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a two-week school on Arithmetic, Groups and Analysis
Galois representations by Luis Dieulefait, Ariel Pacetti and Fernando Rodriguez-Villegas.
Primes, parity and analysis by Harald Helfgott and Adrian Ubis.
It is a common intuition that integers should have an even or odd number of prime factors with equal probability - and that this is true even for integers in short intervals. There are classical results in this direction, but, until very recently, little or nothing was known for very short interval. There has been a row of recent developments, initiated by work of Matomaki and Radziwill on what happens on very short intervals on average. In particular, we shall discuss work by Tao on a weak variant of Chowla's conjecture.
Arithmetic Groups by Emilio Lauret, Ben Linowitz and Roberto Miatello.
We will first give an introduction to algebraic groups over Q and arithmetic subgroups, describing Siegel sets and reduction theory (illustrating via the reduction theory of quadratic forms). We will state and sketch the ideas of some fundamental results like the Borel-Harish-Chandra theorem, Godement's compactness criterion, and Borel's density theorem. If time permits we intend to discuss covolumes of arithmetic subgroups and masses of quadratic lattices.
Equidistribution of points of small height by José Ignacio Burgos Gil and Ricardo Menares.
Curves over finite fields by Cicero Fernandes de Carvalho, Daniel Panario and Miriam Abdon.
After reviewing the finite field theory needed for curves over finite fields and coding theory applications, we consider curves (projective, nonsingular, geometrically irreducible) defined over a finite field that attain the maximum number of rational points (as per Hasse-Weil theorem). We will also give constructions of codes using smooth algebraic curves and the Goppa bound for the minimum distance of these codes.
Abelian Varieties, an introduction, by Marc Hindry, David Roberts and Marusia Rebolledo.
We will introduce the basics of abelian varieties (connected and projective algebraic groups), which are the generalization to dimension bigger than 1 of the elliptic curves.