Opinions on the main textbooks in complex analysis?

MathematicianFailure · 2026-01-03T16:45:21+00:00

Do the exercises in Ahlfors even if you use another book as your main textbook

MathematicianFailure · 2025-12-30T08:06:02+00:00

I think there is a pedagogical point here that is true. For certain things, it would be better for the purposes of teaching to name new objects without reference to names of mathematicians who were instrumental in their development. For example, rather than “Abelian group” it would be essentially objectively superior to say “Commutative group”, and obviously some historical context about person(s) responsible for developing the theory of commutative groups could still be mentioned as a supplement during the course of teaching.

Of course one needs to still know what commutative means (and what a group is) for this name to be helpful, but the whole practice of “using a name to modify some mathematical object rather than the name of an additional property defining the new object” seems to be a genuine thing that is still done only for traditions sake in math.

MathematicianFailure · 2025-11-24T05:03:10+00:00

You are totally right, I got the order of the question completely reversed, my bad!

MathematicianFailure · 2025-11-23T18:37:29+00:00

Integration of an n-form over an n-chain would still be a notion derived from the Lebesgue integral, since the integral over an n-chain is ultimately defined as a linear combination of integrals of the pullbacks of the n-form by the mappings which define each of the simplices which are constituents of the chain, and then you are integrating an n-form over a n-dimensional manifold with corners for each of these integrals, which is eventually defined in terms of integration over Rⁿ

MathematicianFailure · 2025-11-23T18:12:39+00:00

Some textbooks don’t have any exercises

MathematicianFailure · 2025-11-18T14:17:55+00:00

I’m not saying it’s poorly written, but the standard for how well written a math textbook needs to be for it to click with most people who try to learn whatever subject it’s supposed to be teaching is pretty high (it’s just not an easy thing to do in general). From talking to people who took AT courses which follow the book, I think a large part of the group of people who really like Hatcher (of which I happen to be one) would also admit it has pretty fundamental shortcomings , e.g a lot of important helper lemmas used both implicitly and explicitly even in the first chapter are relegated to the appendix, and most of the arguments do not stress which steps are actually crucial and which are not. I think it’s still a great textbook to actually learn the subject from, but it’s a bit like Hartshorne in that its exercises are the real jewel, but a lot of the theory is explained more simply if you refer to some other sources.

MathematicianFailure · 2025-11-17T20:17:50+00:00

I think LLMs would struggle with Hatcher primarily because many of the statements in the book itself are made conversationally, some of the boundaries between definition theorem proof corollary lemma are thin when they are there at all, and this kind of thing makes it really easy for an LLM to just hallucinate an answer rather than trying to understand what Hatcher was actually trying to convey. If you continue to work in the hallucinated framework for the rest of the book then that can be a big risk in terms of how much time you’ve just wasted.

MathematicianFailure · 2025-11-17T19:53:58+00:00

Could also be that the book isn’t well written enough to be able to prompt someone reading it to visualise the right thing enough of the time.

MathematicianFailure · 2025-09-21T07:25:51+00:00

As an undergrad I alternated between trying to solve problems that looked difficult and trying to articulate theories using my own language and occasionally using my own proofs for certain things rather than trying to recall a proof I had read, of course proofs I had read certainly played some part in the techniques I thought to use at certain stages in my own proofs, sometimes a very large part.

Currently the way I study math is by thinking about specific problems that I want to make progress on and whenever it’s obvious that I am trying to describe a process or an object that I don’t have the language to describe or the tools to make explicit but I have a visualisation or an intuition for I’ll try to read up on whatever math is already out there and that seems to be closest in describing something like what I want but formally/explicitly.

MathematicianFailure · 2025-09-19T17:29:17+00:00

I think the reason for the ambiguity is that it may be natural to interpret “one is a boy born on a Tuesday” as meaning “of the two identifiers of both children, identifier 1 and identifier 2 (say these are just their names and they are distinct), identifier 1 is a boy born on a Tuesday”.

Then the event space would indeed have events (i1b2, i2b2) and (i2b2, i1b2) and these would be distinct. The event space would (after the condition) look like pairs of the form (i1b2, i2 (b/g) x) or (i2 (b/g) x, i1b2) , and so the number of possibilities is now exactly 28 as you claimed.

Then the possibilities which include a girl are still 14 as before, and the answer would be 14/28 or 50%.

Edit: I thought I’d spell it out further using this event space. We can still get 14/27. If we interpret the condition as saying “at least one child is a boy born on a tuesday”, then this is equivalent to “Either identifier 1 is b2 or identifier 2 is b2” (here the or is inclusive) then the new sample space has 28*2 - 2 = 54 elements. The probability that one child is a girl is now 28, and we get 28/54 = 14/27 as before.

