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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?

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closed as no longer relevant by Felipe Voloch, Todd Trimble, Andres Caicedo, Joël, Alex Bartel Oct 8 '12 at 8:48

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This is a very good question, but to be useful and not just fun one should look critically at many of the answers below. –  Gil Kalai Nov 8 '09 at 7:54
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Gil, I am very skeptical about the value of this question. I don't think many of the answers given are that useful, because one won't get the punchlines unless one has acquired experience in the subject (and then, why would you need the punchline?). –  Todd Trimble May 20 '11 at 13:27
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@Todd: to get fodder for a cocktail party level conversation.... –  Suvrit Aug 28 '12 at 14:32
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@Suvrit: I guess it would be more of a "Big-Bang-Theory"-kind of party ;-) –  vonjd Oct 7 '12 at 18:37

50 Answers 50

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

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

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

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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!

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1  
Aren't there always infinitely many isoclasses in the infinite case? –  Mariano Suárez-Alvarez Aug 2 '11 at 17:11
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Of course, $L^2(G)$ should be an inifinite dimensional Hilbert space, if $G$ is not finite. Hence Peter Weyl tells you that this is indeed so, but finite groups are compact, so I do not see a wrong statement in my answer. Btw amuch more interesting question, does this imply that every compact infiniten group has infinitely many conjugacy classes? –  plusepsilon.de Aug 2 '11 at 22:14

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

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4  
explained, or approximated? –  Colin Tan May 20 '11 at 10:11

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

*I suppose this goes for most non-linear PDEs

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  • Generating functions are the 19th Century analog of addressable memory.
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Another favorite of mine …

  • Redundancy is the essence of information.
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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.

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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.)

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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)

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

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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

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1  
@Todd Trimble If you study groups by their actions on sets, "$x^g$" is "doing g to x". Natural questions are like "when does $x^g = x^h$?" i.e. when does doing something different on the same thing give the same result? –  Petrus May 20 '11 at 14:48

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.

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Morse Theory: opus dynamicum maxime.

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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)

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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.

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

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

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I thought $K$-theory was! –  Mariano Suárez-Alvarez Dec 13 '09 at 4:15

QFT — every expression converges after a Wick rotation.

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Wick rotation isn't what leads to convergence. A better sentence might be "Large size asymptotics of the moments of regularized path integrals are independent of the choice of regularization." –  userN Oct 24 '09 at 15:07

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