# I need guidance please: what topic/math to study next

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## Summary:

I’m no longer a student but I do self-study still: the internet is my campus, and this post is *your* office hours oh, learned advisors

## Main Question or Discussion Point

@fresh_42 I’m keen to hear your pick.

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Math_QED
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I recommend John Conway's "Functions in one complex variable" for complex analysis. It contains enough details and after 100 pages you know the most important stuff about complex analysis. After that, the book continues studying spaces of analytic functions and more interesting stuff. I would describe the book as "no-nonsense". It comes quick to the point but does contain enough details. Also, you can skip the first 20 pages if you have a metric space/topology understanding of the complex plane already.

Sounds as if you already took some complex variables, but if not, I heartily agree with Math_QED that it is a truly excellent area to study. Incredibly beautiful!

If you're past complex variables, sounds to me as if you're ready for a introductory course in topology, assuming you've been introduced to metric spaces by now. "Topology" by James Munkres is a very good book. (But also the topology book by Hocking & Young has lots of interesting examples as well.)

It's funny, I, too hated algebra (groups, rings, fields) when I was in college, even though I didn't do badly in it. At the time I just saw it as glorified arithmetic, and I have always disliked doing arithmetic.

But now I realize that algebra is endlessly fascinating! It is like the backbone of a great deal of math, and has many beautiful structure theorems. "Topics in Algebra" by I.N. Herstein is a great book, and deep.

benorin, phyzguy and Math_QED
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I will clarify: with regards to complex, I easily aced the undergrad class and subsequently became a reader [HW grader] for it, I enrolled in the grad version but never finished the course (an aside: CCS students are allowed to enroll in grad classes and may drop said class up until the day before the final, partly to encourage us to try hard things w/o serious consequences). But the text I bought for the course was pretty useful (dictionary-sized) it had infinite products and other topics I don’t find in my undergrad text, (but somehow I lost this book). Mind you these events were 20 years ago: use it or lose it (here I’m talking about understanding—not the book), and I lost it. I don’t think I could manage a contour integral without a reference text and time to re-learn. Somethings I still get tho, the basics, analytic continuation, odds and ends really.

I’m presently reviewing my undergrad text on complex to fill in the gaps, but the stuff I’ve doing lately (eg playing with hypergeometric fcns and other special functions of several complex variables) makes me want to get a grad text on it. I learn fast usually.

As for topology, the only experience I have with it is 3 quarters of “baby Rudin” and 3 quarters of “papa Rudin”. Kinda interested, to be honest I view any use of compactness, for example, in a proof as sort of a ninja move. At this point, I’m gonna say sorry for the essay and thank you for your time! Has knowing the content of this post altered your recommendations at all?

You might be interested in this link: https://www.maa.org/sites/default/files/pdf/CUPM/first_40years/1963-PreGradResM.pdf — it is to a comprehensive study (from 1963) by the Committee on the Undergraduate Program in Mathematics. It is probably too ambitious for almost any actual undergraduate math program, but it outlines almost every important thing that (the committee believed) a young mathematician should learn before graduate school.

phyzguy
I second @zinq's recommendation of I.N. Herstein's "Topics in Algebra". I first studied this many decades ago and thought it was pretty trivial, but I now realize there is much more to it than I first thought, and I keep going back to it and learning new things.

Infrared
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I'd give algebra a second chance. I think Galois theory is one of the prettiest topics in the undergrad curriculum (and I'm not at all an algebraist!). I haven't read Herstein, but it has a good reputation. I think Dummit and Foote is the usual reference for undergraduate algebra.

For complex analysis, I like Stein and Shakarchi (Princeton Lectures in Analysis volume 2). It has good coverage of the standard topics in single-variable complex analysis and its exercises are pretty instructive in my opinion.

If you already have a handle on single-variable complex analysis, you might be interested in learning about several complex variables. Many results from the single-variable case carry over, but here are a couple of basic examples to show how things are different when $n>1$:

If $U\subset\mathbb{C}^n$ is open, and $K\subset U$ is a compact set with $U\setminus K$ connected, then any holomorphic function defined on $U\setminus K$ can be extended to all of $K$ (Hartogs). Of course, any function with a pole is a counterexample to this when $n=1.$

Also, for any non-empty open set $U\subset\mathbb{C}$, there is a holomorphic function defined on $U$ that cannot be extended to any larger domain. But in $\mathbb{C}^n$ for $n>1$, this is no longer true, and when it is, we call $U$ a domain of holomorphy.

