Colombia

The aim of this school is to introduce the younger researchers and graduate students from Colombia and north of Latin America to recent results in difference equations. We will focalize on Galois theory for linear systems of difference equations and present it from different perspectives : algorithmic, algebraic, analytic, arithmetical and geometrical. We will insist on the interplay between all these aspects and applications to arithmetic or to integrability of dynamical systems.

Combinatorics is at the center of many areas in pure and applied mathematics. This summer school will focus on the versatility of combinatorics to shed light on problems in algebra, geometry, optimization, and mathematical physics. We will also discuss how algebraic and geometric perspectives can help us understand purely combinatorial objects.

This school is the second of a series of summer schools to take place in Colombia (the last one took place in July1999) on topics on the boarder line between geometry, topology and quantum field theory. The courses are addressed to both physicists and mathematicians with a masters level in either of the fields. The lectures will therefore serve as an introduction to some active areas of research around the following topics.

The objective of the school is to introduce the participants to mathematical tools and particularly geometric ones-used in certain recent developments of Quantum Field Theory.

The topics of this school will be on geometric and homological methods in the representation theory of associative algebras, and their applications. This school will offer five short courses with an intensity of twelve hours each (six hours theoretical and six exercise hours). The lectures will run over the three following main topics:

• Geometric aspects of the representation theory ofalgebras.
• Homological aspects of the representation theory of algebras.
• Applications of the representation theory of posets and algebras.

This summer school is the sixth "Encuentro Colombiano de Combinatoria (ECCO)” (Colombian Meeting on Combinatorics). Previous ones took place in Bogotá, Colombia in 2003, 2008, 2012, and 2014, and in Medellín, Colombia in 2016. The main objectives of the school are to bring young mathematicians from Latin America, Canada, USA and Europe into close contact with each other along with world experts in various fields of combinatorics and related areas and to promote mathematics among young students in a motivating environment.

The notion of a group describes symmetry in mathematics. In recent decades, certain quantum mathematical objects have appeared whose symmetries are better described by group-like objects called tensor categories. Examples of areas of mathematics where tensor categories play a key role include subfactors, quantum groups, Hopf algebras, quantum topology and topological quantum computation. The aim of this is school is to introduce graduate students to tensor category theory and their applications to Topological Quantum Field theory, Subfactor theory and Hopf algebras.

This school, whose topics lie at the threshold of geometry, topology, algebra and quantum field theory, is the eleventh of a series of summer schools organised in Colombia every other year since July 1999. It is addressed to both physicists and mathematicians with a master’s level in either of the fields and offers courses on the following topics:

Combinatorics is at the center of a variety of areas in pure and applied mathematics. In recent years problems arising in algebra and geometry have been better understood and in many cases fully solved by exploiting their relation to certain combinatorial structures. At this school, we aim to introduce the participants to modern geometric techniques that have been used to solve long-standing conjectures in combinatorics and other areas.

Isogenies of elliptic curves, or more generally of abelian varieties, are surjective homomorphism having finite kernel and they play an important role in the study of arithmetic and geometric properties of elliptic curves. Moreover in recent years there has been an increasing interest in isogeny of elliptic curves from cryptographers. The main reason lies in quantum computer as Luca de Feo eloquently puts it: ''The main reason for this is the sudden realization by the cryptographic community of the very possibly near arrival of a general purpose quantum computer.