This summer I’m doing my first SURF! It’s a math SURF, and my mentor studies low dimensonal topology, specificallyHeegaard Floer homology.
I’m working on the SURF with another undergrad, so we’ll be working
together to complete different parts of the project. Our main project is to write a
program to compute Heegaard Floer correction terms from a variety of
input methods. For the first few weeks, I’ll be working on several of
the input methods, including getting input from a drawing of a link.
A quick summary: A knot in math is basically the kind of
knot you tie, except that the ends of the knot are connected so there’s
no loose ends. Here’s a picture of a trefoil knot:
Trefoil knot drawn by Knotilus
A link is what you get when you have multiple knots linked together, like the Borromean rings here:
Borromean rings drawn by Knotilus
An
alternating link is a link with a diagram such that if you start
anywhere on the link and follow the strand you're on, the crossings
alternate over, under, over, under, etc. or under, over, under, over,
etc. The trefoil knot and Borromean rings are both alternating.
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One input method for my SURF project is to take an alternating link
projection, shade regions of the link, and then use the shaded regions
and intersections to create a graph (the kind with nodes and edges). For
example, the program Plink
lets a person draw a link with their mouse. Part of the algorithm is
then to load the drawing and shade regions (while keeping track of
them), like in the picture below.
Link drawn in Plink (left) and shaded regions (right)
Based on the shaded regions and intersections, a graph and quadratic form can be calculated for each link projection. The quadratic form can then be used to calculate the Heegaard Floer correction terms.
Since I'm doing a math SURF, I don't have to go to a lab or anywhere specific on campus to work on the project. Although the view from my dorm room is very nice, it can get lonely in the morning since most people are at labs or JPL. So for the first day, I decided to go work in Annenberg (the computer science building), since they have an undergraduate computer lab.
View from my summer dorm room (the blurriness is from the window screen)
For some of the other inputing methods for the SURF project, I'll first have to learn more about
topology and read a little about homology. My list of topics to learn about include plumbed 3-manifolds, Seifert fibered rational
homology spheres, and Dehn surgery. Fortunately I know several other math majors staying at Caltech over the summer!
Just last week, on an unassuming Thursday, the astro(physics) community got served two humongous cakes at once. Both the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) and the IceCube Neutrino Observatory (IceCube) have ground-breaking scientific findings to share with the class!
A friendly desert community where the sun is hot, the mountains are beautiful, and mysterious sounds echo between our walls while we don’t even pretend to sleep.
This summer I had the incredible opportunity to do a 10-week internship at Gilead Sciences in Foster City, CA. For those unfamiliar, Gilead Sciences, Inc. is a research-based biopharmaceutical company focused on the discovery, development, and commercialization of innovative medicines.
