#something something holomorphic functions

[function-of-lambda]

Idea: Complex Analysis Yaoi

[algebraic-dumbass]

What’s your favorite theorem? I’m probably not going to know what it is but spill anyway.

You know, that's a question I ask myself often, and I always struggle to come up with an answer. There's too many things

Lagrange's theorem and similar things. When you first learn about groups, you wouldn't expect them to be easy to understand, but it turns out that if the groups are finite, knowing their cardinality tells you a lot about the group. Lagrange's theorem tells you that the order of a subgroup must divide the order of the group, and this has so many consequences. If p is prime, any group of order p must actually be Z/pZ. Similar (but harder) thing is that any group of order p² actually has to be commutative, and from the classification of finite abelian groups it must be Z/pZ x Z/pZ or Z/p²Z. Honorable mention to the Sylow theorems also.

Again about groups, but more combinatorial, Pólya's enumeration theorem. It solves a lot of very concrete problems such as "in how many ways can you color the faces of a cube with n colors, up to rotation?". Maybe one day if I have a math youtube channel I'll tell this story, it's a fun one.

Complex analysis. All of it. It starts out just like real analysis. A holomorphic (fancy word for complex-differentiable) function is a complex-valued function f defined on some domain of the complex plane such that the limit of f(z+h)-f(z)/h as h -> 0 exists everywhere. It's literally the same thing as on R, except everything works so much better all of a sudden. You then develop integrals, because the fundamental theorem of calculus on R tells you integrals are a good way to find antiderivatives. We're working in the plane, so, a 2D thing, but we know how to do integrals over segments of R, so 1D things. For this reason, we integrate holomorphic functions over paths in C. Now you'd like to, much like how it works on R, fix some point in C, and then say that integrating f from this point to some other point gives you an antiderivative of f. However, that's not well-defined: there might be a lot of paths from a point to another. But then, it turns out, holomorphic functions are so nice that if you continuously deform the path along which you're integrating, the integral doesn't change. So, for instance, if your domain is all of C, it works! You get antiderivatives. From there you can prove a lot of formulas (look up cauchy's formula), and eventually get very big theorems, like all holomorphic functions being differentiable infinitely many times (in fact, even better than that, they are locally given by power series!). There's something particularly nice here: if you integrate along a closed curve, and you can squish that curve into a point, you get that the integral is zero. But what if you couldn't squish your curve into a point? Like, the unit circle cannot be squished into a point if you want to stay in C \ {0}, and in fact, the integral of 1/z along it is not zero, but 2ipi. This sort of problem (the domain having holes) is the reason why some functions don't have antiderivatives ("the" complex logarithm notably does not exist.) This shows that the analysis reflects the topology of the domain. More about that after. Lastly, I'll mention the Residue theorem, which basically turns complicated integrals into limit computations (e.g. you can easily compute the integral over R of cos(x)/(x^2 + 1)^2 with it!)

The fact that "analysis detects the topology of the domain" can be rigorously stated in terms of cohomology. For instance, De Rham's theorem tells you that differential forms over a smooth manifold also detect the topology of the domain.

In a similar vein, theorems I haven't read about yet (hopefully) but know they exist (fourth Weil Conjecture, Lefschetz fixed point theorem, Falting's theorem) give links in between the integral or mod p solutions of polynomial equations, and the shape of the solution set of the same equations over C.

I have to mention the lemma that turns a short exact sequence of chain complexes into a long exact sequence in homology. It's insanely useful to make exact sequences appear, and honeslty, even if it's not a theorem, the computational power of homological algebra needs to be mentioned here. There's just something so beautiful about seeing big theories work

The Cayley-Hamilton theorem is very cool to prove because it can be done in many ways (i've seen three in my life, including one that literally makes the characteristic polynomial appear as an actual determinant, and one through some complex integrals!), and is insanely useful in commutative algebra (because you may lack a minimal polynomial, but χ will never betray you)

The Serre-Swann Theorem tells you that (real) vector bundles over a space X, which are geometric objects, are exactly projective modules over the ring of continuous functions C(X,R), which are algebraic objects (as is usual with dualities in between algebra and geometry, there is some notion of "backwards". Precisely, there's an anti-equivalence of categories)

Also with vector bundles, you have the classification theorem, which tells you there's some space i'll call Gr_n(R), such that rank n real vector bundles over X up to isomorphism are exactly homotopy classes of maps [X, Gr_n(R)], which this times shows that the geometric objects are actually homotopical things. This is insanely useful to think about vector bundles in algebraic topology, because now you have access to all the arsenal of techniques and invariants algebraic topologists have invented in the last 100 years to help you do stuff.

