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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

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

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@Colin: One wants certain functors to be exact, e.g., the Hom-functor gives Exts, tensoring with a module gives Tor. –  J.C. Ottem Feb 28 '11 at 0:26
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For example, once I was comparing $\overline{I\cap J}$ to $\overline{I}\cap \overline{J}$, where the bar denotes taking the associated graded module with respect to some filtration of $R$-ideals $I$ and $J$. I suspected that there was some homology group which vanished exactly when those coincided, and I was correct (it was a rather complicated $Tor$). –  Greg Muller Feb 28 '11 at 2:05

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

Hold on...

Calculus: Differentiation is approximation by a linear map.

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I like your description of calculus -- I am teaching multivariable calculus this semester, and I think the students have a hard time accepting that the "right" definition of differentiability is that a good linear approximation exists, instead of the more natural-seeming idea that all of the first partials exist. –  Gabe Cunningham Oct 22 '09 at 19:16
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About that description of complex analysis, see Needham's Visual Complex Analysis. –  lhf Nov 8 '09 at 23:39
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@Gabe: I'm teaching multivariable calculus this semester too, but I defined the derivative to be the linear approximation first, and then introduced partial derivatives as a useful computational technique. –  Jeff Strom Oct 27 '10 at 18:26

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

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

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When I was taking complex analysis, I remember someone saying "Complex analysis is the Disneyland of mathematics" because so many incredible theorems turn out to be true. –  John D. Cook Oct 25 '09 at 1:51

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

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Lie groups: Think locally, act globally. ;)

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This applies to many other areas as well. –  Gil Kalai Nov 8 '09 at 7:48
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@Gil I totally agree.In fact,this can be the slogan for topology in general with some slight modifications. –  Andrew L Oct 27 '10 at 20:57
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Less catchy, but: "think at the identity, act globally" is more specific to Lie theory. –  Paul Siegel Jan 7 '12 at 18:08

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

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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Analytic combinatorics: generating functions are awesome.

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

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There is also the book by Flajolet and Sedgewick, which is available at algo.inria.fr/flajolet/Publications/books.html –  lhf Nov 10 '09 at 11:50
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@Andrew L : While I didn't vote this answer up, it is at least (arguably) correct. Your answer, on the other hand, reveals a profound misunderstanding of probability theory. Though probability theory uses many tools from real analysis (eg measure theory), the way it uses those tools and the intuition/philosophical explanation behind them is completely different from those of traditional real analysis. Not to mention that your answer pretends there doesn't exist a giant field of finitary probability that is much more closely connected with combinatorics than with real analysis. –  Andy Putman Oct 28 '10 at 1:56
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I can't believe someone would come along a year later and make this comment. –  Michael Lugo Oct 28 '10 at 3:50
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@Michael : Obviously, you have not been following the saga of Andrew L... –  Andy Putman Oct 28 '10 at 4:18
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@Andy Wise-ass comment to Micheal aside,you made a very fair objection above. Discrete probability is fully half the science.I could counter it by saying combinatorics is essentially analysis on finite sets,but that's a real stretch. –  Andrew L Oct 28 '10 at 4:25

One of my favorites:

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

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

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There's also: The purpose of computing numbers is not yet in sight. — Richard Hamming (1971) –  lhf Nov 3 '09 at 0:52

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

Algebraic geometry: CommRing behaves a lot like Setop.

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

  • "Free" is just another word for nothing to do on the left.
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In algebra, "freedom's just another word for nothing left to lose". :-) –  Todd Trimble May 20 '11 at 13:23

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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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

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

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

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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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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

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

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

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This seems a bit too cryptic for me... –  Yemon Choi Nov 8 '09 at 10:09
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Well, when you try to prove some (non too-far-fetched) fact in Noncommutative Ring Theory, you have roughly two main families of techniques to resort to: 1) Techniques which involve modules. Facts about one-sided ideals, the categorical viewpoint, K-theory over the monoid of finitely generated projective modules, homological tools... 2) Techniques which involve idempotents. Taking corners, rings with local units, rings with enough idempotents, the Peirce decomposition... That's what I tried to comprise by this sentence ;-) –  Jose Brox Nov 11 '09 at 11:00

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

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

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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.")

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Just in case anyone is interested, the paper is ams.org/mathscinet-getitem?mr=164252 by Meyers and Serrin –  Willie Wong May 3 '10 at 13:12
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And the "H" is a cyrillic en, that stands for S.M.Nikolsky. –  Pietro Majer Dec 10 '10 at 23:57

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

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Statistics: every parameter is learnable by sampling.

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

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