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48:15thinker, whatever you want to call it, that I am. I'm going to start with mathematics, and then we're going to go into physics and biology, okay? So, the first mathematics book, Baby Rudin, Principles of Mathematical Analysis. This is called Baby Rudin because I cried like a baby while I was reading this thing. Um this was my first semester at Princeton University. Um there was a course called Math 215. Print Um it was it was called analysis in a single variable. That was the tagline. And then And then the description in the syllabus was analysis in a single variable. The I get I get to
48:57the first lecture, and it's Peter Sarnak, who is a legend in mathematics. He's got this thick uh South African accent, and he goes, "What is a number?" And I'm like, "What What are >> [laughter] >> What are we talking about?" And right? What What's going on? And And And this is the book. This is the textbook for it. Um it was it was it was an insane course. It was one of the most formative courses in my life. Um the first lecture we get into what what it means to be a number, what it means to be a real number, that stuff that I was talking to you about countability, and the fact that real numbers are uncountable, but rational numbers are countable. And And there's functions that are
49:39continuous everywhere, but you can't differentiate anywhere. It's like really, really strange things. Basically, it's it's the foundations of calculus. Because when Newton invented it, like every good physicist, he was very hand-wavy. He was like, "Uh limits approach, you know, and it's like obvious." And for most smooth functions that like we deal with on a day-to-day basis as physicists, it's totally fine, right? And as a physicist, I can totally understand why he would choose to present calculus that way. But pretty soon, like within about 50 years of him saying all that and 100 years of him saying all that, mathematicians start saying, "Well, what about this? And what about that?" And
50:20turns out there's like functions that are completely non-intuitive. Uh there's functions that are continuous everywhere, meaning I can take a pen and I can draw it out theoretically, but I could never find a slope anywhere on that function. That's a really weird thing to do. Baby Rudin, Principles of Mathematical Analysis. Honestly, a great time. Once you Once you get over the trauma and everything, it's it's an amazing amazing book. >> Okay. I'll take I'll take it I'll take it here actually. >> Yes, yes. All right. Next one is actually um it's my favorite course that I ever took at Princeton
51:01University. My favorite course that I ever took at Princeton University was a mathematics course. It was not a physics course, even though I was a physics major. It was complex analysis. The author, Elias Stein, was actually my professor at the time. Um he's also the PhD advisor of Terence Tao. >> Ah, so we've got a lot of Terence Tao in the comments lately. >> Yeah. Um this book is absolutely amazing. It is not like Baby Rudin because this is gentle. This brings you in. >> [laughter] >> And it And it treats you fairly. And it treats you like you're a human being. Okay? And it it goes on to show some of the most beautiful things in
51:43mathematics. It starts with the invention of I, the imaginary number, and it goes on to talk about the Cauchy-Riemann equations, how to take integrals in in the complex plane. It talks about um the the Riemann zeta function. It talks about the prime number function, which is how many how many prime numbers there are as you get higher and higher. It turns out it's like related to the log of how high you get. Um And you know um have you ever heard that like weird math thing that's like all of the numbers added up, like 1 + 2 + 3 + 4, all of them added up to infinity is
52:24-1/12? >> No. >> Okay, that's something that we can get get get into. But it turns out the sum of all the natural numbers is -1/12, and this is the key to understanding that, complex analysis. It has to do with holomorphic functions and the fact that everything is okay in complex analysis. Stuff that stuff that is not allowed in real analysis becomes allowed in complex analysis. It's absolutely beautiful. It's my favorite thing. >> If you would like us to cover it, let us know in the comments. >> Finally, the next one, I lost the cover to this, but this is also Elias Stein. This is part of his analysis two series. This is Fourier
53:04analysis. This was instrumental for physics in general, right? Understanding Fourier analysis. There's a lot of really non-intuitive things about Fourier analysis, like the fact that like I can add up a bunch of sines and cosines, and I can get a convergent function, but then the derivative of that function can be divergent, which is really weird. Like the derivative will blow up, but the function itself will approach like a really nice steady value. Again, really weird things. Th- This was a This was a great course, but this was actually the last course in higher mathematics that I took at Princeton University. I also took at the same time algebra with Galois theory. The textbook is not here because I don't
53:46want to I don't want to talk about that experience. >> It is not getting added to the collection. >> getting added because I don't want to talk about that experience. That's a class that I PDF'd. >> Yeah, yeah, yeah. >> Yeah, yeah, yeah. Remember when we could do a pass or fail? I passed. [laughter] >> I did not get a D or an F. >> D, no F. >> But I I don't want to think about what the letter would have been had I not done that. >> This is not a file that you can open in Adobe. Having a class that's PDF >> Yeah. >> is not maybe a Princeton specific thing, but >> Usually it's just pass fail, but Princeton's >> one of those like, "I'll give you a D." >> [laughter] >> So you got to you got to you got to work a little bit. >> selection process. >> yeah. >> [laughter]