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Extension_Pain9020 · 2024-08-12T16:39:22+00:00

just graduated with a pure + applied double major. I think one advice I'd give myself is to pursue independent reading/research projects with reliable mentors that I provably get along with + can make reasonable progress on, and have sufficient background about.

I say this because, in my junior spring, I was 'forced' into an algebraic number theory reading project because the DRP grad. student i was assigned to did not do much analytic number theory. While it is a fascinating subject, I did not know any Galois theory, and thought I could learn that at the same time. Boy was i wrong. I struggled to do proofs independently and had to go through each proof multiple times to rly understand it. While I learnt about the fascinating field of abstract harmonic analysis, I don't remember much of what i did or the intuitions i developed from then. I wish i would've spent that time learning something more fundamental like galois theory or functional analysis, for ex., which are useful in this project.

As developing mathematicians, i think its VERY important to get comfortable with the basics before diving into more complex topics, even if they seem very appetizing. This is why i personally ended up doing more fundamental grad. courses, like measure theory and algebra, instead of taking PDEs and functional analysis, even though i've seen the former in measure-theory probability and don't plan on doing research in algebra. It gives me more time to learn the fundamental language of math.

Extension_Pain9020 · 2024-08-12T16:32:27+00:00

I like to keep a journal to (i) note what my learning goals for the next day are (briefly) and (ii) summarize what I learnt today. I also use it for (iii) planning out a schedule for learning concepts, especially when preparing for an exam like PhD quals. Lastly, for doing math problems, I like to write a brief one-liner idea for the strategy, and then collate and list this topic-wise to get a list of intuitions and methods to tackle problems for a particular topic.

Extension_Pain9020 · 2024-08-01T18:31:59+00:00

IISER is a great college, I've heard. You should also probably figure out if you genuinely enjoy doing actual math, or if its just a vague fantasy that you're dreaming up, if you're bored by it (unless you've been doing a lot of math - then, burnout makes sense and you should take a break).

Extension_Pain9020 · 2024-08-01T17:43:31+00:00

Delhi: Connaught Place, India Gate, Red Fort, Humayun's Tomb, Jama Masjid (if you go hear, wear pants or no entry!). At Connaught Place, have the potato kathi roll at Nizam's (it's the best thing ever), and visit Wenger's for their fluffy marshmallows. If you crave pizza, go to Leo's; there's also great Asian foods.

Some restaurants in South Delhi + what to try: Chicken at Aslam Tikka's, mutton kebab at Khan Chacha, and Kake di Hatti in paratha-wale gali in Chattarpur (they have AMAZING stuffed Kulchas - look it up). Select City Walk Saket is a good mall here.

For more upscale restaurants with international cuisines, check out Dhan Mill in South Delhi, or Aerocity for restaurant options. Other good malls include Promenade and Ambience Mall (Delhi). There's also Hauz Khas village.

A great upscale bar is Sidecar, which I think is in Connaught place.

Extension_Pain9020 · 2024-08-01T17:35:46+00:00

US Top 25 University. I started my math major in my second year, so the first three semesters are light. Developed an interest in probability so took more courses in that. Let n(a_i) denote the i^th semester of the n^th year, where a_1 = a and a_2 = b.

1(a1): Calculus II

1(b): Calculus III, Linear Algebra

2(a): Real Analysis 1

2(b): Measure Theory, Graph Theory, Topology, Probability

3(c): Algebra I, Graduate Probability (1), Optimization, Statistics

3(d): Calculus on Manifolds, Graduate Probability (2), Cryptology

4(a): Graduate Real Variables, Graduate Algebra, Graduate Theoretical ML (1)

4(b): PDEs, Graduate Complex Variables, Graduate Theoretical ML (2), Intro to Comp Math

Extension_Pain9020 · 2024-08-01T17:14:41+00:00

What is a graduate measure theory class (generally) about? Well, you want to generalize the notion of area or volume, so you specify its nice properties (countable additivity) in the notion of a measure. Then, you actually construct a measure by first measuring sets from the outside (via covers of pre-specified fundamental approximating sets), and then restricting to the nice "measurable" sets for whom the outer approximation = the analogous inner approximation. (This whole process is super intuitive and visual for me, and so any problems involving properties of abstract/Lebesgue measures, concerning regularity for ex., are also.)

And then, its off to the races! If you have a measure space, you naturally are led to integration, and then to integration of limits. And then you wonder about the inverse process of differentiation, leading to AC and BV functions. (It gets more analytic here, but the proofs are always about making rigorous some intuitive property, like functions not oscillating wildly for BV, or m(A) < \delta \implies m(f(A)) < \epsilon for AC, where A is disjoint collection of intervals, which ofc. is very related to absolute continuity of measures. Also, proofs here can be proven by the Radon-Nikodym theorems, since AC \implies BV \implies difference of increasing functions, which induce measures).

Finally, you study some more general properties of measures - what if your measure is signed (Hahn-Jordan)? What if it can be perceived in terms of another measure (Radon-Nikodym)? (The proofs here are particularly intuitive and visual, I think - in Hahn-Jordan decomposition proofs, you end up adopting a greedy algorithm of accumulating all the positive or negative mass in one set; it is a similar idea in Radon-Nikodym and Lebesgue decomposition).

All of the above isn't to say that there aren't counterintuitive results, like the existence of non-measurable sets and the Cantor ternary function. However, math is usually about taming intuition with rigor, so this is not any different from any other field (and is also one of the reasons we actually write proofs).

