Scattered Thoughts

by Mandar Juvekar

Since the start of HSSP many scattered thoughts vaguely related to the program have called my brain home. Some serious, some not as much, these thoughts have helped keep me entertained through an otherwise monotonous summer. Here is a quick highlight reel, with the hope that perhaps one or two of them might entertain you as well!

On teaching over Zoom

A fairly normal subject that I’m sure many people involved in the program must have thought of at some point: I have been thinking a lot about the experience of teaching (and learning) via Zoom. In my opinion, one key thing that Zoom misses that makes a real classroom more exciting is the ability to have small-group conversations during a class, both for learning-related and non-learning-related purposes. Zoom completely takes away the possibility of teachers asking students to “chat with your neighbors”, or for students to clarify small things with those sitting next to them, which I think plays a much bigger role in learning in a classroom than most people would think. I also think that while potentially distracting, the ability to carry out mini side-conversations (even just a few words here and there) among students during a class is important in helping students stay engaged. Being physically located near other students is another thing Zoom misses out on. In my opinion, physical proximity brings two things to a classroom. First, it forces students into each others’ fields of view, which causes an implicit “peer pressure” (for a want of better words) to participate. And second, as a teacher it is so much easier to figure out how students are responding when it is possible to (easily) see the students while teaching. A virtual environment which somehow incorporates these things in a sensible manner would be quite interesting to see: some sort of hybrid between Online Town and Zoom, perhaps?

HSSP as a Splash admin at a different school

I am currently the chair of the Splash at my own school, the University of Rochester, and one of the best things about teaching at HSSP is not having to deal with (or even know of) all the craziness that goes into organizing the program. This is particularly the case right now, just a few weeks after one of our own summer programs ended: after four long, busy weeks of answering emails from students and teachers, rushing around on the days of the event, and dealing with technical difficulties, I now have the chance to be on the other side and just focus on teaching something I love. I think this was a much needed break for me. Being an admin at a much newer (and smaller) Splash, teaching at HSSP has also been a great way to learn from a larger program and get ideas that could potentially be incorporated in our own programs. Having experienced what goes into organizing such an event, I really appreciate all the hard work the HSSP team puts in to keep the program going. You all are truly incredible. Kudos!

On self-similar structures on a plane

A very short and unrelated thought: Highly Self-similar Sharks on a Plane always reminds me of a math research project I’m working on right now which studies extensions of Paul Erdős’s “Distinct Distances Problem” in combinatorics. The problem is that of finding an (asymptotic) lower bound on the number of distinct distances any n points in d-dimensional space can determine. The problem was almost entirely resolved in two dimensions (the plane) by Larry Guth and Nets Katz in 2010 using what came to be known as the Elekes-Sharir-Guth-Katz framework. While I’m not going to go into details (this isn’t a math blog, after all), it turns out that one of the key ideas in this proof was to study quadruples of points such that two of the points can be transformed into the other two via a proper rigid motion: highly self-similar structures in the plane! Yes, I know this connection is a bit of a stretch, but I thought I would put it in here anyway. If you are interested, I really encourage checking out the distinct distances problem, the many fun proofs for different lower bounds that came up over the years, the ESGK framework, and the Guth-Katz result in the plane 🙂

On naming HSSP

Just the other day, I was talking to a friend of mine who is an ESP admin about the naming of HSSP. He explained to me how the trickling of HSSPs into the academic semesters caused the full form to change from High School Summer Program to High School Studies Program, and how that name ran into issues once middle schoolers were allowed into the program. The conclusion of this conversation was that now HSSP is simply named pretty much anything that (a) sounds good, and (b) makes for good artwork. This got me thinking. If the full form of HSSP drives artwork, couldn’t a particularly evil director name the program with the sole purpose of making artwork design as difficult as possible? One way to do this would be to have a name so boring that designing interesting artwork would require a particularly brilliant creative spark. Perhaps something like Hand-drawn Squares on a Sheet of Paper, or Highly Self-Similar Spheres on a Plane. Another way could be to have a name that represents something literally impossible to visualize directly. Perhaps something abstract, or maybe something in 4+ dimensions? Maybe High-dimensional Structures Shine with Purity or Honesty Shows Sincerity and Prudence? Of course, it is very possible to turn these names into beautiful artwork, I just think that might take a lot more effort and creativity. Now don’t get me wrong, I bear no malice to the ESP art team. Quite the opposite, in fact: I really appreciate their work, and am a big fan of this summer’s shark-based art!

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