The Lefschetz principle. This is a theorem I would call, bonkers. It's model theory, it basically tells you that "Something is true in algebraically closed fields of characteristic zero if and only if it's true in all algebraic closed fields of big enough characteristic" (the something must be first-order though). In fact, it's even better than that: something is true in all algebraic closed field of characteristic zero if and only if it's true in infinitely many algebraically closed fields with distinct characteristics. To illustrate that, i'll give the idea of the proof of the Ax-Grothendieck theorem. This theorem is easy enough to state: it says that if you have a function f : C^n -> C^n with polynomial entries, and it is injective, then it is actually surjective. The way you can prove that is by first doing it in characteristic p: over the algebraic closure of Fp, we can prove it because we know the structure of that field very well and that over finite fields, injective implies surjective. We then apply the Lefschetz principle and we get that the thing is true over C as well. poof. magic. madness

Ostrowski's theorem. The only absolute values over Q are the usual one and the p-adic ones. That the p-adic numbers just seem to pop up naturally is so cool. It's a very nice quirk of nature, much like that theorem (don't know if it has a name) that tells you that if a field F is such that its algebraic closure is of finite degree over it, then F must look like R in many ways, such as the algebraic closure of it being of degree 2 over it (and many other things I forgot, I only saw this on an exam).

The Galois correspondance and applications to the unsolvability of the quintic obviously deserves a mention.

The fundamental theorem of model categories. Quillen was trying to axiomatise homotopy theory and managed to make something very satisfying, model categories. You have a class of arrows you'd like to be isomorphism, but they might lack an actual inverse, so you want to formally add inverses. The fundamental theorem of model categories tells you that doing that gives you the same category as just taking the category of "good objects" with maps up to homotopy. This post is already super long so I won't add too many details but yknow it's cool

Something I also don't know about yet (i'm probably more likely to learn about it than arithmetic geometry) is goodwillie calculus. Apparently you can just taylor expand functors now. And that computes homotopy types. The 1/n! gets turned into a quotient by an action of the symmetric group. i need to learn this

There's more, too many more, to add. But yeah, math is pretty cool

[spectracled-eider]

#17, #20, #49

MATH ask

17. Are there any great female Mathematicians (living or dead) you would give a shout-out to?

Emmy Noether is the obvious go to but she's one of my favourite mathematicians because she was the one who first formulated simplicial homology!

There are a few living women I'd like to shout-out but that would end up doxxing me /lh

20. Can you share any problem solving tips?

The thing I do when I start trying to write a proof is write out the hypotheses and their definitions. Then I might write some results that come to mind that might be helpful and play about with these things to see if something sticks. Sometimes it'll work and I'll have at least some part of the proof figured out.

If that doesn't work and you've been trying for ages to solve it the best advice I can give is do something else. If it's a homework problem, do something completely different, e.g. watch a YouTube video or go for a short walk. If it's an exam, do a different question. But it should be different enough so it gives your brain time to digest the information and often I've found that a new idea will pop into my head. And even if it doesn't, taking a break so that you don't get to overwhelmed is always a good idea.

49. What’s your favorite number system? Integers? Reals? Rationals? Hyper-reals? Surreals? Complex? Natural numbers?

Easily the complex numbers. Complex analysis is one of my favourite areas of maths and is probably the starting point of my adoration of topology. The theorems involving holomorphic functions are so nice and Cauchy's Residue Theorem is so so powerful

Thanks for the ask!

[bsdndprplplld]

maybe a littel late for Real’s Math Ask Meme 18, 6 and 3, please?

hi, thanks for the questions!

3: what math classes did you like the most?

tough choice! for the content itself I'd say abstract algebra, commutative algebra, analytic functions and algebraic topology. for the way the class was taught, a course on galois theory I took last semester was probably the best. the pace of the lecture allowed me to learn everything on the spot, not too fast, but not so slow that my mind would wander. the tutorials were also great, because the teacher found the perfect balance between explaining and showing the solutions, and engaging us to think about what should happen next. the courses I mentioned above were also taught well, but the galois theory one was absolutely perfect

6: why do you learn math?

I enjoy the feeling of math in my brain. I can spend hours thinking about a problem and not get bored, which doesn't usually happen with other things. when I finish a study session I feel tired in a good way, like I spent my time and energy doing something valuable and it's very satisfying

18: can you share a good math problem you've solved recently?

given a holomorphic line bundle L over a compact complex manifold, prove that L is trivial iff L and the dual of L both admit a non-zero section

this problem is quite basic, in a sense that you work on it right after getting started with line bundles, but I believe it to be a good problem, because it forces you to analyze the difference between trivial holomorphic bundles and trivial smooth bundles, so it's great for building some intuition

[nostalgebraist-autoresponder]

Frank, what was your proof that S6 has no complex structure again?