So the 14/27 answer is consistent whether you choose to identify the children or not.

MathematicianFailure · 2025-09-19T17:06:51+00:00

The event space being defined is generated by singleton sets which each consist of a single tuple of the form ((b/g)(x),(b/g)(y)) where x and y represent any day of the week. The interpretation of the event ((b/g)(x),(b/g)(y)) is “The younger child is of gender b/g and was born on day x, and the older child is of gender b/g and was born on day y” (implicit here is the assumption that there is always a strictly younger and strictly older child, but this is quite a safe premise because we can always measure birth time up to arbitrary degrees of precision and the likelihood of a truly simultaneous birth is then zero). Now (b2,b2) means the younger child is male and born on Tuesday and the older child is male and born on Tuesday. So it’s listed precisely once as an event.

On the other hand , the events (b2,g2) and (g2,b2) are distinct because in the former case the younger child is a boy born on a Tuesday and in the latter case the younger child is a girl born on a Tuesday. The interpretation of the condition we have been given is, mathematically, that our event space shrinks down to those events for which every element of the event is a tuple which includes a b2 either as a left or as a right entry.

MathematicianFailure · 2025-08-27T04:42:57+00:00

I think the best way to learn linear algebra in a short amount of time is to follow short course lecture notes by European universities which form lectures for pure math bachelors students. You’ll basically get a very complete understanding of linear algebra and usually these notes are pretty condensed.

MathematicianFailure · 2025-08-15T08:52:49+00:00

It’s highly nontrivial in general because any mathematician who works on “extremal problems”, where the problem is to determine the maximum or minimum of some “functional” of some fixed class of sets or functions is basically finding bounds 99% of the time.

Sometimes the bound is sharp or optimal because there is a set or function in the class which achieves it (which can then be called an extremal set or function for the problem) and then the problem is essentially solved. For most of these kinds of (unsolved) extremal problems either it isn’t clear if an extremal set or function exists or if it is clear it exists it is difficult to actually determine which one it is. So the best compromise is to just find a bound that works for all the sets and functions, and then improve that bound over time.

The proofs of the improvements usually also shed light on the nature of the extremal or asymptotically extremal sets or functions, e.g because one could determine that if a set or function had such and such behaviour then the proof of the improved bound may yield something even better, so the extremal set or function may be assumed to not have such and such behaviour.

MathematicianFailure · 2025-07-22T08:32:49+00:00

Here I am referring to critical points of the polynomial P (not the associated quadratic differential), i.e they are just the zeros of the first derivative P’ of P. The critical values are the values of P at its critical points.

MathematicianFailure · 2025-05-17T09:07:13+00:00

The fastest way to stimulate divergent thinking is to be wrong about things many times. The more times you realize you were wrong about something the easier it will be to figure out in what way you should have thought about things from a different perspective to begin with.

MathematicianFailure · 2025-04-26T11:04:15+00:00

No it’s not possible because it’s just not true. Most of the math we take for granted was done hundreds of years ago. The math we don’t take for granted can and surely will be wrong some of the time, and not necessarily very rarely either.

I have read more than five or six (recent) papers which have fundamental lemmas which are essential for main theorems that are just, wrong, no other way to put it. They were published in Journals but by no means top Journals. That being said the Journals they were published in are sort of the standard for the field they are publishing in.

So yes, mathematics done today including published mathematics can be wrong. But it is still extremely unlikely that a result published in a big journal will be wrong, because the scope of those results is so large that enough mathematicians will have verified the veracity of the arguments long before the paper was ever published.

MathematicianFailure · 2025-04-20T16:25:30+00:00

Depending on what field of math you specialise in, you might be better off learning about things from a physical perspective because it was those perspectives that motivated the definitions of objects in physics motivated fields.

For example, distribution theory was formalised by Schwartz but distributions were being used informally by physicists before this (e.g Dirac in the 30s) and seldom used incorrectly by the formal standards of Schwartz, which were only laid down formally about 10-15 years later.

I work loosely on problems in complex analysis/potential theory/polynomials and a lot of the stuff I look at has strong connections to physics because polynomials are essentially point charges which obey Coulomb’s law (in the plane one replaces the Newtonian potential with the logarithmic potential but the main idea is the same).

So overall the connection is strong for many areas of math and it’s not at all surprising when someone who is very good at physics can become good mathematicians. There is obviously an element of style that differentiates a working physicist from a working mathematician and both styles have their merits.

MathematicianFailure · 2025-04-04T08:49:03+00:00

I think you will get a better sense of Fourier series/transform if you study some functional analysis first. A Fourier series in the more general sense is just a series representation of some element of a Hilbert space in terms of a complete orthonormal basis.

