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
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.
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!
Almost a year ago now, I was just about to start my first Summer Undergraduate Research Fellowship (SURF) at JPL. NASA had sent out an email to all of their summer interns containing a social media template to announce that we had been selected as NASA interns. Excited to show my NASA pride, I posted it on my Instagram story, unaware of what would come out of this small action.
Hey hey! We’re starting a series where I walk you through my best finds for food and drinks in the Pasadena region, and in the LA metropolitan area. Diners, Drive-ins, and Dives, if you will (although, for copyright reasons we can’t call it that). As you explore your college options, I firmly believe that food and location are more important than your high school guidance counselor may lead you to believe. And I’m here to share my best finds from my time at Caltech with you.
Over the past several months, I have had the opportunity to intern at the NASA Jet Propulsion Laboratory (JPL) under the mentorship of senior research technologist Dr. Xiaoqing Pi. Dr. Pi’s guidance and mentorship has been instrumental to the development and success of my internship at JPL, where I use machine-learning to enhance the accuracy and integrity of navigation and communication signals. In addition to helping me develop an understanding of atmospheric and ionospheric remote sensing and machine-learning, Dr. Pi has often offered his insights on how to improve my researching skills. Dr. Pi was generous enough to take the time to answer a few questions regarding his research and advice for future student interns. I believe many students can benefit from some of the lessons that he has taught me:
The transition period to remote learning was a very uncertain time, especially for research and the Caltech Summer Undergraduate Research Fellowships (SURF) program. Many hands-on projects had to pivot at the last minute to facilitate off-campus contributions. However, many Techers were able to take advantage of the research opportunities offered at Caltech and JPL to make the best out of remote learning and research. To paint a picture, I’ve interviewed a few talented Techers for some insight on what researching from home looks like for them.