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Analysis polishing

Joined
2/14/23
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Sometime after I started this last semester of my undergrad, I realized analysis was more important than I originally thought it was. I also realized I was on track to be very deficient in analysis knowledge; luckily, I have a very flexible degree program and can shore that up over the next two years.

Now, the point of this post:

To help me along this path I bought and have been working through Abbott's Understanding Analysis, I'm is about two weeks in and I am into the sixth chapter.
My question is, how well polished do my analysis skills really need to be? I don't have any idea since I'm working through it on my own. On Dr. Duffy's continuum of "get it working -> get it right -> get it optimized," I've gotten most everything working, and a decent portion I've got right, but pretty much all of it is far from optimized (of the stuff I've covered). Granted, it's been two weeks, I obviously didn't expect it to be done yet.

I'll use my Math Stat courses as an example, we used Engelhardt and Bain's 'Introduction to Probability and Mathematical Statistics' for the two course sequence (I think, at least we did for the second one). We would only be assigned 1/3-1/2 of the problems as recommended practice, and then the prof would draw on some other source for the HW. Of course, we worked through so many other problems in class that is made up for a lot of the discrepancy, and the course was more getting down main methods that can be applied to many different scenarios. By the end, I could answer almost all problems thrown at me straight up, but I'm honestly not sure if we just skimped on the really tricky problems or not or how she handled that. I suppose I could check. I'm not sure how it should work with this analysis content though, it's a different type of thing.

Should I treat this walkthrough like that, course, or do I need to have every proof presented/asked for polished to a T for future work and it needs to all be down pat? I'm not sure what the standard is. Analysis is more about learning all these concepts so you understand the foundations of mathematics and less about learning a format to figure out sufficient statistics or prove a distribution is a part of the REC. As an example, I know another member on here who said he went through the first four chapters and then just moved into stochastic calc and basic measure theory. Very different options out there.
 
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As an aside, I focused on this heavy at the start of the summer before my internship starts, knowing it will be easier to polish it during that if I've seen everything and read through all the problems/solutions to each section (having worked through several each section). I plan on moving into the C++ course once I get this mostly out of the way, and I'm excited for that, but don't know when the line of diminishing returns on analysis prep is.
 
By 'analysis' you probably mean 'real and complex analysis' and even maybe numerical analysis. If yes, then they are fundamental. Some topics that I think are important:


esp A-D (on the critical path imo)


real useful stuff and you can use it with C++, Python.



You said you were going to St. Andrews (Scotland), so you should try sit in on a course there.
BTW they have a great golf course there.
 
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By 'analysis' you probably mean 'real and complex analysis' and even maybe numerical analysis. If yes, then they are fundamental. Some topics that I think are important:


esp A-D (on the critical path imo)
By this courses framework, I am good (or at least 'working') up to part E and then I've got some assorted portions of the rest down pretty well (Bayesian Statistics, Prob. Rand.Variables, some but maybe not all of the inner-product spaces section). So at least I've got a solid--working ability on the 'critical path'.
I've covered very little of this stuff, but I have already mapped out the courses that cover a lot of it, and I'll be adding them into my schedule.
I've done none of this, I'll be starting the coding track with the C++ courses.
You said you were going to St. Andrews (Scotland), so you should try sit in on a course there.
BTW they have a great golf course there.
I'd love to, but the course on Real Analysis (focus on integration theory, power series, convergence of functions, some metric space) is offered at the same time as a course on Markov chain and processes and I'm not sure which is better at the moment. I'm leaning towards the MC course. The book I'm working through seems to cover what the RA claims to be over (there is a further course on RA but its not offered next semester), and I've got an independent study in stochastic calculus lined up for the spring when I get back, dependent on my self study of 'Understanding Analysis', and taking the MC course would lead in to that really well.

Regardless, I should be able to take courses on differential equations (some ODE and PDE), linear mathematics (algebra and linear operators/special functions), and Bayesian Inference (theory + applications, computational, MCMC), so I'll be doing ok.

My education is a tad unbalanced. On the statistics side I'll have covered all the undergrad level theory I've seen anywhere after next semester (excepting measure theory), so I'm 1.5 years 'ahead'. But on the mathematics side I think I'm 1.5 years behind. Luckily I'm in a flexible degree program so I can shore pretty much all of this up, numerical/computational stuff and analysis.
 
I realized analysis was more important than I originally thought it was. My question is, how well polished do my analysis skills really need to be? I'm excited for that, but don't know when the line of diminishing returns on analysis prep is.
In my opinion, I really don't think real or complex analysis is all too important if your goal is to be a buyside QR and especially if you're trying to be a QT. I feel the focus is much more on practicality and applications (more stats/coding driven) rather than very rigorous and theoretical mathematics. I just can't really see the connection between being really good at analysis and making the firm a lot of money. That being said brushing up on analysis won't really hurt, there's just probably a lot more low hanging fruit with more returns.
 
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In my opinion, I really don't think real or complex analysis is all too important if your goal is to be a buyside QR and especially if you're trying to be a QT. I feel the focus is much more on practicality and applications (more stats/coding driven) rather than very rigorous and theoretical mathematics. I just can't really see the connection between being really good at analysis and making the firm a lot of money. That being said brushing up on analysis won't really hurt, there's just probably a lot more low hanging fruit with more returns.
Well, my stats knowledge is my strong area at the moment (though I will be filling/adding on some areas: for instance, my time series course stoped just before ARCH/GARCH methods) and like I said I'll be starting the coding route this summer as well.

I do think that at least some analysis is needed, maybe not in everyday application but more so that if someone (coworker, academic author of a paper) starts making analysis type arguments I'm not left sitting there twiddling my thumbs and looking useless. The way I see it, if I don't know analysis I'll always be subordinate in a technical conversation to those who do. It might not come up all that often in practice, but if it does and you have to run to your co-worker/boss then everyone knows what the mathematical pecking order is. I don't want to be working with a bunch of Ph.D's who have to dumb it down for me. I need to at least be conversational in this stuff. It will be much, much harder to cover this stuff later. Besides, If I ever want to exit industry and go for a Ph.D or teaching position, then I need to have this stuff down anyway.

Also, I've seen analysis type arguments pop up in samples from interview question books. Haven't gotten into those yet, working on the pre-requisites.

There definitely is lower-hanging fruit (coding, coding is my weak link atm) but if I want a fully well-rounded education in maths/stats I can't skimp on this area. I have two more years and enough degree flexibility to do it, so I might as well.
 
It's good to hear that it won't be the make-or break though. I'm happy with it being good to know but not the bread and butter, that suits my current skill level I guess 😂. I'll be working on understanding it anyway - but don't you worry I'll be staying/straying deeper in the coding/numerical methods/stats side as my focus. My analysis study is planned to be more of a consistent on-the-side thing right now just because I want to be good at it so there isn't a gap in my understanding.
 
Pretty spot on. Good video, not sure who that is but I get the impression he has seen a lot.
Knuth is a giant in the CompSci world, probably best known for his "The Art Of Computer Programming" (TAOCP) books.
Bill Gates once infamously said, if you've read his books, send me your resume.
 
I also like

“It is better to solve one problem five different ways, than to solve five problems one way.”​

― George Pólya

“The heart of mathematics consists of concrete examples and concrete problems. Big general theories are usually afterthoughts based on small but profound insights; the insights themselves come from concrete special cases.”​

Paul Halmos


The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics; and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.

- John von Neumann
 
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