Does analysis do anything other than estimating things and discussing convergence?

Question

Let me first of all declare that till a few months ago, Analysis was the subject I liked most, as I was pretty good at it and the idea was simple: you essentially bound things.

But then, I had these spurts of realizations: has Analysis nothing else to offer? Let's take a quick look at what analysis does.

Fourier analysis: Hell lot of PDE's, heat equations, etc. and hell lot of estimating things. The concept of generalized function I admit, is really nice, but that's essentially all to it. From the beginning to the end, estimate integrals or show certain functions belong to a class.

Complex analysis: This is a beautiful subject, although estimations crop up here as well, quite often. However, I do like things like Cauchy's Theorem, Morera's Theorem, etc. simply because they are NOT estimating things!

Functional analysis: Convergence on arbitrary spaces and some fairly complicated existential theorems. Due to lack of concrete integration, there is lack of estimating things but then, there's always the concept of showing convergence.

Analytic Number Theory: I got bored to death trying to study this. From the first page to the last (probably the book I chose wasn't friendly) I saw integrals being estimated.

I have no grievance towards analysis in particular, and as I have mentioned, I am actually good in it. I can grasp analysis concepts really well and my background is quite strong. However, after a point, you really begin to wonder whether a subject has anything else to offer other than estimating integrals/series and checking convergence. I, unfortunately, haven't been exposed to prospective fields of Analysis which go beyond these. So, at times, the journey has been immensely boring.

I would like to ask the community of mathematicians here: what is your opinion? I would love to know if there are topics in analysis beyond these estimations and computations, so if you know of them please do tell me.

Yes, something I missed is: why did I like analysis? Because it reduced a lot of computations I used to do as a high school student. Look at the Riemann-Lebesgue Lemma. Look at the power of Stone-Weierstrass. These are theorems that really boost my interest. But then, what about the rest?

Answer 1

Well, one might say that algebra consists solely on moving symbols around, and/or solving equations (which would be a gross oversimplification), in the same spirit that analysis consists solely on making estimations.

You seem to be missing the bigger picture here, with the connections between all areas of mathematics. For example, just as you noted, most modern number theory is done with complex analysis, and there a hell lot of (very hard) estimations to be done. I don't think I need to explain why number theory is important, or at least interesting.

Probability theory, and more generaly non-commmutative probability theory (or even more generally all kinds of quantum structures being continuously developed) are very operator algebraic in nature, which is usually thought as a subbranch of functional analysis.

Analysis and estimation is also very important in some duality results, the most famous probably Gelfand's duality theorem for commutative C*-algebras and (locally) compact Hausdorff spaces, which is essentially equivalent to Stone-Weierstrass. This duality result states, in simple terms, that topology and algebra are the same (up to some restrictions).

As in one of the comments, whenever you have any kind of infinite/continuous process, you will need to deal with limits and convergence (say, to study the assymptotic behavior or stability of some process). This basically encompass all "dynamical systems" (with whatever definition you want).

Something interesting that happens, however, is that several kinds of interesting structures have analytical and algebraically structures intimately connected. For example, there are some results that state that the topology of a C*-algebra (an involutive Banach algebra with a few properties) doesn't really matter: The algebraic structure is enough to completely determine it. There are also lots of results results which deal with "automatic continuity" of lots of functions on general topological structures.

This kind of analysis of "general" topological and algebraically interconnected structures is usually called "soft" analysis, while, all this estimation of complicated functions and integrals is called "hard" analysis. The last examples would fall in the "soft" category, and you can't avoid estimation and convergence altogether.

(\end{rant})

Answer 2

A very beautiful application is constructing "newton fractals"

You look to which root the newton method converges if you start with a given number (if it converges at all) and color the point depending on the final root.

This gives very beautiful pictures, unfortunately my java applet did not run when I tried to produce an own picture and I do not know where I can get free software for this.

Another easier beautiful application is the bifurcation of the logistic equation. A rich structure arising from the easy function $f(x)=rx(1-x)$ with parameter $r$.

Answer 3

In a sense, analysis is the art of estimation (like algebra is an art of introducing structures), so don't expect a hardcore analysis paper to look anything but a chain of inequalities on the surface. The "let $\varepsilon>0$" opening sentence is as ubiquitous here as "Once upon a time" in the folk tales.

The point however is that, when facing an analytic problem, we seldom know the answer, so it is almost never in the form "Prove that $A\le \text{Const}B$". Typically it is in the form "Find a non-oscillatory condition equivalent to the boundedness of an operator with a sign changing kernel" or "Figure out what is the largest possible size of a set with some given property" or "Prove or disprove that we have a unique solution of something and that solution is behaving nicely", or "Does every operator in a Hilbert space have an invariant subspace", or.... Now, it is a pure matter of taste if you prefer questions of this type to something like "Is it true that the ring of matrices over a non-commutative nilpotent ring is always nilpotent?" or the other way around. However, keep in mind that in analysis, unlike algebra, we deal with things whose behavior cannot be grasped entirely. The task is always to discern the "main terms" in the behavior (be it an asymptotic formula or a geometric construction) and to convince ourselves that the "error terms" are, indeed, negligible. That latter step usually requires a lot of "boring" estimations.

There are standard tricks, of course, and more often than not about 5/6 of an exposition is just a careful routine check that nothing goes astray where it should not. However that routine check becomes possible only after you figured out where exactly to look and find out what is there at least in the first approximation. Alas, the discussion of how that was figured out is normally left out of the exposition and reduced to "Define ... by ... . We shall show that...", after which some long computation follows leaving the impression that the key to everything is in it, while more often than not it is just one of 15 possible implementations of the idea (and usually neither the slickest, nor the most transparent one).

You say that you feel like you have a fairly good background in analysis by the moment. I wonder if you have ever tried to solve a problem about which you wouldn't be sure which tools to apply, which way the answer goes, and whether the solution might be within your reach or not. If yes, I'm curious what was it.

If not, you haven't really tried it yet and just judge analysis from the perspective of a spectator on a soccer game, who wonders if there is more to the game than hitting a spherical object by foot so that it would go into a rectangular frame. You cannot appreciate soccer unless you play it yourself and the same is with analysis (or algebra, or geometry, or probability, or computer science, or...) So, in that case I suggest you try now. The problem I'll offer has the classical flavor of "estimate an integral" just to leave no doubt that it is the very same "classical analysis" you got so bored with, but I'm not telling you anything about it either in terms of the degree of difficulty, or in terms of where it came from.

Let $P(x,y,z)=xyz+Q(x,y,z)$ be a polynomial of $3$ real variables of degree $5$ such that all coefficients of $Q$ are less in absolute value than some fixed positive $\varepsilon$ (which you are free to fix to be any small number you want). Let $\alpha\in(0,1)$ be close to $1$ (say, $0.99$). Can we find a universal constant $C$ such that $$ \int_B \frac{dx\,dy\,dz}{|P(x,y,z)|^\alpha}\le C $$
where $B$ is the unit ball in $\mathbb R^3$ centered at the origin?

Let's see if you can figure it out.