**Lie Algebras in a Topological Setting.**

Because of the nature of Mathematics
Research opportunities available to an undergraduate
student, I have applied to a number of Research Experience
for Undergraduates (REU) Programs in Maths. A partial
list of these programs is:

1. RIPS/IPAM Applied Mathematics Program (UCLA)

2. Small Research Program in Mathematics (Williams
College)

3. Summer Mathematics REU at University of Minnesota

4. Summer Mathematics REU at University of Notre Dame

5. AMSSI Summer Math Program by Pomona & Loyola
Colleges

6. Summer Mathematics REU at Lafayette

7. Summer Mathematics REU hosted by the Claremont
Colleges

Most of the above programs pertain
to research in the field of Topology and/or Algebra.
I am interested in these fields for future research
in my graduate studies, and have done my best to gain
the most extensive knowledge of these subjects. I
am currently enrolled in the Graduate Algebraic Topology
course taught by Lisa Traynor and am in the process
of writing my thesis on Lie Algebras and Representation
Theory with Paul Melvin as my advisor. I will be continuing
my thesis (started as an undergraduate honours project)
next year and hope to develop into my Masters thesis
for the A.B./M.A. in Mathematics.

Over the summer, I hope to gain further
experience and knowledge regarding the subject, and
perhaps even learn of the interpretation of Lie Algebras
in a Topological setting. I feel that this would give
me a great start to expanding my thesis to a more
mature Mathematical project.

I was fortunate enough to be granted
the Horizons Off-campus Research Scholarship last
summer, which I used allow me to attend the REU at
Cornell University. For a serious Mathematics undergraduate
student who wishes to continue studying the subject
in graduate school, this was one of the best hands-on
research experiences one could have asked for. Since
I am not eligible for the majority of the funds provided
by the REU programs (due to my international-student
status), this grant is my only option at gathering
the experience that would best prepare me for future
research

Note:

Aditi Vashist was accepted to Williams College, MA
to the

SMALL PROGRAM June 11 – August 9th 2007.

**Summary:**

**Moduli Spaces of Punctured Poincaré
Disks**

By Aditi Vashist

SMALL Mathematics REU, Williams College, Williamstown,
MA

This summer at Williams College, I worked
with three other students investigating the structure
of the Poincaré disk with n punctures on the
interior and m restricted to the boundary. With certain
basic rules to define the structure of the object,
we are able to analyse the Configuration Space of
this construction – namely the space of all
possible arrangements of the punctures on the disk
that satisfy the necessary conditions. We then looked
at these spaces under certain group actions that we
defined, to give us what are referred to as Moduli
Spaces. Since there are certain holes in this space
(i.e. corresponding to configurations that are not
legal based on the defined rules), we use the Fulton-MacPherson
Compactification of these spaces to get higher-dimensional
closed topological spaces. This involves taking the
configurations not allowed, for example, two punctures
occupying the same space on the disk, and doing a
series of Blow-ups where these particles escape onto
different copies of low-dimensional manifolds and
where their interactions can be studied more carefully.
We then analysed the topological nature of these spaces
in relation to other algebraic varieties.