# BPS state counting, stability structures and derived algebraic geometry

**Summer School, University of Hamburg, 31.8.–4.9.2009**

In physics a D-brane is an allowed boundary condition for open strings, and a BPS-brane is a critical D-brane, typically fulfilling a first order partial differential equation. Depending on the physical model these have various mathematical incarnations, for example as a special Lagrangian submanifold or a holomorphic vector bundle. The physical interpretation suggests that BPS-branes can be counted, and it is an ongoing mathematical challenge to define and investigate the expected BPS invariants. Definitions that work in some cases are due to Donaldson and Thomas, Joyce, Behrend, Bridgeland and Toledano-Laredo, Pandharipande and Thomas.

A recent paper by Kontsevich and Soibelman links the problem of counting BPS states with several other objects of current mathematical interest, such as motivic integration, quiver representations, stability structures and cluster transformations. The idea of the summer school is to unravel this whole circle of ideas, notably by providing the necessary background and by illustrations via special cases.

The summer school is aimed at PhD students as well as young post-docs with a background in algebraic or complex geometry.

## Invited Speakers

- Daniel Huybrechts (Bonn)
- François Loeser (Paris)
- Andrew Neitzke (Harvard)
- Markus Reineke (Wuppertal)
- Yan Soibelman (Manhattan, KS)
- Balazs Szendroi (Oxford)
- Yukinobu Toda (Tokyo)

## Programme

Click here for the programme.

## Venue

The summerschool will take place at the Physics Department of the University of Hamburg, Jungiusstrasse 9, 20355 Hamburg, Germany. The lectures are held in the lecture theatre II, which is in front of the building, towards the Jungiusstraße. For getting there consult the webpage of the First Intitute for Theoretical Physics and the map.

You have access to the library of the Department of Mathematics, which is located at the ground floor of the nearby Geomatikum, Bundesstraße 55, between 9.00 am and 8.00 pm.
Click here to see how to get there from the Physics Department.

## Application

The deadline for application has expired.

A conference fee of 20 Euro will be due upon arrivial in Hamburg.

## Accomodation

We have reserved a number of rooms in some inexpensive hotels for you.

If you interested in us making a hotel reservation for you, please note so on the application form.

## Financial Support

Partial financial support will be available for younger participants.

## Organizers

S. Mohrdieck, M. Pumperla, B. Siebert