## Intel STS, Day 3

(9:36) I just came back from my first round of judging. I had a mathematician and two biologist. The mathematician asked me two questions. The first was to compute $\pi_3(S^2)$. I said that it’s generated by the Hopf fibration, $S^3\to S^2$. Then I said that the fiber is $S^1$, and we get the long exact sequence in homotopy groups, and by the homotopy groups of $S^1$ and $\pi_n(S^n)\simeq\mathbf{Z}$, we get the desired isomorphism $\pi_3(S^2)\simeq\mathbf{Z}$. He then asked me to construct the Hopf fibration. I didn’t recall the exact construction, so he gave me a hint: namely, consider the dimension of $\mathbf{CP}^n$. I said that in the case $n=1$, the dimension is $2$, and by the CW-structure, it’s basically $S^2$. Then recognize that $\mathbf{CP}^1$ is the collection of lines through the origin in $\mathbf{C}^2$, and $\mathbf{C}^2$ is $\mathbf{R}^4$. By definition, $S^3\subset \mathbf{R}^4$, which leads to the Hopf fibration. Then he asked me the following question: if you have $100$ switches, all turned on, and you turn on all the switches labeled as multiples of $2$, then toggle those which are multiples of $3$, etc. once the whole thing is done, which ones are left? I wasn’t able to think properly, and he asked me to consider $25$; I then realized that each prime factor must appear at least twice in the prime factorization. He asked me to consider $27$, and I then realized that the ones left turned on are the squares.

I then had a biologist, who asked me how many legs a dog has. (4, obviously) Then he asked me about the origin of life. I described about how life went from unicellular stuff to things living on land. I was then asked about why oxygen was important for life, and I said that you need oxygen to get energy for organisms to function. (I should have mentioned ATP, but I didn’t.) My last judge told me that people had a concern for vaccines affecting autism, and I had to design an experiment to test that, without worrying about ethics. I said to take as control babies and not inject them with the claimed vaccines, and then inject into other babies and inject them with the claimed vaccines. The other biology judge was shocked; but the other judge reminded him that I was told to not worry about ethics. He then asked me which part of the cell contains genetic information (the nucleus). He then asked me under what circumstances would vaccines not be given, and I said that if there’s bacteria which are potentially helpful but we didn’t know about their helpfulness. I actually finished well before time, which was quite awesome, and we laughed a lot at the end.

(11:35) Just came back from my second round of judging. My first judge was from NASA, and he asked me to explain why space suits are important, and key features about them. I was also asked to explain improvements which I’d make to the current design – I just said that the bulk of the weight was probably electronics, and the size would decrease and speed would increase, although I didn’t actually say any key additions I’d make (oops). I was then asked about tidal waves, and he asked my why they occured. I talked about the gravitational pull of the moon, and then explained high and low tides. He then asked me why there are two high tides a day. Think of the earth’s water bodies as being deformed from a $2$-sphere to an ellipsoid; this gives the required answer. He then asked me about snowflakes, and why they were formed. I told him that water, in gaseous state, would condense. When he asked me what object they would condense on, I told him that water condensing on dust particles gives fog, but I didn’t know about clouds – and he said that they did indeed condense on dust, and we laughed a little. He then asked me why snowflakes form a hexagon, and I said that it’s maybe because of the crystalline structure of ice, which also explains why ice is less dense than water.

My second judge began by pulling out a big tube filled with a liquid, which she said I could assume was water, and inside the tube there were small glasses in the shape of tops,filled with colored liquid. These glasses had a small ring attached to them, which had temperatures labeled on them. She asked me why this worked; what I said was that it was maybe because of vapor pressure, and she told me that what was inside the glass didn’t matter. I then suggested that perhaps as temperature increases the volume of the air inside the glass would increase, which was maybe the reason for why they’d rise or fall. I think that was absolutely wrong, though, and she just boxed some words that she wrote on her paper. Whoops. She told me, “OK, let’s go to another question”. She asked me if there were any theoretical limits to a human’s age. I said that I know that brain cells cannot regenerate, and so eventually there’d be too less of brain left for the human body to continue functioning. I then said that maybe other organs could be regrown, via stem cells (she became very happy once I said that). She asked me to explain stem cells, and I told her that they could be used to regrow damaged parts of certain organs, and explained their limitations and how they were different from human embryonic cells.

(3:08) Came back from my third interview, which went really bad. My first judge asked me to describe modules over $\mathbf{C}[x]$, and I blanked out. She then suggested that I should consider $\mathbf{C}^n$; I was able to define the multiplication, but made a very foolish mistake which I then fixed. I then had a judge who asked me some ecology questions, namely whether biodiversity is more at the poles or at the equator. I said that it would be at the equator, but I was probably wrong. She then asked me what about comparing the equator and the temperature zones, and I said the temperate zones because of temperature variation. My last judge asked me about optics – and here I failed miserably. I forgot absolutely everything about optics, and mumbled a lot, probably false things, about light and refraction. At least tomorrow we will be on territory more familiar to us.

(10:27) After the whole judging process, we voted for our Seaborg winner. We had about 4 ties, and in the end, I ended up winning! It’s really exciting; I’m very happy to be given the opportunity to represent the STS class, and on the 75th anniversary of STS. We then went to an awesome Italian restaurant, where Demetri had a toast (with water, of course) to celebrate me winning Seaborg. It was really fun, and the food was really filling. (Michael Li and I were joking around, asking people for martinis.) We then visited the World War II memorial, and the Martin Luther King statue. Then we visited the Lincoln memorial, which was quite ethereal (to use a friend’s phrasing of the whole area). We came back at about 10 pm; and we have to get up at 6:30 tomorrow – and daylight saving only reduces the amount of sleeping time, so I guess I’ll go to sleep now.

(technically day 4, but 1:07 am) I’ve been reading up on a whole lot of homotopy theory, but there’s just so much! I chatted with my roommate, Josh, for a while, and he explained his project to me, while I tried to explain mine to him. I always thought that the straightening of (co)Cartesian fibrations was crazy technical but it really isn’t so bad :P.