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# Quotients and Moduli

**Schedule**: Mo 16-18 (If this is problematic, we can try to nd a different time.)

**For:** Bachelor and Master students of Mathematics.

**Prerequisites**: Algebra 1 and some basic knowledge of algebraic geometry will be helpful.

We can adjust the level of the seminar depending on the background of the participants: The first part of the seminar could also be done using only elementary methods and conversely, the last part can easily be expanded if needed.

**Distribution of the talks:** Tuesday 7.4.2015, 16:00 WSC-O-3.46

*You can register earlier by email !*

**Overview:**

One of the basic questions in many parts of mathematics is: Can we get an overview of all possible objects of a certain kind? You have seen this for example in linear algebra: Vector spaces are classied by their dimension. Looking at all linear maps f : V -> V is a bit more daunting, but in the end we found that when we put f in Jordan normal form the geometry of the map f becomes quite transparent.

Equivalently this could also be formulated as finding good representatives for matrices up to conjugacy.

Surprisingly, many other algebraic objects admit a similar description: often one can describe the “space of all objects” as a quotient of a large space by a symmetry group. Classically one could phrase this procedure as trying to simplify systems of equations with variable coefficients by a clever change of coordinates. In particular this often allows to reduce the number of coefficients (classically often called the number of “moduli”) on which a problem depends.

In this seminar we want to learn about a very successful algebraic approach to this kind of classication problems, called geometric invariant theory. As a main source for the seminar we will use the first chapter of [1]. Towards the end we aim to give a further outlook on applications and more recent approaches to moduli problems.

A preliminary program for the seminar.

**References:**

^{1} A. Schmitt: Geometric Invariant Theory and Decorated Principal Bundles. Zürich lectures in advanced mathematics, EMS Publishing House, 2008

^{2} S. Mukai: An Introduction to Invariants and Moduli. Cambridge University Press 2003.