## 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.

(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.
(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.