21–30 of 102 posts
7 non soft classes in a single summer is ambitious to say the least.
Yeah, it is, its more of a loose goal, I'm doing research and working on finishing 2 Master's alongside it, so we'll see what happens! I really just want to learn and grow, so if it takes months or if it takes years, I'm in!
Earlier quoted context omitted.
Hey, author here, I really appreciate the feedback. Are there any specific prerequisites to Real Analysis?
Technically speaking, no. Real analysis is pretty self-contained. You basically start out by constructing the reals from scratch and deducing continuity as a consequence of the completeness of the real field. You use a tiny bit of set theory to establish notation and define bounds, and then from there you go into limits, derivatives, integrals and maybe the Lebesgue measure. I wouldn't expect a real analysis course t…
Then take a course in graduate probability which is based on measure theory. Good authors are Loeve, Neveu, Breiman, Chung, among others. So, learn about the cases of convergence including almost sure convergence, a good version of the central limit theorem, proved carefully, the weak and strong laws of large numbers, ergodic theory, and martingale theory.
Learn stochastic differential equations.
Learn some potential theory.
Then do some work in optimization and optimization under uncertainty.
Earlier quoted context omitted.
Hey, author here, I really appreciate the feedback. Are there any specific prerequisites to Real Analysis?
Technically speaking, no. Real analysis is pretty self-contained. You basically start out by constructing the reals from scratch and deducing continuity as a consequence of the completeness of the real field. You use a tiny bit of set theory to establish notation and define bounds, and then from there you go into limits, derivatives, integrals and maybe the Lebesgue measure. I wouldn't expect a real analysis course t…
Not necessarily -- a lot of books just take the existence of a unique set with certain properties as an axiom and call it R.
The main topics of basic real analysis IMO are differentiation, integration, (uniform) continuity, compactness, convergence, etc.; how to construct the reals from the rationals is a side point at most.
This could be personal bias as I just don't personally think that the exercise of constructing the real numbers is very interesting.
Suggestion: Set aside complex variables and do measure theory instead. For the Fourier and Laplace transforms, cover those via measure theory. For measure theory, H. Royden, Real Analysis and the first half of W. Rudin, Real and Complex Analysis. Then take a course in graduate probability which is based on measure theory. Good authors are Loeve, Neveu, Breiman, Chung, among others. So, learn about the cases of conver…
Speaking frankly, it's ambitious to complete even one of these courses in a single summer. Most students would be taking one or two of these in 12 - 16 weeks while doing nothing else but being a student. Accomplishing the same in about eight weeks from video lectures will be more difficult. I would strongly urge the author to slow down, choose a single course they have the background for and work from there. I can't…
https://www.scotthyoung.com/blog/myprojects/mit-challenge-2/
It's amazing and wonderful you can get an MIT education from watching youtube videos of their lectures. I've used them to fill in gaps in my education. Highly recommended, and it's to MIT's great credit that they're doing this.
While at it pick up Julia, another MIT product. esp for applied math it's invaluable compared with python or matlab. Juno is a nice ide, it's gotten a lot better over the last two months.
Can someone please explain how MIT course numbers work? I get that "18" means maths, but why do specific courses sometimes have two digits after the "." and sometimes three, and if the level of study is encoded in it, why does it seem to jump to starting with "6"?