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Quotients and Moduli

Schedule: Mo 16-18 (If this is problematic, we can try to nd a di fferent 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 classi ed 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 fi nding 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 classi cation problems, called geometric invariant theory. As a main source for the seminar we will use the fi rst 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.