Has Abstract Algebra ever been of service to Analysis?

Question

I’m not saying that it ought to be. I was just wondering whether it has. What I have in mind is that it would have been of material help in proving, say, the Hahn-Banach Theorem, or some such. If it has, what is the most important/impressive instance of this?

Answer 1

Speaking as someone who is basically an algebraist, I think of algebra as using structure to help understand or simplify a mathematical situation. The common algebraic objects (groups, rings, Lie algebras, etc.) reflect common structures that appear in many different contexts.

Now analysis often seems to have a certain slipperiness that makes it hard to pin down precise structures that meaningfully persist across different problems and contexts, and hence seems to have been somewhat resistant to methods of algebra in general. (This is an outsider's impression, and shouldn't be taken too seriously. But it does honestly reflect my impression ... .)

On the other hand, there do seem to be places where algebra can sneak in and play a role. One is mentioned by Qiaochu: Wiener proved that if $f$ is a nonwhere zero periodic function with absolutely convergent Fourier series, then $1/f$ again has an absolutely convergent Fourier series. Wiener's proof was by hand harmonic analysis, but a conceptually simpler proof (in a much more general setting) was supplied by Gelfand (I believe) using the theory of Banach algebras, which hinges on algebraic concepts such as maximal ideals and radicals. Wiener proved his result as a step along the way to proving his general Tauberian theorem, and this result again admits a conceptually simpler proof, and generalization, via Banach algebra methods.

Another, more recent, example is the work of Green and Tao on asymptotics for the Hardy--Littlewood problem of solving linear equations in primes. Here they introduced algebraic ideas related to nilpotent Lie groups, which play a key role in understanding and analyzing the complexity and solubility of such equations.

Answer 2

Yes, differential equations is, intuitively speaking, probably about as far as you can get from abstract algebra in the realm of analysis. Yet algebra still manages to rear its ugly head there :).

One example is the entire subject of D modules (see also Sato's algebraic analysis that Akhil mentioned in his comment.)

But my favourite example in the somewhat surprising application of algebra to analysis is that of lacunas for hyperbolic differential operators. Given a constant coefficient hyperbolic partial differential operator $P(D)$ on $\mathbb{R}^n$, it has associated to it a fundamental solution $E$, which solves the equation that $P(D)E = \delta$, the Dirac distribution. Using the fundamental solution we can write the solution to the inhomogeneous initial value problem $P(D)u = f$, $(u, \partial_t u)|_{t=0} = (u_0,u_1)$ in integral form. A Petrowsky lacuna of $P(D)$ is a region in which the fundamental solution $E$ vanishes.

Now, by hyperbolicity, the fundamental solution is supported within a bi-cone with vertex at the origin, a condition known as finite speed of propagation. So that forms a trivial region in which $E$ vanishes. But other than that the situation can be complicated. Just consider the fundamental solution for the linear wave equation. In odd number of spatial dimensions, the fundamental solution is supported precisely on the set $\{t^2 = r^2\}$. So the region $\{t^2 > r^2\}$ form lacunas for $P(D)$. On the other hand, in even number of spatial dimensions, the fundamental solution is supported on the whole set $\{ t^2 \geq r^2\}$: the equations look almost exactly the same, but the difference between even and odd dimensions is huge!

Petrowsky wrote a paper in 1945 giving precise "topological" conditions for the existence and characterisation of lacunas. Later on, Atiyah, Bott, and Garding wrote two papers revisiting this problem, in which the theorem(s) of Petrowsky are proved using an algebraic geometric framework. One can read more about this in Atiyah's Seminaire Bourbaki notes.

Answer 3

I am not sure how to respond to this question. Functional analysis is a big part of analysis, and functional analysis is largely considered with topological vector spaces; do vector spaces count as "abstract algebra"?

Does Fourier analysis count as "service"? It and its more general companions surely count as the most important intersection of algebra and analysis both in pure and applied mathematics.

How about the theory of Banach algebras? They are the natural setting for spectral theory, which is surely an important analytic topic, and they can also be used to prove an important lemma of Wiener and study quantum mechanics and all sorts of other things.

