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

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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.

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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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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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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

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It's a question of where you put the quantifiers. For almost every point, the value is almost the same as it is at almost every nearby point. –  gowers Oct 28 '10 at 16:59

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

I'll offer two punchlines for Galois Theory.

  1. There's a one-to-one, order-reversing correspondence between intermediate fields of a finite, normal, separable extension $K$ of $F$, and subgroups of the group of automorphisms of $K$ fixing $F$.

  2. A polynomial is solvable in radicals if and only if the Galois group of its splitting field is a solvable group.

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

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

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

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Deeper than it looks like at first sight, you shouldn't vote it down so easily! –  Jose Brox Nov 8 '09 at 2:19

Morse Theory: opus dynamicum maxime.

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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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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? –  Marc Palm Aug 2 '11 at 22:14

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?

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I'm sure at least some people would reverse the last one... –  Ketil Tveiten Oct 28 '10 at 11:56
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D Grinberg, surely you meant ‘Lie group’ rather than ‘Lie algebra’ in the finite-group classification? –  L Spice May 21 '11 at 6:38
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@n-category theory: I would definitively watch that movie! :-D –  Johannes Hahn Oct 7 '12 at 18:50

Terry Tao, in a post on Google Buzz, has given an overview of mathematics in the form of multiple "punch-lines" of the requested variety.

Here are three examples from Tao's post:

  • Algebra is the mathematics of the "equals" sign, of identity, and of the "main term"; analysis is the mathematics of the "less than" sign, of magnitude, and of the "error term".

  • Algebra prizes structure, symmetry, and exact formulae; analysis prizes smoothness, stability, and estimates.

  • Most of geometry would not be classified as either algebra or analysis, but simply as geometry.

Definitely Tao's aphorisms are thought-provoking and inspiring ... but are they useful ? Don't ask me! :)


Partly inspired by Tao's essay, here is a one-sentence definition of quantum mechanics (as optimized for systems engineers)  …

  • Quantum mechanics is the algebraic geometry of $n$-particle Hamiltonian flows and Lindbladian compressions as pulled-back onto the natural $r$-indexed stratification of $r$'th secant varieties of $n$-factor Segre varieties whose $r\to\infty$ limit is … $n$-particle Hilbert space.

… and it turns out to be very useful (and great fun) to rewrite standard quantum physics texts like Charles Slichter's Principles of Magnetic Resonance based upon this one sentence definition.

Joseph Landsberg's recent Bull. AMS review "Geometry and the complexity of matrix multiplication" (2008), which has been praised in multiple MathOverflow posts, provides an overview of the broad utility—despite their unwieldy name—of stratifications of secant varieties of Segre varieties (which extends far beyond quantum physics).

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Quantization: First quantization is a mystery, second quantization is a functor (Edward Nelson)

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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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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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@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

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

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Analysis: Allez en avant, et la foi vous viendra (D'Alembert, to a student who had difficulty in believing the calculus of infinitely small. Translation: go on, and faith will be bestowed on you :)

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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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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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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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Topological Vector Spaces: You can make an infinite dimensional space have every nice property of finite dimensional spaces- but not all of them at once.

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Algebraic Topology: Geometry is hard, and Algebra is easy so...

(I am sure this applies to many other fields, and certainly algebra is hard.)

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Homotopy theory is an attempt to do homological algebra in non-abelian categories.

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really andrew? If anything most answers are vast oversimplifications. This one does in fact have some meat to it though. Quillen's theory of model categories is precisely what (I think) is being referenced. With a simplicial model structure you can do resolutions like you would in homological algebra. –  Sean Tilson Nov 28 '10 at 17:20
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Quillen's first paper on model categories was called "Homotopical Algebra" to emphasize this analogy. But homotopy theory was a subject before that sort of abstract homotopy theory came in, and although derived-functor methods are an important tool in the homotopy theory of spaces they aren't what it's all about. –  Tom Goodwillie May 21 '11 at 14:38

Renormalization in quantum field theory: "just because something is infinite doesn't mean it is zero". (Explanation: this was said in about 1950 when regularization/renormalization was discovered as a way of getting sensible non-zero values for formally infinite expressions.)

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

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Like MO points are the end-all and be-all of existence. –  Ketil Tveiten Oct 28 '10 at 11:57
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One difference is that whereas most linear algebra concepts generalize nicely to, say, Banach spaces, differentiation, perhaps the most basic concept of calculus, doesn't make sense in a topological space. –  gowers Oct 28 '10 at 14:17
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I like this one because despite its tautological flavor, it is not. –  Pietro Majer Dec 30 '10 at 17:41
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... differentiation being just another linear operator.... under the right conditions. :) –  paul garrett Oct 7 '12 at 17:31

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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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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Statistics: every parameter is learnable by sampling.

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