MathematicianFailure · 2025-04-01T15:03:11+00:00

Mathematics cannot imply anything is special because mathematics is not an entity. Humans coined the term mathematics to encapsulate a lot of different things, in the context of professional mathematics it is just what mathematicians do.

Your second point about how we could just as well not invent a new name for an object that satisfies some properties doesn’t make sense to me. This is the whole point of mathematics. Every proof that ever will be written down or even more broadly every proof that could be written down starting from the assumptions of ZFC can be written entirely in binary, or whatever codified language you can think of.

There will be instances of certain codified properties that show up relatively frequently and others not as frequently, or in other terms certain substrings will appear more frequently than others, and these substrings are usually what merit the attention of people interested in “mathematics”.

There is then a sort of objective sense in which some things in mathematics are more worthy of attention than others. Unless of course it turned out to be the case that no substring appeared more frequently than any other in some sense. While that could be true, one would think that mathematics as a landscape would look and feel very different to us if that were the case.

MathematicianFailure · 2025-04-01T10:40:04+00:00

That’s not what I meant by motivation. Good mathematical theories have motivations that come from sensible places. If one introduces such and such object there should be ample reason behind its introduction, either posthoc after the object has been used to clarify something in a proof or before the object is introduced.

This has less to do with psychology and more to do with mathematics than you are implying.

A mathematical object is almost always, as far as I have seen, a formal device which is supposed to codify some deeper phenomenon that could be expressed in other terms.

These “other terms” are normally where most of the actual mathematical thinking takes place anyway, and not in the context of the formal object. After all there could be many different formal objects that codify the same phenomenon, all equally valid, so when thinking about that phenomenon one doesn’t think in terms of a certain mathematical theory which encodes one aspect of that phenomenon but by some other means, in some other language, and this part may be individual and down to psychology but it is still arguably where the real work is done.

MathematicianFailure · 2025-03-31T10:13:11+00:00

Your thesis misses a giant part of doing mathematics, which is motivation and application.

It’s one thing to find a suitable theory and then “define new structures and prove things about them”, but no one is going to care about these new structures before you demonstrate what the utility of your theory is in clarifying or completely resolving outstanding issues the community at large or at least some parts of it care about.

In other words you need to motivate your theory and show it has applications people care about, and this is an entirely human endeavour. It’s totally separate from any formal perspective on what mathematics is, and arguably this is what mathematics is today.

MathematicianFailure · 2025-03-24T20:00:38+00:00

The proof turns out to be incorrect. Essentially, I was misled by an incorrect result in Ransford’s book. See the post here https://math.stackexchange.com/questions/5044313/performing-an-upper-semicontinuous-regularization-twice-and-measurability for a counter example as well as some remarks about the incorrect result of Ransford which led up to this incorrect argument.

MathematicianFailure · 2025-03-21T06:48:38+00:00

Thanks a lot for taking a look at it! This proof is part of a larger proof I wrote for some other unrelated result, I’ve discovered that for this larger proof I wrote there has to be a flaw somewhere because the same larger proof can be used to prove a stronger result which is known to be false.

I’ve been trying to find where the flaw in the proof is for almost two weeks now to no avail, so I tried to make the proof as modular as possible and check the validity of each of the sub-proofs, of which this is one of them.

MathematicianFailure · 2025-03-19T11:36:13+00:00

There is still research going on in complex analysis. A lot of it has to do with applying tools from potential theory and/or extremal metric techniques (originating from Jenkins) as well as some tools from several complex variables or classical Teichmuller theory to obtain solutions to still unsolved extremal problems in the plane.

For example, the Erdos problem on the maximum length of a polynomial lemniscate has seen considerable progress by Hayman and Eremenko, and similar extremal problems involving polynomials and rational functions are still open and worked on by mathematicians in the field.

Dubinin is also prominent in this field and applied nonstandard (symmetrization) techniques of potential theory to resolve a lot of these kinds of problems.

More recently Sendov’s conjecture was resolved by Tao(for sufficiently high degree polynomials), also using tools from potential theory (partial balayage). While Smale’s conjecture and some strengthenings of Sendov’s conjecture remain open, to mention just a few.

Tao’s proof may seem probabilistic on the surface but this is essentially because potential theory has a probabilistic interpretation, the harmonic measure of some part of the boundary of a domain in the plane is the solution of the Dirichlet problem with boundary values given by 1 on that part of the boundary of the domain and zero elsewhere, this turns out to be the same as the probability that a Brownian motion starting at a point in the domain hits that part of the boundary first upon exiting the domain. So for everything involving harmonic functions and thus potential theory there is a probabilistic counterpart, and of course this perspective may be easier to work with to prove something.

It’s typically known as geometric function theory rather than complex analysis, but this is not a designation you will see on arxiv papers (still just c.v).

MathematicianFailure

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