Abstracts

 Sacsayhuaman Michael Harris, Roberto Miatello, Ariel Martin Pacetti and Gonzalo Tornaría
Modular forms: constructions and applications

Modular forms appeared naturally in the context of holomorphic differentials on certain quotients of the upper half plane. Their study developed considerably with Hecke's work on what are now called Hecke operators acting on the space of modular forms, and with L-series associated to modular forms.

Side by side with advances in the general theory of modular forms, there is a substantial body of work on how to construct them. It is there that the relation between modular forms, quadratic forms and modular symbols first appeared. In this course, we will start with the definition and basic properties of modular forms, and proceed toward constructions and applications.

  1. Modular forms: motivation, definition, construction
  2. Binary quadratic forms: definition and basic properties
  3. General quadratic forms: basic properties. Relation with modular forms (theta series). Basis problem.
  4. Quadratic fields and their relation to binary quadratic forms. The class group and the Hilbert class field.
  5. Modular functions; analytic construction of the Hilbert class field. The class number 1 problem.
  6. Hecke operators in modular forms. L-series: Euler product and functional equation.
  7. Modular elliptic curves (modular parametrization). Geometric construction of the Hilbert class field (Heegner points).
Associated talks:
  1. Modular forms and elliptic curves for totally real fields: definitions and results. Shimura curves.
  2. Construction of the class field for real quadratic fields: Stark-Heegner points.
Pablo Candela, Javier Cilleruelo, Juanjo Rué and Julia Wolf, assisted by Ana Zumalacárregui
Additive Combinatorics

Additive combinatorics can be defined as the study of structures in subsets of groups (classically, abelian groups). Even though this is not a new discipline, it has undergone dramatic developments in the last few years, and has become a very fruitful and active area of research thanks in part to its interactions with other areas of mathematics, such as ergodic theory, functional analysis and probability theory. We shall present an overview of the abelian theory. We will introduce central themes in the area and show their interaction with several areas in mathematics, such as probability, graph theory and algebraic structures over finite abelian groups, among other subjects.

  1. Structure in sumsets
  2. The sum-product phenomenon
  3. The polynomial method
  4. Sidon sets
  5. Fourier-analytic methods
  6. Removal lemmas, with applications

Marc Hindry and Marusia Rebolledo
 Invited talks by Fernando Rodriguez Villegas and Ricardo Menares
Introduction to elliptic curves; zeta functions and L-functions coming from geometry

This course is an introduction to the geometric and arithmetic theory of elliptic curves and their associated L-functions. Such curves appear naturally in the study of Diophantine equations: they are the first example where one cannot apply systematically the Hasse principle (as one can do for conics). The diverse structures of these curves, as well as their ties, via L-functions, with algebraic objects (Galois representations) and analytic objects (modular forms) are at the heart of many results and open questions in arithmetic geometry. Among these results, the best known is certainly Fermat's Last Theorem.

Cusconuit

L-functions are given by Dirichlet series, just like the Riemann zeta function is. Dirichlet series and the Riemann zeta function were introduced in order to prove the main theorems on the distribution of prime numbers. The success of this approch has led to the introduction of analogues called Hasse-Weil L-functions associated to elliptic curves. We will define these L-functions and explain their main properties – including some that are only conjectural, such as the “Riemann hypothesis” - as well as their relation with the arithmetic elliptic curves.

  1. Elliptic curves over a field: definition and group law.
  2. Elliptic curves over a local field. Reduction.
  3. Rational points and torsion points.
  4. Classical zeta functions and L-functions
  5. Hasse-Weil L-functions
  6. Value at s=1

Misha Belolipetsky and Harald Helfgott
Groups and expanders

An expander graph is a graph in which any set of points is connected to many others. Any such graph has several desirable properties: for example, a random walk on an expander graph becomes uniformly distributed very quickly.

While it is easy to prove that expander graphs are very common, proving that a given graph is an expander is non-trivial. Various strategies involve algebra, geometry, number theory, combinatorics...

The first half of the course will be dedicated to a study of combinatorial techniques (Helfgott, Bourgain-Gamburd, etc.). We will then go into modular techniques, with a special focus on a construction of Gromov's involving quotients of the upper half-space \( \mathbb{H}^3\).

Andrzej Zuk
Analysis and geometry on groups

The following classical problems in group theory will be discussed:

  1. Burnside problem: infinite, finitely generated torsion groups.
  2. Milnor problem: groups of intermediate growth.
  3. Day problem: exotic amenable groups.
  4. Siegel problem: finite generation of lattices.
  5. Atiyah problem: closed manifolds with irrational L2 Betti numbers.
  6. Margulis construction: expander graphs.
  7. Gromov problem: uniform exponential growth.

Along presenting solutions to these problems we will introduce the following subjects:

  1. Groups generated by automata.
  2. Amenability and property (T).
  3. Random groups and graphs.
  4. L2 invariants of groups and manifolds.
  5. Random walks on groups and graphs.
Cusconuit