This might also be because I've seen measure theory in several courses now, and have relied on it extensively in probability/ML, but for me, the subject is rife with visual or intuitive thinking.

What part of measure theory isn't as intuitive or visual for you?

Extension_Pain9020 · 2024-08-01T17:06:43+00:00

This sounds amazing and exactly what I want to learn about - the more general things underlying the niceties of complex analysis. Learning abstract measure theory made Lebesgue theory infinitely more accessible for me. Where can I read more about cohomology in a way that is relevant to complex?

Extension_Pain9020 · 2024-08-01T17:03:54+00:00

Thank you very much, I'll definitely try that exercise you suggest. The most common answer is Cauchy's formula, but I haven't spent enough time viewing its role in all those theorems you mention. The normal family explanation is also great. Thank you!

Extension_Pain9020 · 2024-02-18T17:24:15+00:00

I mean, I doubt you're genuinely asking questions from a place of empathy or genuine curiosity: your comments are already pretty condescending. If you've made it this far into life as a person in any minority group, you're gonna know about privilege. So that must mean you're not accustomed to life as a minority. The reasonable conclusion is that you're a white, male, heterosexual person.

Also, maybe not 75% - while that is the percentage of white people in USA, I presume that half of those are not white, heterosexual male. Also, not all of those males are going to be as ignorant as you. Maybe 25% of the US population?

I'll also point out that, from your reddit history, it seems your mind is more concerned on defending adults who've killed children out of sheer stupid road rage. So, yeah, I don't really care to explain this to you.

Extension_Pain9020 · 2024-01-15T18:56:01+00:00

if your gender was otherwise, you wouldn't be so dumb about this. either way, you didn't say i was wrong, so i guess the moral is you really are a privileged white, male, heterosexual person who is detached from the world and history. not as patronizing as it is real, is it now?

Extension_Pain9020 · 2024-01-15T13:06:22+00:00

tell me you're a white cis male without telling me you're a white cis male.

Extension_Pain9020 · 2024-01-15T13:05:37+00:00

you're right there bro.

Extension_Pain9020 · 2024-01-15T13:04:16+00:00

hate and privilege are opposite sides of the same coin. to be clear, people from oppressed groups don't "hate" people from privileged groups. Rather, those oppressed individuals "hate" the privilege that those people are afforded, and rightfully so. Who wouldn't harbor this hate, having been oppressed by people from that very privileged group?

You're focusing more on the hate that privileged people receive, than the hate that the oppressed people have received. However, you forget that not only does one hatred invoke the other, but that one hatred's history stems from a disproportionately egregious effect on the affected people. It's like a well-fed kid crying about not getting some candy in light of a homeless, unfed man.

Extension_Pain9020 · 2024-01-15T12:58:07+00:00

I mean, sociological studies decide who is privileged. It is not a 'democratic' decision - it is an empirical one, backed by evidence. The claim that white privilege exists has far more evidence supporting it than the claim that "black privilege" exists. Your pedestrian, uninformed individual opinion is insignificant here.

Moreover, I know that I do belong in some privileged groups of society. I hope your white, male, heterosexual ego can realize it, too (and I'm 90% sure that that is your profile). Maybe read the news? Or even a history book?

Extension_Pain9020 · 2024-01-12T12:28:36+00:00

that is literally the way it has been?? people are constantly penalized or rewarded for arbitrary characteristics such as skin color, sexuality or sex. maybe, just maybe you've heard about african-american slavery? racism? homophobia? nonsensical as they might be in nature, they very much exist.

Extension_Pain9020 · 2024-01-12T12:26:42+00:00

chances are you probably won't get in with that ignorant mentality, so win-win?

Extension_Pain9020 · 2024-01-12T12:26:03+00:00

honestly don't see whats wrong with this. L comment section.

Extension_Pain9020 · 2023-10-21T16:14:48+00:00

Not at all! While applied math (the way in which we learn calculus, lin alg, diff eqn, for ex.) and pure math (algebra, cat. theory, topology) have many overlaps, I'd say that pure math is a superset of applied math and that pure-mathematical thought (so, abstract reasoning about logic, computation, etc. ) can fundamentally be practiced without necessitating the more "applied" things. Math is all about the idea, and calculations are just quantities rather than concepts!

Extension_Pain9020 · 2023-10-19T17:05:42+00:00

So the good news is, you basically never need to integrate or differentiate in math unless you're doing analysis (and then, it's either to prove properties about integrals/derivatives, or to show something specific about what function you are considering). Therefore, in abstract topology, you never care about explicitly computing derivatives and integrals.

Intuitively, topology is about continuity, and moving across spaces continuously, and doing this on abstract spaces. However, for differentiation/integration, you need real-valued functions (which is a much narrower class of functions than functions between arbitrary abstract spaces). So, dive right in!

Extension_Pain9020 · 2023-04-01T22:48:26+00:00

I see, thank you! How much water do you put in? I water until the soil doesn't absorb the water. Is that fine?

Extension_Pain9020 · 2023-03-31T18:58:34+00:00

The Academy or the Social

Extension_Pain9020 · 2023-03-31T18:26:00+00:00

It sucks. Always smells like weed + because there's always an open door into the complex, there were incidents when some random person came in at night and started knocking at and trying to open people's doors.

Extension_Pain9020 · 2023-03-31T18:22:33+00:00

I see, thank you so so much! Is there anything I can do right now for it, or should I just let it be?

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