The notion of topos, introduced by Alexandre Grothendieck in the early sixties in the context of algebraic geometry, has an intrinsically eclectic nature; indeed, a topos can be seen as a generalized space, as a mathematical universe but also as a theory. In fact, the interplay of all these different points of view is precisely what makes Topos Theory a unifying subject in Mathematics with important, both actual and potential, applications in the whole realm of knowledge.

The lecture course aims to give an introduction to this fascinating subject with a specific emphasis on the recent applications of interdisciplinary nature. At the end of the course, the participant will have acquired tools and methods to study mathematical theories from a topos-theoretic perspective, extract new information about mathematical dualities, and establish new and fruitful connections between distinct fields.

**Structure of the course:**
The course will consist of five lectures for a total of 12 hours, taking place in the afternoons of the days 22nd - 26th February 2010. The topics of the lectures are the following:
1) Some basic categorical notions
2) The interpretation of logic in categories
3) Sheaves on a site
4) Classifying toposes
5) One topos, many sites: the role of topos-theoretic invariants and the interdisciplinary applications

**Prerequisites:**
Some mathematical maturity will be of help in understanding the lectures; anyway, all the necessary technical background will be reviewed on the first day.
Discussion:
There will be plenty of opportunities for questions and informal discussion with the speaker, each day after the scheduled lectures.

**Language:**
The course will be given in Italian.

For further information, partecipants may contact the speaker at the address: olivia.caramello@sns.it

The deadline for the registration is now extended to the 18 of February.