This summer, I had an opportunity to work as a Research Student for a professor in the Department of Mathematics and Statistics, University of Saskatchewan, with a focus in Combinatorics (Yes, the same kind of math in Good Will Hunting!). My professor has been doing research for a few years in a surprisingly quite large area of Combinatorics which is related to Self-Avoiding Walks and Self-Avoiding Polygons.
What is a Self-Avoiding Walk?
A Self-Avoiding Walk (SAW) is a sequence of moves with a fixed length which does not intersect with itself. It can be of any dimensions, but we mostly work with two or three dimensions.
Make the end of a Self-Avoiding Walk meet with the starting point, and you have a Self-Avoiding Polygon (SAP)!
Self-Avoiding Polygons in three-dimensions can be seen as DNA strands and hence, can be used as a model to study the behaviour of a real DNA.
In May, I attended an Applied Combinatorics Summer School, which was organized by University of Saskatchewan, Saskatoon. In the school, I learned about various applications of Combinatorics in Physics, Chemistry and Biology. Being the youngest among the students attending the summer school was quite overwhelming, since this school was for masters and PhD students. I got quite a bit of exposure to senior level mathematics, which broadened my perspective. My undergraduate courses do not seem as hard as they used to, now that I have seen more complex mathematics!
June and July, 2015
Besides taking a couple of summer courses at the University, I spent quite a bit of time during these two months to learn about the Pivot Algorithm, a very common method used to calculate Cn, the number of possible walks/polygons, given a fixed length/perimeter.
I then tried to implement what I had learnt by coding my own simulation for small lengths of 2D polygons.
By August, I was knowledgeable enough to assist my professor’s PhD student with his project, “Modelling DNA and type 2 topoisomerase via Self Avoiding Polygons in Different Lattices”. I ran simulations of Self-Avoiding Polygons with different salt concentrations to see if and how different salt concentrations change the probability of entanglements and knots in DNA.
Well, it turns out that DNA tends to stick together and form knots when the solution that it is in, contains more salt.
Working as a summer student at the university and taking three university Economics and Computer Science courses, I definitely feel satisfied with what I have achieved this summer.
To take a break before going back to school in September, my family and I took a road trip to BC, which involved one day in Victoria and one day in Vancouver. I saw a couple of Orcas and dolphins, experienced a 4D movie in Vancouver and swung by Lake Louise in Banff!
Now that September is here, I cannot wait to get back to school and begin my second year. Besides my courses, I plan to take on the role of Associate Editor for my university’s undergraduate research journal. I am also hoping my next summer brings me even more exciting opportunities: I shall be looking for an internship with an IT-based enterprise.