Lots of cool classes

Hello again!

I think this week is the last blog post which makes me very sad. I’ve really enjoyed these last 10 weeks doing research (and not attending school). It’s crazy to think that my time at BASIS has already come. But ok, I know what you’re thinking right now. Where is the math?

I have had quite the hectic week. Differential Geometry on Tuesday was difficult for me and I have not had any time to review.

Starting Wednesday, I left for my visit to Stanford. It has been really fun as I’ve been meeting up with all my friends here and sitting in on amazing math lectures. I’ll give a brief summary of the classes and what I learned in them.

Graph Theory: I had seen most of the material before in math class in ninth grade, but I learned a lot of cool theorems about Hamiltonian cycles ( paths that go through every vertex exactly once in a graph) and their relationship with the degree ( the number of edges attached) of vertices. Graph theory has a lot of cool applications in social networking and computer science as well for those interested!

Abstract Algebra: I had already seen all the material before (thanks to Ms. Bailey), but the lecture was nice. It focused on relations (homomorphisms) between mathematical objects called groups.

Algorithms: Technically this is a computer science class but it’s actually a math class. I learnt about different sorting algorithms and compared their worst case times. I learnt about Big-O and Big-Omega notation which would actually end up helping for my next class.

Analytic Number Theory: After feeling good about understanding the previous three lectures well, I (stupidly) decided to attend a much more difficult class. This class was hard. I understood sentences and bits of proofs, but for the most part I was very lost. Apparently math uses Big-O and Big-Omega notation as well.

Representation Theory: I normally would not understand anything in this class, since it has a lot of prerequisites. However, luckily for me, today was the first day of category theory. I now (2 years after category theory) know the definition of a category. Things Ms. Bailey said way back when make sense now.

All these classes were really cool  (even the ones I did not understand) so feel free to ask me if you want to know anything about them.

As for nim, I’ve wrapped up and cleaned up my program. My equation is still ugly, but my result is somewhat neat. J

That’s all for now!

Thanks for reading~

Inspiration from Science Fair

So this week at ASU was pretty standard. I'm falling a bit behind in differential geometry again. It looks like I gotta hit the books hard this weekend. We've been learning really cool and general theorems about surfaces like Isometries and the Theorema Egregium, which literally translates to " The Remarkable Theorem" (it is actually very cool). Surprisingly this week, there was a lot of computation in proving various theorems, which was something I had not seen too much of in Differential Geometry previously. The lecturer actually took time out of the lecture to tell us the importance of computation. A lot of the theorems we had this week, including the Theorema Egregium, had very computational proofs that ultimately led to a beautiful and abstract statement. This weekend really opened me up to the possibility that sometimes the ugly road in math can lead to the best results.

Another cool thing I did was visit the Arizona Science and Engineering fair this week. I got to see the cool high school projects, which as always were super awesome. I also got to see the middle schoolers and the lower schoolers from our school. Seeing the younger kids projects was cool, since I got to see how developed their creative thinking skills in math already are. I visited a lot of my friends from science fair last year and saw how they've been doing in life and research. The most interesting part, however, was seeing a friend (Abi) and his project. He worked on the secretary problem (https://en.wikipedia.org/wiki/Secretary_problem). Also Wikipedia is not very bad for math so its ok that I used a wikipedia link. In his paper, he tackled different variations of the secretary problem similarly to how I was solving different variations of nim. On his poster, he had fairly complicated looking recursive equations that allowed for picking the optimal secretary in certain cases. One neat trick he did, however, was to convert some of his summations into integrals. This not only made his equations look much less intimidating, but also gave some intuition in his equation. Another point he focused on was looking at the results as the number of candidates in his problem got larger.

I now have a couple of things that I want to try to do to my equations. Maybe this will give the nice results I've been looking for all this time. Maybe it wont ( like all the other things I've tried). I'll keep you updated with how my equations look next week.

Thanks for reading~

The Second Midterm

This week has been interesting at my Senior Research Project site. I spent most of my time studying for differential geometry instead of looking at nim since we had a midterm this week. For nim, this week, I primarily looked at trying to convert general theorems about impartial games into nim as well as trying to look at theorems about other games and how they would translate into nim. I had a lot of general hunches about different games, but proving things has been harder than I originally expected. With a lot of proofs I've attempted, the intuition for the general outline of the proof has been there, but I have been largely unsuccessful at converting these ideas into rigid proof-writing. Theorems such as the Sprague Grundy theorem that make statements of different games often dont provide very simple proofs. Rather than showing directly that a certain aspect of a game holds, the theorems use proofs like contradiction, induction, and casework. Some of the proofs also seem to follow a circuitous path only to get a nice proof at the end. Thats been nim research for me.

Now for most of the week, I have been frantically cramming differential geometry. After falling behind a bit in the past few weeks, I caught up this week and felt somewhat ready for the midterm. Thankfully, this was not actually for a grade. I took the practice midterm and was able to solve about three of the five problems. With the book, I could solve four of them. But one problem took me too long to solve. It wasn't too long, but it just had a proof that went in a direction that I never bothered to look at ( like all the game theory proofs I had been reading).

I kept studying for the midterm, thinking that I wasn't prepared. Suprisingly, the midterm went well (at least for my standards) and I got four of the five problems. The last problem seemed obvious from reading it and turned out to have a simple proof. I just overthought it.

So now I feel like I'm back on track (for real this time) with differential geometry. I've really enjoyed the college experience and schedule. It has taught me to manage my time better, study independently, and read higher level textbooks better. The class has definitely taught me skills I'll need in college next year.

As for nim... I'm a bit behind. I've hit a few roadblocks, but I'm ready to keep trying to solve some of the cool problems I have.

That is all for now. Thanks for reading~