## Describe a topic in one sentence. [closed]

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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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 2009 at 7:54
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 2011 at 13:27
@Todd: to get fodder for a cocktail party level conversation.... – S. Sra Aug 28 at 14:32
@Suvrit: I guess it would be more of a "Big-Bang-Theory"-kind of party ;-) – vonjd Oct 7 at 18:37

## closed as no longer relevant by Felipe Voloch, Todd Trimble, Andres Caicedo, Joël , Alex BartelOct 8 at 8:48

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

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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 2009 at 15:07

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 2009 at 19:16
About that description of complex analysis, see Needham's Visual Complex Analysis. – lhf Nov 8 2009 at 23:39
@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 2010 at 18:26

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 2009 at 1:51

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 2011 at 0:26
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 2011 at 2:05

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 2009 at 11:50
@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 2010 at 1:56
I can't believe someone would come along a year later and make this comment. – Michael Lugo Oct 28 2010 at 3:50
@Michael : Obviously, you have not been following the saga of Andrew L... – Andy Putman Oct 28 2010 at 4:18
@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 2010 at 4:25

Lie groups: Think locally, act globally. ;)

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This applies to many other areas as well. – Gil Kalai Nov 8 2009 at 7:48
@Gil I totally agree.In fact,this can be the slogan for topology in general with some slight modifications. – Andrew L Oct 27 2010 at 20:57
Less catchy, but: "think at the identity, act globally" is more specific to Lie theory. – Paul Siegel Jan 7 2012 at 18:08
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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 2010 at 13:12
And the "H" is a cyrillic en, that stands for S.M.Nikolsky. – Pietro Majer Dec 10 2010 at 23:57
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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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Linear algebra: everything can be explained by a linear system.

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explained, or approximated? – Colin Tan May 20 2011 at 10:11

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 2009 at 0:52

Algebraic geometry: CommRing behaves a lot like Setop.

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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 2009 at 2:19

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 2009 at 10:09
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 2009 at 11:00

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

*I suppose this goes for most non-linear PDEs

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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 2010 at 16:59

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 2011 at 13:23

Another favorite of mine …

• Redundancy is the essence of information.
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• Generating functions are the 19th Century analog of addressable memory.
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One of my favorites:

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

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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 2009 at 4:15

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

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

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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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Statistics: every parameter is learnable by sampling.

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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 2010 at 11:56
D Grinberg, surely you meant ‘Lie *group*’ rather than ‘Lie algebra’ in the finite-group classification? – L Spice May 21 2011 at 6:38
@n-category theory: I would definitively watch that movie! :-D – Johannes Hahn Oct 7 at 18:50

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

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

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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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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 2010 at 11:57
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 2010 at 14:17
I like this one because despite its tautological flavor, it is not. – Pietro Majer Dec 30 2010 at 17:41
... differentiation being just another linear operator.... under the right conditions. :) – paul garrett Oct 7 at 17:31