2007 Off Campus Research Internship Awardee

Aditi Vashist (Mathematics and Physics)



Mentor: Cesar Silva, Director, SMALL Williams College, MA.



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

Aditi Vashist was accepted to Williams College, MA to the
SMALL PROGRAM June 11 – August 9th 2007.


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.

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