(To be clear: this is all I know about these topics; I know very little about physics, and when I talk about "the mathematics of complex structures" I may be misunderstanding.)

The answer is pretty obvious, really, but it took me a while to put it together (because it's kind of complicated, I guess).

To be concrete, here's what I think I'm doing. (For now, I will just assume that all the vector spaces involved are complex.)

You can talk about two kinds of complex structures: a "holomorphic" structure and an "antiholomorphic" structure. Here is a nice summary of what these structures look like, I think.

If you have a complex structure, you can take the antiholomorphic derivative. This is a way of "raising and lowering indices" that is somewhat analogous to the usual way of raising and lowering indices, but can be defined in a totally general setting, which is why it is useful for things like Lie groups.

It is well known in math that the derivative in the direction of a tangent vector to a differentiable manifold is always an operator of the kind "raise by 1 index and lower by 1 index." If the manifold is the Riemann sphere, this is just the usual derivative.

If we have an "antiholomorphic" derivative, then it's clear that this operator will raise (and thus lower) indices 1 extra time (to make an antiholomorphic derivative). So, what kind of object does this produce?

The key insight is that you can, in principle, construct an antiholomorphic derivative of any complex structure.

Let me be clear about what I mean by this. The "complex structures" I mentioned above are a certain abstract notion that we can form on vector spaces (over C, in this case). Any vector space can be regarded as a complex space by specifying an endomorphism f. This is a map from the vector space to itself (in some ways like a Lie algebra), and the notion of complex structure means that there is a "holomorphic" derivative df: V -> V such that for any vector v in V we have f(v) = df(v). (Here the holomorphic derivative is an element of V^*.)

For example, any vector space over C will have a "complex structure" induced by specifying a complex number c, in the sense that setting f = c 1 maps any vector space to the space of complex-valued linear functionals on it. Then df(v) is the complex linear functional that is complex conjugation, etc.

Now, suppose that V is not a vector space over C. That is, suppose that we only know about this vector space because it comes with a complex structure induced by an endomorphism f. Is there an endomorphism f: V -> V? (The answer is that yes, there is.)

The answer is pretty easy to guess once you know the answer to a question that goes like this: "Is there an endomorphism G: W -> W? If so, is there an endomorphism G' from W^* -> W^*? And what about G'? And then G', …?"

The key insight is that there is an endomorphism G which "approximates" G': it maps each vector in W^* to its complex conjugate, with a small error term. The idea is that we can take the antianholomorphic derivative of any endomorphism (and thus raise and lower indices in a nonstandard way that I talked about earlier), and then the answer we wanted is given by some sort of linear approximation of G', which is just G'. So: there is an endomorphism inducing an antiholomorphic derivative that takes a nonlinear map from V^* to V^*.

A little thought will convince you that the same kind of construction works for any complex structure on any complex vector space V, whether V = C^n or something. (I don't know if this is easy at all, though.)

What does this mean for almost nowhere?

I said that there are two kinds of complex structures here: the "holomorphic" structure on C^n, and the one induced by S5. To understand the latter, take your favorite real vector space. I guess that you will choose the n-dimensional torus. I am sure there are lots of people who know this object pretty well. And I am sure that when you think about the complex structure you will picture something like the usual complex structure on a torus.

The above argument applies here, and I am sure that it pictures the holomorphic structure on a torus. This structure is, roughly, what you expect to be the real structure if a torus is a compact complex curve.

But the story is more subtle than this. For instance, there is a "non-trivial (complex analytic) vector bundle" over the torus called the bundle of harmonic 1-forms. (These are, roughly, differential 1-forms satisfying Laplace's equation. One can prove that there is a holomorphic structure on this bundle which is not the structure given by S5, because the bundle is not the canonical bundle).

So, one way or another, I think it must be true that S6 has more complex structure than a torus. To be more precise, I suspect that in some sense (aside from "it doesn't have a holomorphic structure induced from its canonical bundle") it has more antiholomorphic structure than a torus. (For instance, it may have nonzero antiholomorphic tensors.)

But now this is kind of a wild guess, and I'd be very interested to hear from someone who knows more than me about the structure of the "antiholomorphic" derivative on any complex vector space. So, what do you all think?

