## Intel STS, Day 4

(21:31) I just came back from a very exciting and fun day. Today we had our poster judging, and mine actually went pretty well. I waited for a judge for about an hour, and my first judge was a mathematician. He began by asking me whether $\mathfrak{q}(\mathscr{C})_n$ (in my paper) was trivial, and I asked if he meant contractible. I told him that it indeed was in the version of the draft that I submitted to STS, but I actually ended up fixing that problem.

He then asked me about what the algebraic K-theory of a general ring is. I went into my usual mode, and said that if $R$ is a ring, then we can consider $\mathrm{Mod}^\mathrm{proj}_R$ is the category of projective $R$-modules. Then this is an abelian category (whose definition I reviewed), and you can choose an exact category structure. Then you do the Quillen Q-construction, and taking the geoemtric realization of the nerve of this category gives a topological space, whose homotopy groups are the K-groups.

I was then asked to compute $K_2(\mathbf{Z}/p\mathbf{Z})$, and I couldn’t do it on the spot, so he asked me to compute $K_1(\mathbf{Z}/p\mathbf{Z})$, and I initially said $\mathbf{Z}/p\mathbf{Z}$, but then later said that it was actually $\mathbf{Z}/(p-1)\mathbf{Z}$, which comes from Quillen’s computation. Then he asked me what $K_1(\mathbf{Z})$, but when I couldn’t do that on the spot either, he pointed to a table I made on my poster and asked me what it meant (it was a table comparing $\mathbf{Z}$-algebras, $\mathbf{S}$-algebras (algebras over the sphere spectrum), and $\mathrm{Sp}$-algebras).

Once I began talking about the sphere spectrum, he asked me what a spectrum was. I told him the usual definition, and he asked me why spectra were “stable”, and I replied the Freudenthal suspension theorem. He also asked me for an example of a spectrum that’s not $\mathbf{S}$, and I said $\mathrm{H}A$ for an abelian group $A$. He asked me to define Eilenberg-Maclane spaces (I told him the definitions via cohomology and the usual homotopy groups definition), and show that they form a spectrum (which was easy). Then he asked me to prove that Eilenberg-Maclane spaces are unique up to homotopy equivalence, which was slightly harder since I didn’t recall the proof. I said that perhaps we should start with a general construction of Eilenberg-Maclane spaces, choose a $(n-1)$-connected space (I realized now that I should have said that this could be a wedge of $n$-spheres), and attach cells to kill higher homotopy groups – denote this space by $X$. He then told me that that was very good, but that I hadn’t proven that they’re unique up to homotopy equivalence. I said that we might have to use Whitehead’s theorem, which says that if $X,Y$ are CW-complexes and the map $X\to Y$ induces isomorphisms on all homotopy groups, then $X$ and $Y$ are homotopy equivalent (he asked me to find $X$ and $Y$ such that their homotopy groups are all isomorphic but $X$ and $Y$ are not homotopy equivalent, and since I wasn’t able to think of such an example fast, he asked me to continue); so in this case, we would have to define a map $X\to K(A, n)$, which he then asked me to do. Very soon after asking me the question, though, he said that he had run out of time, and that I shouldn’t worry because he was asking me incredibly hard questions. (After I had my next judge, I went up to him (this judge) and told him $K_2(\mathbf{Z}/p\mathbf{Z})$ is $0$ since $2$ is even, and that $K_{2n}(\mathbf{Z}/p\mathbf{Z})$ is isomorphic to $0$ in general.)

My next judge was also a mathematician, who immediately asked me if every category is an $\infty$-category, and I said yes, canonically. I told her that this is analogous to every set being a category (although a better example might have been every category being a $2$-category); but it was a very easy question. She then asked me what differed between my definition and Barwick’s definition of K-theory, which was also a straightforward question. She asked me to compute $K(\mathrm{Fin}_\ast)$; a simple argument about $\mathrm{Fin}_\ast$ being the free symmetric monoidal category on a point, and since $\mathbf{S}$ is the free symmetric monoidal groupoid with inverses on a point, it makes sense to say that $K(\mathrm{Fin}_\ast)$ is $\mathbf{S}$. She then asked me how abelian categories relate to stable $\infty$-categories, and I described the derived $\infty$-category construction. I was then asked to explain my project as I would to the media, and both of us laughed a little (this was nice, because I had messed up very bad in the questions in her room yesterday). After I began comparing the letters P and O and saying that they’re “homotopy equivalent”, and saying that O can be viewed as being a “topological ring” in a very crude sense, etc. she said that that was a very great explanation, because it conveyed my project, and showed that I had intuition for these structures.

