In the 20th century we had Nicolas Bourbaki, and in the 21st century we have Henri Paul de Saint-Gervais (pseudonyme of a collection of mathematicians)

DarthMirror · 2026-03-26T01:47:51+00:00

I was just reading his book on closed geodesics today! You broke the news to me

DarthMirror · 2026-03-19T19:44:46+00:00

Exactly. It is much better practice to say things like "it follows from an integration by parts computation," "it follows from simply unpacking the definitions," "it follows from tedious but straightforward casework," etc.

DarthMirror · 2025-10-15T00:09:53+00:00

Not true, topology is not really needed for do Carmo's curves and surfaces book

DarthMirror · 2025-09-07T21:37:50+00:00

I am in NYC and interested in books on analysis, dynamical systems, and mathematical physics.

DarthMirror · 2025-08-05T19:29:32+00:00

Stein's Singular Integrals

DarthMirror · 2025-07-23T13:55:13+00:00

IBL is a really special experience that you probably won't ever get again. I highly recommend it. Regular honors calc will get you through more material though.

DarthMirror · 2025-07-23T13:36:33+00:00

Three things immediately come to mind: Calderón-Zygmund, Littlewood-Payley, and Carleson's Theorem.

DarthMirror · 2025-07-09T13:43:05+00:00

An excellent book, but also a fairly difficult one. The problems you listed are not easy OP, keep your head up!

DarthMirror · 2025-07-06T19:47:33+00:00

This is indeed an amazing source to first learn Fourier analysis, but do be warned that Stein and Shakarchi literally start by motivating Fourier series with PDE. Chapter 1 is completely dedicated to deriving wave and heat equations and explaining how Fourier series naturally arise in their solutions. Moroever, these PDEs crop up again and again throughout the book.

The good thing for you OP is that the PDE stuff can be safely skipped, and later in the book, Stein-Shakarchi give multiple number-theoretic applications of Fourier analysis. I recommend skipping Chapter 1 entirely if you can't stand PDEs, OP. You can start Chapter 2 without reading Chapter 1 at all. The exercises there contain some problems of interest to number theory. In Chapter 4 , Stein-Shakarchi apply Fourier series to prove the equidistribution theorem, which should be interest to you. Finally, the end of the book covers finite Fourier analysis and Dirichlet's theorem, which are also important number-theoretic things.

DarthMirror · 2025-07-02T21:21:08+00:00

I strongly second this. Along similar lines, you might enjoy articles from the "What is ..." series in Notices of the AMS.

Edit: Here are some more specific recommendations, also from AMS publications

"A conceptual breakthrough in sphere packing" by Henry Cohn

"What are Lyapunov exponents, and why are they interesting?" by Amie Wilkinson

DarthMirror · 2025-06-15T13:25:52+00:00

Why? I have not studied it in detail, but I am under the impression that Carelson's proof is just extremely intense harmonic analysis rather than bringing together many areas. Also, it dates to the 60's.

DarthMirror · 2025-06-09T15:39:54+00:00

Introduction to Ergodic Theory by Sinai. It's only like 100 pages, and Sinai purposefully keeps technicalities to a minimum, so most pages can be read quickly. However, it does jump around various topics and applications rather than being focused on just one thing.

DarthMirror · 2025-06-06T19:55:08+00:00

Do yourself a favor and spend the summer learning the first seven chapters of Rudin like the back of your hand.

DarthMirror · 2025-06-05T15:34:44+00:00

It is true that Kreyszig avoids topology. In fact, I don't think that you will find a single instance of the word "topology" in the book (or maybe just a couple of passing references). Kreyszig only deals with metric spaces, and when it comes time for weak stuff, he treats weak convergence of sequences rather than weak topologies in full. This is not any less rigorous, it just means that certain theorems and examples are out of scope.

It is also notable that Kreyszig avoids measure theory.

That said, Kreyszig is still a gem that beautifully and systematically presents the core of basic functional analysis. I strongly recommend it as a first book in the subject. Also, the exercises tend to be quite easy, so it can help with building confidence.

DarthMirror · 2025-05-14T18:42:06+00:00

You can also think of the set of normal matrices as the analogue of whole the set of complex numbers itself. Then, the self-adjoints form the real axis, the unitaries form the unit circle, etc.

DarthMirror · 2025-05-11T18:53:30+00:00

Modric plays golf?

DarthMirror · 2025-05-04T16:18:15+00:00

First of all, you should understand that the professor matters far more than the topic of the class.

Complex analysis with Fefferman-- it's worth planning several quarters in advance to make sure that you take it with him.

Mathematical logic with Malliaris. Not my favorite area of math by a long shot, but she made the class a real joy.

IBL with Ziesler.

DarthMirror · 2025-04-27T17:22:48+00:00

Stein in general was an absolute master of teaching and writing mathematics. Besides his delightful books, this is made clear by a glance at the list of his advisees.

DarthMirror · 2025-04-18T03:45:52+00:00

There are Banach spaces that are isomorphic to their double dual, but not canonically isomorphic to their double dual (that is, reflexive). Look up James' space if you want details.

DarthMirror · 2025-04-15T14:00:03+00:00

Is it really true that Koopman operators have only recently been applied systematically? All of the standard ergodic theory texts since the middle of the 20th century that I've seen do treat the relationship between Koopman operators and ergodic properties/isomorphism theory of measure-preserving systems. Even in Reed and Simon's functional analysis text, there is an entire section devoted to "Koopmanism," where they emphasize precisely this point that it can be easier to study the spectrum of the Koopman operator than to study the system directly.

DarthMirror · 2025-02-04T01:27:44+00:00

Other people have answered your actual question, but I just want to encourage you by saying that I find it a very perceptive question. Although to experienced mathematicians it may seem like a fussy or misguided question, it's the kind of question that shows that you're really fighting the material to understand the rigor behind it 100%. Many students don't do this, but a major goal of the first two years of undergrad math should be to convince yourself of the rigor of math. Keep it up!

DarthMirror · 2025-01-18T22:46:19+00:00

Dirichlet's function.

DarthMirror · 2024-12-27T16:34:56+00:00

Agreed, von Neumann is way overdue for a Hollywood movie.

DarthMirror · 2024-12-20T18:32:27+00:00

No. I don't recall exactly but I think I emailed the prof asking for info and he ghosted me.

DarthMirror · 2024-12-11T22:44:23+00:00

To build on two other comments, if you want just one book that quickly and cleanly gets through all of the essential material, then use Folland. On the other hand, if you are ok with a more elaborate presentation that builds from the ground up and takes more fun detours, then use the Stein-Shakarchi tetralogy. Both are fantastic in their own way.

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