The subject gets pretty technical, but if you like analysis a lot, it might be for you. If geometry appeals to you more, you could try reading about Riemann surfaces/complex manifolds, but you might want more differential geometry background first (especially in higher dimensions).

mathwonk
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I used to say complex analysis was a unique subject since almost all books on it are good ones, maybe because the basic subject is so clean and beautiful. But my favorite complex book is probably the one by Henri Cartan. The most helpful one for beginners (i.e. me, at that time) is probably that by Frederick Greenleaf.

FactChecker
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It looks like pure math has been your emphasis. You might want to consider some applied math topics. There is a whole world of them: numerical analysis, computer science (many subjects here), statistics (many subjects here), linear, nonlinear, and dynamic programming, stochastic processes, control laws, artificial intelligence, neural networks, etc.

S.G. Janssens and sysprog
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@zinq I went with your recommendation for "Topology" by James Munkres and ordered it just now, I also pre-ordered A Course of Modern Analysis, 3rd ed. by Whittaker & Watson, and got a replacement copy of Papa Rudin, and Theory of Functions vol. I & II by Knopp. Hope all those round me out for a while. The algebra book will have to wait as I was up to almost \$300 with what I had: maybe next time. Thanks again all who sat in here for office hours to advise me!

lavinia
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If you are looking for an intro to Geometric Topology you might try the Shape of Space by Weeks then move on to Three Dimensional Topology and Geometry by William Thurston.

Personally, I never found it inspiring to learn a subject for itself. I always wanted to see how it is used and how it answers questions. For instance, learning linear algebra by itself nearly killed me but seeing how it is used in Calculus and Representation Theory made it come alive.

The same is true of Complex Analysis. Learning it in order to understand Elliptic Curves made it come alive. In conjunction with a Complex Analysis book you might look at Elliptic Curves by McKean and Moll which for me is a deliciously wonderful book. Hard though.

The great exception to this for me was William Fuller's first book on Probability Theory which is still the best math book I have come across. He develops the theory gradually starting from simple ideas about counting and shows how probability theory emerges from its simple beginnings.

It is a matter of how you like to learn I suppose.

Parenthetically, I would add that learning math topics is not the same as learning to think mathematically. Mathematics involves a mental discipline that transcends any particular subject. Learning this discipline makes learning particular subjects easier and brings the absolute joy of thinking about things on one's own.

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mathwonk and berkeman
mathwonk
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maybe your spell checker didn't like Feller's name?

lavinia, Math_QED and FactChecker
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Feller's books are like other classics that I love in that he really tries to expound thoroughly on each subject so that it becomes intuitively clear. They belong on the same bookshelf as Knuth.

lavinia
lavinia
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maybe your spell checker didn't like Feller's name?
I wish I could blame it on the spell checker,

S.G. Janssens
For me, complex analysis came to life in conjunction with spectral theory for linear operators and applications to dynamical systems and control. Sometimes electrical engineering researchers are better at this than mathematicians. Another motivation for complex analysis comes from elliptic PDEs and the study of harmonic functions.

As you can see from some of the other replies, people have different motivations to study complex analysis, ranging from the pure to the applied (or: applicable). This makes it a very rich subject indeed. I second the Checker of Facts in post #9 that it's good not to forget about the "applications", even when your interest is primarily pure. (This is for the same reason that I don't think it makes sense to study topological vector spaces without learning about their motivation via linear differential equations.)

While I am not such a fan of the second part, I agree that the first part of Feller is really beautiful. (Feller had an interest in and made contributions to both pure and applied areas, for example: evolution semigroups and weak topologies, but also population genetics.)

lavinia
lavinia
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For me, complex analysis came to life in conjunction with spectral theory for linear operators and applications to dynamical systems and control. Sometimes electrical engineering researchers are better at this than mathematicians. Another motivation for complex analysis comes from elliptic PDEs and the study of harmonic functions.

As you can see from some of the other replies, people have different motivations to study complex analysis, ranging from the pure to the applied (or: applicable). This makes it a very rich subject indeed. I second the Checker of Facts in post #9 that it's good not to forget about the "applications", even when your interest is primarily pure. (This is for the same reason that I don't think it makes sense to study topological vector spaces without learning about their motivation via linear differential equations.)

While I am not such a fan of the second part, I agree that the first part of Feller is really beautiful. (Feller had an interest in and made contributions to both pure and applied areas, for example: evolution semigroups and weak topologies, but also population genetics.)
I agree that Feller's second book falls short of the first and it omits Martingales which have wide application.

An application of probability theory that I used during trading years was the pricing of derivatives contracts.

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S.G. Janssens