Examples of advance via good definitions

Question

In my research I came across a case where I could derive a known theorem with rather straightforward way by choosing "non-standard" definitions using my knowledge from a related field. This particular case does not seem to be interesting to wider audience, but I would expect that there are cases where the real crux is in the definitions and the later deductions are just "the necessity" to establish some results. Good definitions do, after all, "compress" prior knowledge in a succinct form and often make the following work easy.

This motivates to ask if there are concrete examples where the real progress is in the definitions and the following (potentially interesting) results are just illustration of the power of the definition(s)?

Answer 1

Not sure if this qualifies since it is a whole theory, but distribution theory is in essence the consequence of a particularly well chosen definition.

Answer 2

A really good example is the definition of a scheme by Grothendieck. Previous attempts (e.g. "Foundations of Algebraic Geometry" by André Weil) were extremely complicated in comparison and didn't have half of the good properties.

Grothendieck basically took the correspondence between classical affine varieties and nilpotent-free finitely generated algebras over a field (given by taking maximal ideals) and started removing adjectives... Pursuing it to the ultimate conclusion, the end result was a theory of geometric spaces that locally "look like" commutative algebra (compare: differential manifold locally "look like" calculus). This theory:

1. Makes a lot of geometric sense once you get over some initial weirdness (embedded points? really??)
2. Unites number theory, commutative rings and algebraic geometry.
3. Gives a rigorous foundation to the classical methods of the 19th century Italian algebraic geometers.

For example, there is a really good theory of degenerations (in fact, much better than that of complex manifolds!) because we have concrete infinitesimals - in the local model, they are actually the nilpotent elements of your ring.

Answer 3

Other examples are "normal family" in the theory of analytic functions (Montel), and its generalization to the rest of mathematics, "compact set" (Aleksandrov and Uryson).

Such examples are really abundant: Hilbert space, Banach space, Harmonic measure, defining all these things really led to new areas of research and revolutionized the existing areas.

For example from the areas other than Analysis, let me mention the "ideals" defined by Dedekind. This definition really led to new ways of thinking in many areas of mathematics.

Some recent striking examples are "fractals" and "attractors". To be sure, there is no standard mathematical definitions of these things and they belong to wider area of science rather than only mathematics. Which probably makes them even more influential.

EDIT. Definition. A definition is called good if it leads to a progress in mathematics:-)

Examples: subharmonic function, conformal map, quasiconformal map, Riemann surface, manifold, Riemannian metric, group, vector space, ... and all examples listed above.

Answer 4

Manjul Bhargava defined the concept of a $p$-ordering of a subset of a Dedekind ring, where $p$ is a prime. In the case of $S\subseteq \mathbb{Z}$, the definition is as follows. Let $a_0\in S$ be arbitrary, and for $i>0$, let $a_i$ be any element of $S$ that minimizes the highest power of $p$ dividing $$(a_i - a_0)(a_i - a_1) \cdots (a_i - a_{i-1}).$$ Stating this definition almost automatically prompts us to examine the sequence of highest powers that arise, and the fundamental theorem of the subject is that this sequence of highest powers is independent of the choice of $p$-ordering. This theorem quickly led to huge progress on some long-standing and difficult questions about polynomial mappings on subsets of $\mathbb{Z}/n\mathbb{Z}$, and also led to a generalized notion of factorials whose theory is an ongoing topic of research.

Answer 5

The definition of combinatorial species, due to Joyal and developed in a book by Bergeron, Labelle and Leroux, makes enumeration and also the computation of the Frobenius character of a permutation action often trivial.

More focused, the main observation is that the plethysm of symmetric functions (or, if you prefer, the composition of polynomial functors) has a very natural "categorification" in terms of composition of combinatorial structures.

As a toy example, to illustrate the above, the sentence

A perfect matching is a set of two-element sets

becomes a theorem with immediate corollaries

The exponential generating function of perfect matchings is $\exp(x^2/2)$.

and

The Frobenius character of the sequence of permutations representations given by the symmetric group acting on perfect matchings is the plethysm $\sum_n h_n\circ h_2$.

