Jan 25, 2024

We started our busy day after breakfast with the first of our two individual contest rounds. In these rounds, students are given seven minutes each to solve difficult questions working on their own. There are five questions in each round and each question is worth 10 points. The fun for those of us watching the students comes when we collect their answers, announce the correct answer, and hear them cheer if they get the question correct. The questions are quite challenging, but our participants continued to impress us with their ability to solve many of them.

We took a break from contest activities to hear a presentation, and the first up today was Mourya Reddy Narasareddygari, who started her lecture with a discussion of virtual reality and augmented reality. Mourya told the students about how many of the virtual/augmented reality applications - games (such as Pokemon Go), medical (such as running through/rehearsing surgical procedures), and retail simulation (such as applications allowing potential customers to see how furniture would look in a room) - have undergone major changes in last 2-3 years alone.

The main question was: How is math related to virtual and augmented reality? One primary way is that coordinate geometry is used to locate and characterize moving objects in space - including angles and rotation. Students saw a demonstration of Tinkercad, an application used to build 3-D objects for animation (or 3-D modeling) on the x, y, and z axes. Starting from a 2-D rendering, in this case of blue stars, z-axis depth was added to create a 3-D representation. Tinkercad is a very basic rendering tool that does not require any specific computer coding knowledge or proficiency to be able to use it. If you’re curious, please check it out at tinkercad.com!

After lunch, we returned to the Social Sciences building to complete another set of 5 challenging individual contest questions. It was an exciting round!

Next up was our final lecture of the day, presented by Paul Ellis, professor of mathematics at New Jersey’s own Rutgers University. Paul’s presentation was on knot theory - if you are not familiar with the topic, it is fascinating, but not easy to explain in this newsletter format without extensive graphics and/or video exhibits. Paul started by posing two immediate questions: 1) can you take a trefoil knot and turn it into the “unknot” (a simple closed loop of rope); and 2) is it possible to turn a right-handed trefoil knot into a left-handed trefoil knot? The students had mixed opinions, so Paul gave them actual ropes to try - nobody could pull off either of the desired transformations, so the answer to both questions seemed to be no. But how to prove it mathematically?

Things get pretty complicated at that point, including a discussion of tricolorability, which is present if, in a diagram of a knot (strands and crossings) there are 1) at least two colors used; and 2) at each crossing there are 1 or 3 colors but never only 2. Paul explained that tricolorabilty is an “invariant” characteristic, so no matter what moves (RM - Reidemeister moves) are made it will not change. The discussion continued deeper, into the Jones polynomial among other things. The Jones polynomial provides a mathematical way to prove that two knots are different; if one compares 2 knots and the Jones polynomials are different, the knots are definitely different. (Though if Jones polynomials are the same that doesn’t prove they are same knots.) If your head isn’t spinning at this point, you must be a true connoisseur of the mathematics of knots!

Finally, after dinner this evening was the first night of our two-evening Talent Show. I am the master of ceremonies for these shows, and I introduced a variety of acts this evening---we had singers, pianists, guitarists, a violinist, and even a piano piece for four hands played by two brothers who are at camp this summer. I am always amazed at the many talents our students possess besides their mathematical talents. I myself try to inject a little humor into the show by telling jokes between student performances (some might say VERY little humor!), and whether I get laughs or groans it is fun to see the students’ reactions. I am happy to report that the overall atmosphere was both festive and supportive, and that on more than one occasion the students and counselors in the audience saluted a singer with synchronized swaying of their cell phone flashlights to the rhythm of the song being performed.

One other great piece of news - they fixed Decker Hall! Tomorrow’s first contest will be in Social Sciences as planned, and tomorrow’s first lecture will be in the Brower Student Center, but after that we will generally be back in Decker, including for the second night of the Talent Show.