• Rings of integers, Dedekind rings
• Splitting of prime ideals (inertia degree, ramification), ideal class group and
class number, case of Galois extensions (decomposition group, inertia group),
• Fundamental examples : number fields, quadratic fiels, cyclotomic fields
• Hilbert class field
Christian Maire
Summary: We will present local/glboal aspects:
• p-adic valuation and completion of Q,
• analysis in Q_p,
• algebraic extensions of Q_p,
• Hilbert symbol,
• Hass-eMinkowski Theorem.
Frédérique Oggier
Summary: In this course, we taught show how lattices and codes, both independently and jointly, are used in the context of communication systems. The course was structured as follows :
Frédéric van Wijland
Summary: High dimensional data are everywhere, from physical systems to the spreading of epidemics, the world of economic agents or that of neural networks and machine learning. The course aims at stressing the basic techniques underpinning our understanding of the emergence of collective behaviors in very large assemblies of interacting agents.
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