# Moduli spaces, stability conditions and birational geometry

**Lecturer:**Christian Lehn**Date:**Friday 10:00 - 12:00**no lecture on Juni, 30th****Room:**MPI MiS, G3 10**Language:**English**Target audience:**MSc students, PhD students, Postdocs**Keywords:**Moduli spaces, stable sheaves, geometric invariant theory, stability conditions, birational geometry, wall-crossing**Prerequisites:**Undergraduate course in algebraic geometry, e.g. scheme theory, divisors, line bundles, coherent sheaves. Also basic knowledge of (algebraic) topology would be helpful.- Remarks: If necessary a small repetition of basics from algebraic geometry at the beginning is possible. See website of the lecture

## Abstract:

This course introduces students with basic knowledge of algebraic geometry to the theory of moduli spaces of sheaves. While curves and surfaces may often be studied via explicit equations it becomes more and more difficult in higher dimensions to detect interesting geometric properties from equations. Moduli spaces of sheaves (or stable objects) are an important class of examples of higher dimensional algebraic varieties and their geometric properties are studied through the geometry of the objects they parametrize.

During the course we will encounter the following topics

- (semi-)stability of sheaves
- algebraic groups and geometric invariant theory
- derived and triangulated categories, Bridgeland stability
- birational geometry and wall-crossing

Many of the notions and techniques relevant to the theory of moduli spaces have already been introduced by Mumford in the late sixties. In the last twenty years moduli spaces have played an important role in mathematical theories coming from theoretical physics and in the light of the derived approach they have recently reentered the focus through the fascinating connection to birational geometry.

## Regular Lectures (Summer 2017)

**Basic Notions in (Geometric) Group Theory**- Karim Adiprasito
- Monday, 11:00 - 13:00, University A-314

**Information Theory II**- Nihat Ay
- Tuesday, 11:00 - 12:30, MPI MiS A3 02

**The variational approach to stochastic PDE**- Benjamin Gess
- Wednesday, 11:00 - 12:30, MPI MiS A3 01

**Mathematical methods in the neurosciences**- Jürgen Jost
- Friday, 13:30 - 15:00, MPI MiS A3 01

**Moduli spaces, stability conditions and birational geometry**- Christian Lehn
- Friday, 10:00 - 12:00, MPI MiS G3 10

**Tropical Probability**- Rostislav Matveev
- Thursday, 11.00 - 12:30, MPI MiS A3 02

**Partial Differential Equations**- Felix Otto
- Wednesday, 9:15 - 11:00, MPI MiS A3 01

**Tropical Geometry**- Yue Ren
- Monday, 9:00 - 10:30, MPI MiS A3 02

**Interpolation Spaces**- Jonas Sauer
- Tuesday, 9:00 - 10:30, MPI MiS A3 01

**Navier-Stokes Equations**- László Székelyhidi:
- Thursday, 9:00 - 11:00 and Friday, 9:00 - 11:00, University SG 3-11

**IMPRS-Ringvorlesung: Geometry and Dynamical Systems**- Tanja Eisner, Hans-Bert-Rademacher, Matthias Schwarz
- Tuesday, 13:15 - 14:45, MPI MiS E1 05 (Leibniz-Saal)

**Seminar on Ergodic theory of group actions**- Tanja Eisner, Tobias Finis, Artem Sapozhnikov
- Tuesday, 09:15 - 11:45, University SG 3-11