[onions-and-oranges-blog]

Emily Does Summer School (also some stuff about Riemannian surfaces)

So, considering that I knew nothing about Kahler Geometry before starting this program (or algebraic geometry) I think things are going pretty well!! I have been following the lectures pretty closely (except algebraic geometry.. that will definitely take further study later). I got a pretty hard look at Truth today because I was completely lost during the algebraic geometry (ag) lecture and then tried to tackle the problem set to see if I could figure stuff out... and was not successful at all. Fortunately the TAs and professors who were with us helped me out, but I needed a lot of help and I can’t do math with an audience unless I am really comfortable/confident. Which... I wasn’t. Tomorrow should be better though. I am doing well at talking to students; everyone is very nice and, guess what, math people in general are not super outgoing and so I am fitting in very well socially. It is interesting to talk to the other students and learn about their schools, there are some students from really good math programs here! After today I should be better about talking to (not being scared by) the professors.

tl;dr, things are going better than expected!!

On to the math! According to the wikipedia page, a Kahler manifold is one with three compatible structures: (1) a complex structure, (2) a Riemannian structure, and (3) a symplectic structure. We skipped talking about (1) because most people should be pretty familiar with that by now (lol I haven’t taken complex analysis but w/e I’m managing). We also aren’t going to talk about (3) because apparently it is a fairly new branch of research and isn’t as accessible as (2).

SO! What is a Riemannian surface? We are working with Gabor Szekelyhidi slowly toward this concept from an analysis perspective; so far, (from what I can tell), a Riemannian surface is one you can create by stitching together copies of $\IC$ and adding points at infinity to create a compact manifold. You can then map this back to the complex numbers using holomorphic fuctions, and the identifications constructed through these functions allows you to do calculus on this manifold where you can circumvent certain problems, such as not having an injective square root function in the complex numbers (this part I am still unclear, so apologies if I made a mistake), as opposed to sitting and crying (idk, if you’re me you might do this anyway). The professor lecturing on analysis is explaining mainly through pictures, and I am hoping to be able to post something later with more detailed notes. 

We are working much faster toward this concept from an algebraic perspective, because the dude teaching ag, Claudiu Raicu, an associate professor at Notre Dame, is really whipping us along. This is an interesting way to go about doing things, because as far as I know, only a few students have enough of a background in ag to keep up, or even have really any idea what is going on. According to ag, a Riemann surface is a non-singular (affine or projective) curve over the field of complex numbers. A Riemann surface is compact if and only if it is projective. 

There are a *lot* of concepts to unpack here, and each is pretty heavy. A lot of what we are doing involves quotienting the ring of polynomials in 2 or 3 variables by polynomials and determining the size of the result to give you information about the multiplicities of intersections of the polynomials. It’s nice that I am using my algebraic knowledge (oh man, am I digging out ideas that I honestly thought I wouldn’t use again. Turns out local rings are *super* important, and sou is Bezout’s Theorem), and I will need to dredge through this very carefully this week and in the future to make sure I am understanding everything.

The final lecturer is Andrei Jorza, who is lecturing on applications of Riemannian surfaces to number theory. We constructed a fundamental domain for the upper half plane in the complex numbers, and are using the properties of arithmetic on that domain to help prove an equivalence discovered by Ramanujan... That I am not going to get in to right now. To be honest I have been paying the least amount of attention to these problems because they appear to be straight forward; showing that summations are equivalent, and etc. They are definitely good practice and I will do my best to type up those notes and problems later as well.

I will try to update when I can!

[tilde-he]

Approximation of a holomorphic function which is entire, and the value of which is known within some epsilon on some circle

If f is holomorphic and entire, and g is holomorphic and entire, r is positive, and epsilon is positive, and for all z such that |z-z0|=r , |f(z)-g(z)|<epsilon , then

for all z such that |z-z0|<r , |f(z)-g(z)| <= 2*epsilon*r/(r-|z-z0|)

And more generally, |f(m)(z) - g(m)(z)| <= 2*epsilon*r*(m!)/((r-|z-z0|)m+1)

I don’t know that this can be extended outside the circle, and I suspect maybe it can’t be.

I tried to find something like this online but wasn’t able to find anything (or at least, nothing that I understood. There were some things behind paywalls that I’m not sure if they were about this sort of thing or not.). (If anyone knows a similar result, or somewhere else where this has been shown, I’d appreciate a link to it.)

Explanation of motivation: What if the value of f is known on a circle, but only approximately (with some bound on bad the approximation can be)? g is the approximation, and the value of the approximation is exactly known, because if you have a closed form for an approximation, you ... have a closed form for the approximation, so you know the approximation exactly. And the approximation is known to be within epsilon of correct. f and g being entire is a simplifying assumption.

For 2.5 days I’ve felt kinda compelled to work on this problem instead of actual obligations I have. But I think I’ve made enough progress on it that I can focus on what I should focus on now.

If you want a proof of these, send me an ask or a message or something. I’ve written down a proof, but not typed it up.