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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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    $\begingroup$ This is a very good question, but to be useful and not just fun one should look critically at many of the answers below. $\endgroup$
    – Gil Kalai
    Commented Nov 8, 2009 at 7:54
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    $\begingroup$ 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?). $\endgroup$
    – Todd Trimble
    Commented May 20, 2011 at 13:27
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    $\begingroup$ @Todd: to get fodder for a cocktail party level conversation.... $\endgroup$
    – Suvrit
    Commented Aug 28, 2012 at 14:32
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    $\begingroup$ @Suvrit: I guess it would be more of a "Big-Bang-Theory"-kind of party ;-) $\endgroup$
    – vonjd
    Commented Oct 7, 2012 at 18:37

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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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  • $\begingroup$ May I ask, what is it that one wishes to be true (in an abelian category)? $\endgroup$
    – user2529
    Commented Feb 27, 2011 at 13:11
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    $\begingroup$ @Colin: One wants certain functors to be exact, e.g., the Hom-functor gives Exts, tensoring with a module gives Tor. $\endgroup$
    – J.C. Ottem
    Commented Feb 28, 2011 at 0:26
  • $\begingroup$ Those are certainly the examples that give the most classical derived functors, but I think this platitude is more helpful when applied to a question which isn't obviously measured by a homology group. $\endgroup$
    – Greg Muller
    Commented Feb 28, 2011 at 2:03
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    $\begingroup$ 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 $Tor$). $\endgroup$
    – Greg Muller
    Commented Feb 28, 2011 at 2:05
  • $\begingroup$ What would be the difference measured by cohomology groups? $\endgroup$
    – Camilo Sarmiento
    Commented Oct 8, 2012 at 15:32
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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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  • $\begingroup$ And he gets 1 point for THIS. Might as well say,"Topology.Everything you know from calculus is true under the right conditionn.Otherwise it's false," I give up......... $\endgroup$
    – The Mathemagician
    Commented Oct 28, 2010 at 7:10
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    $\begingroup$ Like MO points are the end-all and be-all of existence. $\endgroup$
    – Ketil Tveiten
    Commented Oct 28, 2010 at 11:57
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    $\begingroup$ 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. $\endgroup$
    – gowers
    Commented Oct 28, 2010 at 14:17
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    $\begingroup$ I like this one because despite its tautological flavor, it is not. $\endgroup$
    – Pietro Majer
    Commented Dec 30, 2010 at 17:41
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    $\begingroup$ ... differentiation being just another linear operator.... under the right conditions. :) $\endgroup$
    – paul garrett
    Commented Oct 7, 2012 at 17:31
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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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    $\begingroup$ 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. $\endgroup$
    – Gabe Cunningham
    Commented Oct 22, 2009 at 19:16
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    $\begingroup$ About that description of complex analysis, see Needham's Visual Complex Analysis. $\endgroup$
    – lhf
    Commented Nov 8, 2009 at 23:39
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    $\begingroup$ @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. $\endgroup$
    – Jeff Strom
    Commented Oct 27, 2010 at 18:26
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    $\begingroup$ How can we describe a second-order derivative? $\endgroup$
    – user2529
    Commented Feb 27, 2011 at 13:13
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    $\begingroup$ Colin: when I think of Hessians, I think of billowing clouds. (I'm not sure how easy that is to grasp. Think of an expanding cloud of smoke, and how molecules are pushing away from each other as seen from the perspective of projecting their trajectories down onto the tangent plane at a given molecule at a given instant. The behavior is locally given by fundamental ellipsoids, corresponding to a diagonalization of a matrix of second partials.) $\endgroup$
    – Todd Trimble
    Commented May 20, 2011 at 14:22
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Lie groups: Think locally, act globally. ;)

