People
Kim Ruane Professor, Chair
Contact Info:
Tufts University
Department of Mathematics
200 College Ave
Science and Engineering Complex
Room LL009
Medford, MA 02155
Email @tufts.edu:
kim.ruane
Phone: 6176272006
Papers
Expertise:
Geometric Group Theory/Topology
Research:
To understand what I study, it is important to understand how I came
to study the subject. I was originally recruited to graduate school
to work on a project with Professor DeWitt Sumners that involved
applications of topology to DNA research. The idea is that
biologists can take pictures of DNA using a special microscope but
once they have the pictures, they have to be analyzed. Such a
picture is a two dimensional projection of a configuration of
strands in three dimensional space. It is a well developed theory of
mathematics called knot theory which studies such configurations. In
particular, the question of whether two such configurations are the
same is a difficult math problem. Imagine taking just one very long
piece of string that has been bunched up very tight. If you pull the
two ends of the string, you could end up with a straight piece of
string or one with a big knot in the middle. How do you know which
one of these you will get only looking at the original bunched up
string? To a given configuration of strands, one can associate a
mathematical object called a group. It is an important theorem that
says two configurations are the same if and only if they have the
same associated group. So while studying under Professor Sumners, I
encountered the subject of group theory and became completely
fascinated with this subject on its own.
The concept of a group arises quite naturally in the study of
geometric objects. The set of symmetries (or rigid motions) of a
geometric object form a group because if you do two or more of these
in succession, you get another. For example, there are 8 symmetries
of the square but all 8 can be constructed as a combination of a
rotation by 90 degrees and a flip through the center axis. Thus you
only have to know these two to understand all 8. Likewise, any rigid
motion of the standard Euclidean plane can be constructed using a
combination of horizontal translation and rotation about the origin.
These two motions allow you to describe the infinitely many rigid
motions of the plane. Computing the symmetry group of two geometric
objects can also distinguish between the objects as in the case of
the knots. If the groups are different, then the spaces are
different. Showing two groups are different is sometimes an easier
problem than directly showing the spaces are different.
During the second summer of graduate school, I took a course on
fuchsian groups. These are groups of rigid motions of the noneuclidean
(or hyperbolic) plane. In this course, we used a combination of
geometry, group theory, and topology which were my favorite subjects
from my elementary graduate work . I asked the Professor if he could
suggest some research papers for me to read which would take me
beyond the material in the course. He gave me several articles which
outlined a new approach to studying a class of groups that are
similar to fuchsian groups only much more general. In particular,
this class includes many of the knot groups that first got me to
graduate school in the first place! This area of research is called
Geometric Group Theory and it has been a very popular and productive
area of research for the past 2530 years. As with many areas of
mathematics, the beauty of the subject comes from many simple ideas
from different disciplines woven together to reveal powerful new
tools for solving problems.
Courses:
Math 250