Clearly all this was known before - to the experts. However, the "combinatorial" definition of plethysm (and scalar product) renders puzzles that were originally non-obvious into relatively straightforward exercises, and solutions thereof easily adaptable to similar problems.

Answer 6

Here is an answer which interpolates between those by Piero and Nati.

In his Ph.D. thesis Grothendieck introduced the notion of nuclear topological vector space. Essentially, it is the correct definition of "finite-dimensional-like" infinite dimensional space. Namely, all theorems in multilinear algebra and probability which hold for finite dimensional spaces also hold verbatim for nuclear spaces. Although this is speculation on my part, I suspect that one thing Grothendieck learned from Laurent Schwartz is how powerful a mathematical idea a good definition can be, e.g., the definition of distribution. However, this theory would have been rather limited were it not for the second definition of nuclear spaces that Grothendieck provided. He then went on to repeat this trick again and again in algebraic geometry, and the rest is history.

Answer 7

Karol Borsuk's definition of a retract (1931), initially motivated by investigations concerning the fixed-point property of certain compact metric spaces, resulted in the development of the theory of retracts and some far-reaching generalizations in group henry and category theory, see https://en.wikipedia.org/wiki/Retract and https://en.wikipedia.org/wiki/Retraction_(disambiguation) .

Answer 8

The definition of a fibration by Serre is by far my favorite example. There, if I recall correctly, leaders in the field were actively looking for the right definition and knew that it would "quickly lead to the lifting property." Serre's definition was the lifting property - "quickly" became in no time, by definition! (Sorry, I don't recall and now can't find the exact reference for that approximate quote.)

Answer 9

There were numerous errors in work on algebraic number fields before researchers finally hit upon a satisfactory definition of an algebraic integer as the root of a monic polynomial with integer coefficients.

Answer 10

In a more applied field: The definition of extended real valued convex functions (instead of convex functions defined on a convex set) and their subgradients (instead of, for example, directional derivatives) has advanced the field of convex analysis and convex optimization quite a bit. Also, the notion of lower semi-continuity is really handy in this field.

Answer 11

An answer from fiction, so perhaps it doesn't count. In Walter Miller's science fiction classic A Canticle for Liebowitz a character comments on the power of good notation discovered in an ancient text:

"Fragments from a twentieth century physicist! The equations are even consistent."

Kornhoer peered over his shoulder. "I've seen that," he said breathlessly. "I could never make heads or tails of it. Is the subject matter important?"

"I'm not sure yet. The mathematics is beautiful, beautiful! Look here– this expression– notice the extremely contracted form. This thing under the radical sign– it looks like the product of two derivatives, but it really represents a whole set of derivatives."

"How?"

"The indices permute into an expanded expression; otherwise, it couldn't possibly represent a line integral, as the author says it is. It's lovely. And see here– this simple-looking expression. The simplicity is deceptive. It obviously represents not one, but a whole system of equations, in a very contracted form. It took me a couple of days to realize that the author was thinking of the relationships– not just of quantities to quantities– but of whole systems to other systems. I don't yet know all the physical quantities involved, but the sophistication of the mathematics is just– just quietly superb! If it's a hoax, it's inspired! If it's authentic, we may be in unbelievable luck. In either case, it's magnificent. I must see the earliest possible copy of it."

https://7chan.org/lit/src/A_Canticle_for_Leibowitz_-_Walter_M__Miller,_Jr__4.pdf

Answer 12

If you look in the literature, there are multiple different definitions of Euclidean domain. A lot of the differences has to do with how a Euclidean norm is defined at zero. People seem unnecessarily averse to the idea that zero should have largest norm (corresponding to the fact that in the reversed lattice of ideals in a ring, the zero ideal is at the top), and thus the image of a Euclidean norm shouldn't be restricted to $\mathbb{Z}_{\geq 0}$.

In Lenstra's unpublished Lectures on Euclidean Rings, if this type of ordering is used, then the "minimal norm" on a Euclidean domain is shown to be logarithmically superadditive (see Corollary 2.5). Many other natural properties and results flow from this more natural definition.