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    $\begingroup$ This applies to many other areas as well. $\endgroup$
    – Gil Kalai
    Commented Nov 8, 2009 at 7:48
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    $\begingroup$ @Gil I totally agree.In fact,this can be the slogan for topology in general with some slight modifications. $\endgroup$
    – The Mathemagician
    Commented Oct 27, 2010 at 20:57
  • $\begingroup$ A slight variant of it has been proposed elsewhere: arxiv.org/abs/0710.5295. $\endgroup$
    – LSpice
    Commented May 21, 2011 at 6:34
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    $\begingroup$ Less catchy, but: "think at the identity, act globally" is more specific to Lie theory. $\endgroup$
    – Paul Siegel
    Commented Jan 7, 2012 at 18:08
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Complex Analysis: Taylor series behave the way you want them to in real analysis.

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    $\begingroup$ 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. $\endgroup$
    – John D. Cook
    Commented Oct 25, 2009 at 1:51
  • $\begingroup$ Every complex differentiable function is complex analytic. So we call this common concept as "holomorphic". $\endgroup$
    – Colin Tan
    Commented Dec 16, 2022 at 13:06
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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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  • $\begingroup$ Not to disparage the quote, but I find it funny that a function that is nowhere continuous (such as x=0 if rational, 1 otherwise) can be called 'nearly continuous'. $\endgroup$
    – Ketil Tveiten
    Commented Oct 28, 2010 at 11:51
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    $\begingroup$ 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. $\endgroup$
    – gowers
    Commented Oct 28, 2010 at 16:59
  • $\begingroup$ and finally someone precisely explains why I don't like real analysis ;) $\endgroup$
    – David Corwin
    Commented Dec 28, 2012 at 6:58
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One punchline in algebraic geometry is that all commutative rings are actually the ring of functions on some space.

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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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    $\begingroup$ There is also the book by Flajolet and Sedgewick, which is available at algo.inria.fr/flajolet/Publications/books.html $\endgroup$
    – lhf
    Commented Nov 10, 2009 at 11:50
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    $\begingroup$ @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. $\endgroup$
    – Andy Putman
    Commented Oct 28, 2010 at 1:56
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    $\begingroup$ I can't believe someone would come along a year later and make this comment. $\endgroup$
    – Michael Lugo
    Commented Oct 28, 2010 at 3:50
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    $\begingroup$ @Michael : Obviously, you have not been following the saga of Andrew L... $\endgroup$
    – Andy Putman
    Commented Oct 28, 2010 at 4:18
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    $\begingroup$ @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. $\endgroup$
    – The Mathemagician
    Commented Oct 28, 2010 at 4:25
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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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Numerical analysis: The purpose of computing is insight, not numbers. — Richard Hamming (1962)

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    $\begingroup$ There's also: The purpose of computing numbers is not yet in sight. — Richard Hamming (1971) $\endgroup$
    – lhf
    Commented Nov 3, 2009 at 0:52
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One of my favorites:

