Colombia

The purpose of the school is to introduce the students to some advances and new trends in Applied Mathematics. The main emphasis will be to reveal the important role of applied mathematics for other sciences (engineering branches, image processing, biology, and economics) and to provide students and young researchers with basic tools for modeling and simulation of real-life problems.

This school is the third of a series of summer schools to take place in Colombia (the two previous ones took place in July 1999 and 2001) 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 objectives of the school are:

• To introduce graduate and advanced undergraduate students to the current research in the field, facilitate students .
• To get connections with international researchers which help them to get funding or at least acceptance for further studies abroad.
• To promote the development of the subject through academic exchange among Southamerican researchers.
• To update the knowledge of the local mathematical community by inviting senior local and international researchers.

This school, with some 50 odd expected participants, aims at introducing master or Phd students and post docs from Colombia or nearby South American countries, to active topics of research lying at the threshhold between theoretical physics and mathematics.

It runs over three weeks, so as to encourage interactions between the speakers and participants, enabling them to set up long term scientific interactions.

Real algebraic geometry has greatly expanded its horizon in the past few years, in great part due to the invention of new tools and the use of methods from other fields (complex algebraic geometry, tropical geometry, symplectic geometry, algebraic topology, combinatorics, etc.). The goal of this school is to present several of these aspects through 5 courses of 5 hours each. The school will end with a 3-day conference, destined to give the students the widest possible and most up-to-date vision on real algebraic geometry.

Differential Galois Theory is a subject of great development in the last years, with important ramifications that cover algebra, analysis, geometry, algorithmic or applied mathematics. It represents a useful technique in order to treat a wide range of problems related with differential equations, mainly linear, many of them of application, for instance, in physics.

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.