My last judge was from NASA, and he asked me to explain my project at a high level, and dumb it down when I needed to. He then asked me what applications there were, and I said something about quantum field theories (which was all that popped into my head at the moment). After my whole talk, he said that everything was very eloquent, and said that he looked forward to meeting me again in the future.

We then went to a very nice sandwich place on the next block, which was very good. I enjoyed eating the sandwiches, although I couldn’t recall whether I had ordered a sandwich or a salad (we fill in forms for lunch the day we check in), and had to wait for quite a while for them to finish making the sandwiches. It was a lot like Subway’s, but a lot cooler because all 40 finalists were hanging out and high-fiving each other because judging was finally over! (But that wasn’t entirely true; some of the judges came in to ask finalists questions during public exhibition.)

And it was finally time for public exhibition! I was very excited, but expected to get relatively few people since I have a technical math project, which was only true for the first 15 minutes. Many people came up to me just to pick up a card, and others talked with me for a long time. All of them had varying levels of mathematical knowledge, and I met a lot of MIT people (incoming freshmen and those already at MIT), and Po-Shen Loh. The bulk of people were generally asking me questions like where I was going to college (I really don’t know, because I got into Caltech during STS). Some asked me if I was ready to go to college at the age of 15, and I said that I think I was, especially if I was going to a school with people who are as awesome as the other finalists, because I’m very comfortable around everybody else.

One of the most memorable and happy memories from the public exhibition was a small six or seven year old, who came with her mother and her younger brother. She was initially very scared to talk to me, but then asked me how and what I did my project on; although there were at least fifteen such kids who asked me such questions, she was especially memorable because she had a passion for science in her. After she finished talking to me, her brother began pulling on her mom’s leg, asking her if they could go buy food; but the girl (whose name I forgot) asked her mom if she could go and see more projects. It made me very happy to see such a young kid with a passion for learning! (Soon enough, though, my throat started hurting, because by 4:00 pm, we had already talked for about 5 hours.)

When checking the schedule, I saw that I’d be meeting with Caitlin Sullivan at 9:30 to discuss my speech (since I’m the Seaborg winner); however, they hadn’t told me how long the speech was, and whether I needed a full draft due by 9:30. One of the SSP staff told me that it was a 10 minute speech (ten minutes?!) and I needed a full draft by 9:30, which Caitlin then told me was not true (phew). I did end up writing a draft though, with a lot of help from my roommate, Josh.

We then changed into casual clothes, and went bowling, which was truly one of the most memorable events of STS (although I think I say that about all the events, which is true!). I am very bad at bowling, but the whole experience was very fun(ny) because there werre others who had little to no experience, and near the end, we just randomly bowled (because nobody else actually was playing) and missed everything. Our lane had someone who was much better than us; by the sixth round, all of our scores averaged about 20, and hers was 65, so it was really a (very hilarious) competition for the inexperienced.

After coming back to the hotel, Caitlin told us about how we’d practice for the gala tomorrow, and certain rules, etc. After she was done, and she had left to the other office with a draft of my speech, 10 of us gathered in a circle to play Cards against Humanity – but we had to close it down because it would have looked really bad if some SSP staff walked in and saw us playing such a game. We then played Taboo, which was really fun – I was partnered with Vikul, and although this was the first time playing the game, we ended up coming second (by one point!). Tomorrow we’ll be visiting Arlington, VA, for a TechShop to help middle-schoolers build stuff. I think it’ll be fun, but with my clumsy hands, it’ll probably be more funny. (I don’t know if I’ll be able to post anything for the remaining days because I’m usually incredibly tired after the day’s events.)