Answer 4

Lie-theoretic methods play a big role in the group-theoretic approach to special functions and separation of variables in differential equations. For example, Willard Miller showed that the powerful Infeld-Hull factorization / ladder method - widely exploited by physicists - is equivalent to the representation theory of four local Lie groups. This Lie-theoretic approach served to powerfully unify and "explain" all prior similar attempts to provide a unfied theory of such classes of special functions, e.g. Truesdell's influential book An Essay Toward a Unified Theory of Special Functions. Below is the first paragraph of the introduction to Willard Miller's classic monograph Lie theory and special functions

This monograph is the result of an attempt to understand the role played by special function theory in the formalism of mathematical physics. It demonstrates explicitly that special functions which arise in the study of mathematical models of physical phenomena and the identities which these functions obey are in many cases dictated by symmetry groups admitted by the models. In particular it will be shown that the factorization method, a powerful tool for computing eigenvalues and recurrence relations for solutions of second order ordinary differential equations (Infeld and Hull), is equivalent to the representation theory of four local Lie groups. A detailed study of these four groups and their Lie algebras leads to a unified treatment of a significant proportion of special function theory, especially that part of the theory which is most useful in mathematical physics.

See also Miller's sequel Symmetry and Separation of Variables. Again I quote from the preface:

This book is concerned with the relationship between symmetries of a linear second-order partial differential equation of mathematical physics, the coordinate systems in which the equation admits solutions via separation of variables, and the properties of the special functions that arise in this manner. It is an introduction intended for anyone with experience in partial differential equations, special functions, or Lie group theory, such as group theorists, applied mathematicians, theoretical physicists and chemists, and electrical engineers. We will exhibit some modern group-theoretic twists in the ancient method of separation of variables that can be used to provide a foundation for much of special function theory. In particular, we will show explicitly that all special functions that arise via separation of variables in the equations of mathematical physics can be studied using group theory. These include the functions of Lame, Ince, Mathieu, and others, as well as those of hypergeometric type.

This is a very critical time in the history of group-theoretic methods in special function theory. The basic relations between Lie groups, special functions, and the method of separation of variables have recently been clarified. One can now construct a group-theoretic machine that, when applied to a given differential equation of mathematical physics, describes in a rational manner the possible coordinate systems in which the equation admits solutions via separation of variables and the various expansion theorems relating the separable (special function) solutions in distinct coordinate systems. Indeed for the most important linear equations, the separated solutions are characterized as common eigenfunctions of sets of second-order commuting elements in the universal enveloping algebra of the Lie symmetry algebra corresponding to the equation. The problem of expanding one set of separable solutions in terms of another reduces to a problem in the representation theory of the Lie symmetry algebra.

See Koornwinder's review of this book for a very nice concise introduction to the group-theoretic approach to separation of variables.

Answer 5

Here is how Galois theory is applied to differential equations. Although I know nothing about this subject here is an excerpt from Rotman's book on Galois theory.

From Rotman's Galois theory book.

• There is Galois theory in differential equations, due to Ritt and Kolchin. A derivation of a field $F$ is an additive homomorphism $D: F \to F$ with $D(xy) = xD(y) + D(x)y$; an ordered pair $(F,D)$ is called a Differential Field. Given a differential field $(F,D)$ with $F$ a (possibly infinite) extension of $\mathbb{C}$, its differential Galois group is the subgroup of $\text{Gal}(F/\mathbb{C})$ consisting of all $\sigma$ commuting with $D$. If this group is suitably topologized and if the extension $F/\mathbb{C}$ satisfies conditions analogous to being a Galois extension (it is called a Picard - Vessiot extension), then there is a bijection between the intermeditate differential fields and the closed subgroups of the differential Galois group. The latest developments are in "$\text{A. Magid}$ Lectures on differential Galois theory" published by the American Mathematical Society.

Answer 6

Developing analysis beyond Euclidean space (e.g., on manifolds that don't a priori have a natural embedding into $\mathbb R^n$) requires a fair dose of multilinear algebra in order to define differential forms, which are the objects that are integrated. I think integration on manifolds counts as something of basic interest to analysts. :)