Course Announcement - Math 911 - Spring 2012 - Theory of Groups

Instructor:  Susan Hermiller
               304 Avery Hall, smh@math.unl.edu

The goal of this course will be an introduction to the theory of
groups, with an emphasis on infinite groups.

Groups are a useful mathematical tool which originated in the study
of symmetry.  For example, the set of reflections and rotations
of the plane that map a square back to itself form a dihedral
group.  In the same way, reflections, rotations, and translations
of the plane, R^n, etc. that preserve infinite symmetric objects
(eg tilings) form examples of infinite groups.

Several methods for studying groups, including algebraic,
topological, and geometric techniques, will be discussed in
this course.  Many of these methods have applications in other
areas of algebra and mathematics in general, as well.
There will be two main topics for the course:

1) Nilpotent, polycyclic, and solvable groups:
   Nilpotence and solvability are algebraic measures of how close a
   non-abelian group is to being abelian.  They're special cases of
   a more general construction, known as poly properties.  Studying
   groups with poly properties is particularly tractable, because
   the poly structure allows for many proofs by induction.  We'll concentrate 
   on finitely generated poly-P groups, including the relationships among 
   the finitely generated nilpotent, polycyclic, and solvable groups.

2) Geometric group theory:
   Geometric group theory encompasses a wide variety of geometric,
   topological, and algorithmic methods in the study of finitely
   generated (and finitely presented) groups.  The specific focus
   of this half of the course (which may depend on the interest of
   the students) will include growth functions, connections between
   group theory and dynamical systems (note:  _no_ prior knowledge
   of anything in dynamical systems will be assumed) via
   "self-similar groups", and measuring algorithmic complexity
   (particularly for the conjugacy problem).

In all of the topics covered, I'll discuss applications to finding 
computational algorithms for answering questions about the groups, 
and to finding bounds on how efficient those algorithms can be.

Prerequisites:  Math 817 and Math 872 (Math 872 may be taken
concurrently, or may not be needed - I'll eventually assume some
familiarity with finitely presented groups and fundamental groups),
or permission of instructor.

Text:  There will be no formal text for the course, but
much of the material we will cover in the first topic of
the course can be found in the book "A Course in the Theory
of Groups" by D.J.S. Robinson (Springer, 1996), and material
for the second part of the course can be found in "Topics in 
Geometric Group Theory" by P. de la Harpe (U. Chicago Press, 2000).

If you'd like more information, please feel free to write me
(smh@math.unl.edu) or stop by.

Susan Hermiller