Torelli groups

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Not regularly scheduled

Special topics course offered in Spring 2018 by Dan Margalit on "Torelli groups"


Algebra I  and  Algebraic Topology , or permission of the instructor.

Course Text: 

The basic references will by the book A Primer on Mapping Class Groups
by Farb and the Instructor, as well as selected papers.

Topic Outline: 

The mapping class group is the group of symmetries of a surface.  This
is a basic and beautiful object in mathematics.  It describes hyperbolic
(or algebraic, or complex, or conformal) structures on a surface.  It
also describes all ways of constructing a surface bundle over a given
space.  The topic also has deep connections to many areas of mathematics
and physics, including low-dimensional topology, group theory,
representation theory, number theory, dynamics, and quantum field
theory, to name a few.

In this course we will focus on a particular subgroup of the mapping
class group called the Torelli group.  This is the subgroup acting
trivially on the homology of the surface.  We can think of this group as
encoding the non-linear, or more mysterious, aspects of the mapping
class group.

We will give a panorama of the most important theorems about Torelli
groups, including classical work of Johnson that gives a finite
generating set and the recently announced work of Church, Ershov, and
Putman, which shows that the kernel of the Johnson homomorphism is
finitely generated.

Each theorem is a gem and each brings some new tools and ideas into the
picture.  As such, the course should be useful, not just for those who
want to know about the Torelli group, but for those who want to learn
how to think about and do mathematics.