Question

When you study a topic for the first time, it can be difficult to pick up the motivations and to understand where everything is going. Once you have some experience, however, you get that good high-level view (sometimes!) What I'm looking for are good one-sentence descriptions about a topic that deliver the (or one of the) main punchlines for that topic.

For example, when I look back at linear algebra, the punchline I take away is "Any nice function you can come up with is linear." After all, multilinear functions, symmetric functions, and alternating functions are essentially just linear functions on a different vector space. Another big punchline is "Avoid bases whenever possible."

What other punchlines can you deliver for various topics/fields?

Answer 1

One punchline in algebraic geometry is that all commutative rings are actually the ring of functions on some space.

Answer 2

QFT — every expression converges after a Wick rotation.

Answer 3

Complex Analysis: Holomorphic functions are just rotations and dilations up to the first order.

Hold on...

Calculus: Differentiation is approximation by a linear map.

Answer 4

Complex Analysis: Taylor series behave the way you want them to in real analysis.

Answer 5

Homological algebra - In an abelian category, the difference between what you wish was true and what IS true is measured by a homology group.

Answer 6

Analytic combinatorics: generating functions are awesome.

("generating functions are awesome" is actually the title of a talk I gave a couple weeks ago.)

Answer 7

Lie groups: Think locally, act globally. ;)

Answer 8

Sobolev spaces: H = W

(There are ostensibly two kinds of Sobolev spaces, denoted with H's and W's, plus some superscripts and subscripts. Someone wrote a paper showing that the two kinds were equivalent and entitled their paper "H=W.")

Answer 9

Operator theory: all separable infinite-dimensional Hilbert spaces are isomorphic, but they aren't all the same and moving your problem between them works wonders.

Answer 10

Linear algebra: everything can be explained by a linear system.

Answer 11

Numerical analysis: The purpose of computing is insight, not numbers. — Richard Hamming (1962)

Answer 12

Algebraic geometry: CommRing behaves a lot like Set^op.

Answer 13

Logic teaches us that (untrained) intuition is often wrong, but that when it's right, it's for the wrong reason.

Answer 14

Noncommutative Ring Theory: If it is not modules, then it is idempotents.

Answer 15

Navier-Stokes Equations: Energy estimates and more energy estimates.

*I suppose this goes for most non-linear PDEs

Answer 16

Real Analysis: Get your hypotheses right, or suffer the counter-examples!

Measure Theory: "Every [measurable] set is nearly a finite union of intervals; every [measurable] function is nearly continuous; every convergent sequence of [measurable] functions is nearly uniformly convergent." -- J.E. Littlewood

Answer 17

The bonniest mot I can ever recall — from some graduate algebra course:

• "Free" is just another word for nothing to do on the left.

Answer 18

Another favorite of mine …

• Redundancy is the essence of information.

Answer 19

• Generating functions are the 19th Century analog of addressable memory.

Answer 20

One of my favorites:

"Algebraic topology is the "art" of Not doing the integral"

Answer 21

Linear Algebra is the correct generalization of dimension. (This came from Hubbard)

Answer 22

"set theory is the study of well-foundedness" - A.R.D Mathias

Answer 23

Geometric group theory: the large-scale geometry of a group is invariant under quasi-isometry.

Answer 24

Configuration space integrals: Don't take limits- compactify!

Dror Bar-Natan explained this punchline to me when I was just starting grad school.

Answer 25

Statistics: every parameter is learnable by sampling.

Answer 26

Representation theory of Lie groups: there is a whole world between $\mathrm{Sym}^n V$ and $\wedge^n V$. (Okay, this is an oversimplication - I am talking about the representations of $\mathrm{GL}\left(V\right)$ here, but this is the fundament of all other classical groups.)

Constructive logic: if you can't compute it, shut up about it. (At least some forms of constructive logic. Brouwer seemed to have a different opinion iirc.)

Homological algebra: How badly do modules fail to behave like vector spaces?

Gröbner basis theory: polynomials in $n$ variables can be divided with rest (at least if you have some $O\left(N^{N^{N^{N}}}\right)$ of time)

Finite group classification: what works for Lie groups will surely be even simpler for finite groups, right? ;)

Algebraic group theory: In order to differentiate a function on a Lie group, we just have to consider the group over $\mathbb R\left[\varepsilon\right]$ for an infinitesimal $\varepsilon$ ($\varepsilon^2=0$).

Semisimple algebras: The representations of a sufficiently nice algebra mirror a structure of the algebra itself, namely how it breaks into smaller algebras.

$n$-category theory: all the obvious isomorphisms, homotopies, congruences you have always been silently sweeping under the rug are coming back to have their revenge.

Modern algebraic geometry (schemes instead of varieties): let's have the beauty of geometry without its perversions.

How many of these did I get totally wrong?

Answer 27

Algebraic geometry is the study of the intrinsic properties of any mathematical object which can be locally described by polynomial equations.

Or

Algebraic geometry is not about solving systems of polynomial equations, rather it's about studying the intrinsic properties thereof.

Answer 28

Analytic Number Theory: log log log log log...

Did I see that quote in Havil's book Gamma?

Answer 29

Dirichlet forms: a symmetric Markov process is a self-adjoint operator is a closed symmetric form is a Markovian semigroup.

(I've left out a lot of hypotheses, but the essence is that all these are in correspondence, and the properties of any one appear in the others.)

Answer 30

Functional analysis: Everything you know from linear algebra is true, under the right conditions; otherwise it's false.