This school is the fifth of a series of summer schools to take place in Colombia (the three previous ones took place in July 1999, 2001, 2003 and 2005) on topics on the border 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.

The meetings are part of a long term scientific project in Colombia and the northern part of South America supported by the Universidad de los Andes ( Bogotá , Colombia ) and the Université Blaise Pascal ( Clermont-Ferrand , France ) with the help of other organizations. The idea is to encourage and promote research in that part of the world on topics at the border line of mathematics and theoretical physics. In these schools we have focused on the interactions between Quantum Field Theory, Geometry and Topology since we feel these are particularly active areas of research which can attract young researchers both in Colombia and the neighbouring countries.

Encouraged by the French authorities, we have set up a PREFALC (Programme Régional France-Amérique Latine) network at the master's level between the Universidad de los Andes , the Université Blaise Pascal, the Université de Metz, the CINVESTAV (México) and the Universidad Central de Venezuela. Within this framework we will get partial support for the school and, additionally, for two mini-workshops in Geometry, Topology, and Algebra, one in Caracas in July 2006 and the other in Mexico in 2007 (the exact month has not been settled yet).

In the long run, we hope to set up a research centre for both Colombian scientists and scientists from neighbouring countries of the northern part of South America .

The school will be three weeks long, during which the professors (leaders in their own areas) will give both mini-courses and specialized talks on their fields of expertise for students at the master's and PhD levels. The length of each course will range from 6 hours to 8 hours and they will serve as introductions to active areas of research these days. Moreover, notes of each course will be provided by the lecturers, and these, in due time, will be published.

The lecture notes will be posted on the school web site and then published in a volume of proceedings as were the lectures given at the past schools (World Scienti¯c for the 1999 and 2001 schools, Springer Verlag for the 2003 school and AMS for the 2005 school).

There will be the opportunity for the participants to present their research, and we will publish their research papers in the proceedings prior the approval of a referee. Also, there will poster sessions where students will be able to present their master's work or their partial results on their Ph. D. work.

Last but not least, due to the fact that the village of Villa de Leyva is very small, there has been plenty of scientific interaction between professors and participants in the past. We expect the same to happen in this forthcoming opportunity.

	Academically, the Département de Mathématiques Appliquées at the Université Blaise Pascal, Clermont-Ferrand, France.
	Financially and logistically, the Universidad de los Andes, Bogotá, Colombia, through its Rectory, Science School, Physics Department and Mathematics Department.
	Financially and academically, the Clay Mathematics Institute, ECOS-Nord and PREFALC