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

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

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  • $\begingroup$ Hmmmmmm-any of the topologists want to take issue with this one as an oversimplification? $\endgroup$
    – The Mathemagician
    Commented Nov 28, 2010 at 6:19
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    $\begingroup$ 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. $\endgroup$
    – Sean Tilson
    Commented Nov 28, 2010 at 17:20
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    $\begingroup$ 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. $\endgroup$
    – Tom Goodwillie
    Commented May 21, 2011 at 14:38
  • $\begingroup$ It's actually one of the better slogans here. $\endgroup$
    – Todd Trimble
    Commented Oct 7, 2012 at 18:10
  • $\begingroup$ Although I'd apply it more to model category theory. $\endgroup$
    – Todd Trimble
    Commented Oct 7, 2012 at 18:11
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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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    $\begingroup$ In algebra, "freedom's just another word for nothing left to lose". :-) $\endgroup$
    – Todd Trimble
    Commented May 20, 2011 at 13:23
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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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  • $\begingroup$ I was going to make a snarky comment about typos, but then your explanation made me think it wasn't a typo after all. As someone who doesn't know anything about renormalisation, I ask snark-free: is the quote really "… doesn't mean it is zero" rather than "… doesn't mean it isn't zero"? $\endgroup$
    – LSpice
    Commented Oct 8, 2012 at 14:39
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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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    $\begingroup$ I'm sure at least some people would reverse the last one... $\endgroup$
    – Ketil Tveiten
    Commented Oct 28, 2010 at 11:56
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    $\begingroup$ D Grinberg, surely you meant ‘Lie group’ rather than ‘Lie algebra’ in the finite-group classification? $\endgroup$
    – LSpice
    Commented May 21, 2011 at 6:38
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    $\begingroup$ @n-category theory: I would definitively watch that movie! :-D $\endgroup$
    – Johannes Hahn
    Commented Oct 7, 2012 at 18:50
  • $\begingroup$ I like how the smiley ";)" is key to the description of finite group classification ;) $\endgroup$
    – David Corwin
    Commented Dec 28, 2012 at 7:03
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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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    $\begingroup$ Deeper than it looks like at first sight, you shouldn't vote it down so easily! $\endgroup$
    – Jose Brox
    Commented Nov 8, 2009 at 2:19
  • $\begingroup$ I disagree that most logic is true for the wrong reasons. I think many people just haven't developed good intutitions about logic. $\endgroup$
    – Richard Dore
    Commented Nov 11, 2009 at 19:43
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    $\begingroup$ Studying logic seems to be equivalent to developing good intuitions about logic. I'll edit this to read "untrained intuition"; maybe that is a better way to put it. $\endgroup$
    – user1241
    Commented Nov 12, 2009 at 7:43
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Noncommutative Ring Theory: If it is not modules, then it is idempotents.

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    $\begingroup$ This seems a bit too cryptic for me... $\endgroup$
    – Yemon Choi
    Commented Nov 8, 2009 at 10:09
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    $\begingroup$ 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 ;-) $\endgroup$
    – Jose Brox
    Commented Nov 11, 2009 at 11:00
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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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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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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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  • $\begingroup$ I find Tao's slogans much better than yours -- sorry. $\endgroup$
    – Todd Trimble
    Commented Oct 7, 2012 at 18:19
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"set theory is the study of well-foundedness" - A.R.D Mathias

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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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  • $\begingroup$ This is intriguing. I'd like to see more examples. $\endgroup$
    – Student
    Commented Nov 26, 2019 at 21:15
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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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About Sobolev spaces, Meyers and Serrin theorem H = W

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    $\begingroup$ Just in case anyone is interested, the paper is ams.org/mathscinet-getitem?mr=164252 by Meyers and Serrin $\endgroup$
    – Willie Wong
    Commented May 3, 2010 at 13:12
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    $\begingroup$ And the "H" is a cyrillic en, that stands for S.M.Nikolsky. $\endgroup$
    – Pietro Majer
    Commented Dec 10, 2010 at 23:57
  • $\begingroup$ But this slogan explains nothing. $\endgroup$
    – Todd Trimble
    Commented Oct 7, 2012 at 18:14
  • $\begingroup$ @PietroMajer Do you have a reference for this fact? Thanks in advance! $\endgroup$
    – ACL
    Commented Apr 10, 2019 at 16:42
  • $\begingroup$ See Willie Wang's comment. I added a link. $\endgroup$
    – Pietro Majer
    Commented Apr 10, 2019 at 20:15
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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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Etale cohomology - you can apply fixed-point theorems from algebraic topology to Galois actions on varieties.

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  • $\begingroup$ Though I could imagine thinking of a more eloquent rephrasing. $\endgroup$
    – David Corwin
    Commented Oct 7, 2012 at 22:18
  • $\begingroup$ It's not bad though. Better than a lot of other answers. $\endgroup$
    – Todd Trimble
    Commented Oct 7, 2012 at 22:22
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