Tanya Khovanova's Math Blog

PRIMES-USA: A new MIT program for talented math students from across the country.

I’ve been working as a math coordinator for RSI, the most competitive summer program for high school juniors. RSI arranges for these select students to do scientific research. I only work with kids who do math, and usually we have a dozen of them. Every student has an individual mentor, usually a graduate student, with whom they meet daily. I supervise all the projects and meet with each high school student about once a week. My job was described as “going for the biggest impact”: when the project is in trouble, I jump in to sort it out; when the project is doing well, I push it to further limits.

RSI is a great program: kids enjoy it and we produce interesting research. My biggest concern is that the program is too short. The kids do math for five weeks and they usually approach a good result, but at the end of RSI we generally see just a hint of what they could truly achieve. Kids who continue to work on their own after the program ends are more successful. Unfortunately most of the students stop working at the end of the program just as they are approaching a big theorem.

I discussed this dissatisfying trend with Pavel Etingof and Slava Gerovitch and we decided to do something about it. Pavel and Slava conceived and found funding for a new program called PRIMES that is similar to RSI, but runs for a year. From February through May, PRIMES students meet with their mentors weekly. In fact, we require on the application that the students commit to coming to MIT once a week, thereby limiting us to local students. Theoretically, someone from Detroit with a private jet who can fly to MIT weekly would be welcomed.

Before the first year, we wondered whether the smaller pool of local students would be weaker than national and international RSI students. To our delight, that wasn’t the case. In the first year we got fantastic students. One explanation is that PRIMES is much more flexible. We do not mind when our students go to IMO in the summer or to math camps or when they go away on vacation with their parents. As a result, we get students who would never apply to RSI because of their summer schedules. Our PRIMES students have won so many prizes that I do not remember them all. We post our successes on the website.

Our success in PRIMES suggests that there are likely many talented kids in other states who never even apply to RSI because of a scheduling conflict. This led us to try to adapt PRIMES to national needs. So we created a new program called PRIMES-USA that will accept students from across the country. We will work with them through Skype. These students must commit to travel to MIT for a PRIMES conference in May. Because this is our pilot program, we will only accept five students.

Thank you to everyone who helped me to find a host for my Number Gossip website. Some readers and friends even offered me free hosting on their servers. I decided to pay for hosting because I have many specific requirements and that might be a burden on my friends.

On the basis of my readers’ recommendations, I chose Dreamhost as my new webhosting provider. I apologize for the interruption in the flow of the gossip. I know that many people use Number Gossip for birthday gift ideas. I can tell you that on my previous birthday, you could have congratulated me on becoming prime and evil.

I visited Raymond Smullyan on my way home from Penn State. We went for lunch at Selena’s Diner. What do two mathematicians do during lunch? Exchange magic tricks and jokes, of course. Here is a story Raymond told me:

Raymond: What is the date?
Stranger: I do not know.
Raymond: But you have a newspaper in your pocket!
Stranger: It’s no use. It’s yesterday’s.

Consider central symmetry: squares and circles are centrally symmetric, while trapezoids and triangles are not. But if you have two trapezoids, which of them is more centrally symmetric? Can we assign a number to describe how symmetric a shape is?

Here is what I suggest. Given a shape A, find a centrally symmetric shape B of the largest area that fits inside. Then the measure of central symmetry is the ratio of volumes: B/A. For centrally symmetric figures the ratio is 1, and otherwise it is a positive number less than 1.

The measure of symmetry is positive. But how close to 0 can it be? The picture on the left is a shape that consists of five small disks located at the vertices of a regular pentagon. If the disks are small enough than the largest symmetric subshape consists of two disks. Thus the measure of symmetry for this shape is 2/5. If we replace a pentagon with a regular polygon with a large odd number of sides, we can get very close to 0.

What about convex figures? Kovner’s theorem states that every convex shape of area 1 contains a centrally symmetric shape of area at least 2/3. It is equal to 2/3 only if the original shape is a triangle. That means every convex shape is at least 2/3 centrally symmetric. It also means that the triangle is the least centrally symmetric convex figure. By the way, a convex shape can have only one center of symmetry.

After I started writing this I discovered that there are many ways in which people define measures of symmetry. The one I have defined here is called Kovner-Besicovitch measure. The good news is that the triangle is the least symmetric planar convex shape with respect to all of these different measures.