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

Etale cohomology - you can apply fixed-point theorems from algebraic topology to Galois actions on varieties.

Answer 2

Nonlinear optimization: Newton's method beats everything else (when it works); when it doesn't, do something that looks like Newton's method.

Answer 3

Four-Dimensional Smooth Manifolds: Whitney's trick gone wrong.

Answer 4

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

Answer 5

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

*I suppose this goes for most non-linear PDEs

Answer 6

I think this belongs on this list too:

The theory of groups is a branch of mathematics in which one does something to something and then compares the results with the result of doing the same thing to something else, or something else to the same thing. – James Newman

Answer 7

Representation theory of compact groups: The representation theory is the same as for finite groups, only that there might be infinitely many isomorphism classes of irreducible representations.

(That's the Peter Weyl Theorem!)

Perhaps it would be a much better question, to interpret a well known theorem in one sentence!

Answer 8

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

Answer 9

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 10

Set theory without choice: You have no choice, but to wonder...

Forcing: If it doesn't not fit, force it.

Large cardinals: "If you want more you have to assume more." (Dana Scott)

Answer 11

Additive combinatorics: Any two attempts to define what it means for a finite set to be `additively structured' will be approximately equivalent.

Answer 12

Morse Theory: opus dynamicum maxime.

Answer 13

Harmonic analysis: The integral operator with the kernel (blank space to fill in) is bounded from (blank space to fill in) to (blank space to fill in).

(communicated by Mark Rudelson)

Answer 14

Another favorite of mine …

• Redundancy is the essence of information.

Answer 15

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 16

Number Theory : Arithmetic properties (such as number of rational solutions) of geometric objects (such as elliptic curves) are often reflected in analytical functions (such as L-functions) associated to those objects i.e. geometry reveals its arithmetic analytically.

Answer 17

Probability/Statistical mechanics:

Take a probabilistic model (possibly complicated, involving huge state space, describing a complex system) and rescale it suitably, such that in the limit a simpler "macroscopic" object emerges;

if the latter is still random it's a central limit theorem, if it's deterministic it's a law of large numbers, if you look at fluctuations from the latter it's large deviations; if it is largely independent on the details of the starting probabilsitc model, you have a universality phenomenon (and are happy because when modelling your real system you were forced to add some assumptions just for mathematical comfort); if it changes qualitatively when playing with a parameter of the original model you have a phase transition and want to know the critical values of the parameter.

Answer 18

Geometric representation theory: keep translating the problem until you run into Hard Lefschetz, then you are done.

Answer 19

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

Answer 20

QFT — every expression converges after a Wick rotation.