Colombia

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.

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 organized 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:

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.

Combinatorics is at the centre of a variety of areas in pure and applied mathematics. For
instance, 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 discrete objects.
At this school we aim to explore modern geometric techniques that have been used in recent
years to solve long standing conjectures in combinatorics and other areas.