most recent 30 from http://mathoverflow.net 2013-05-24T15:46:18Z http://mathoverflow.net/feeds/question/1890 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence Gabe Cunningham 2009-10-22T16:28:18Z 2012-10-07T22:16:28Z

<p>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.</p> <p>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."</p> <p>What other punchlines can you deliver for various topics/fields?</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/1895#1895 Dinakar Muthiah 2009-10-22T17:25:26Z 2009-10-22T17:25:26Z

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

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/1900#1900 Ilya Nikokoshev 2009-10-22T17:47:36Z 2009-10-22T17:47:36Z

<p>QFT &mdash; every expression converges after a Wick rotation.</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/1902#1902 Sonia Balagopalan 2009-10-22T17:54:19Z 2009-10-22T18:26:59Z

<p>Complex Analysis: Holomorphic functions are just rotations and dilations up to the first order.</p> <p>Hold on...</p> <p>Calculus: Differentiation is approximation by a linear map.</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/1905#1905 Qiaochu Yuan 2009-10-22T18:10:36Z 2009-10-22T18:10:36Z

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

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/2287#2287 Greg Muller 2009-10-24T14:40:46Z 2009-10-24T14:40:46Z

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

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/2296#2296 Michael Lugo 2009-10-24T15:35:00Z 2009-10-24T15:35:00Z

<p>Analytic combinatorics: generating functions are awesome.</p> <p>("generating functions are awesome" is actually the title of a talk I gave a couple weeks ago.)</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/2298#2298 Vectornaut 2009-10-24T15:39:36Z 2009-10-24T15:39:36Z

<p>Lie groups: Think locally, act globally. ;)</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/2407#2407 John D. Cook 2009-10-25T01:49:45Z 2009-10-25T02:23:55Z

<p>Sobolev spaces: H = W</p> <p>(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.")</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/2444#2444 Yemon Choi 2009-10-25T09:53:26Z 2012-10-07T18:55:40Z

<p>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.</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/3880#3880 lhf 2009-11-03T00:49:34Z 2009-11-03T00:49:34Z

<p>Linear algebra: everything can be explained by a linear system.</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/3882#3882 lhf 2009-11-03T00:51:20Z 2009-11-03T00:51:20Z

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

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/3904#3904 Reid Barton 2009-11-03T06:15:13Z 2009-11-03T06:15:13Z

<p>Algebraic geometry: CommRing behaves a lot like Set^op.</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/4073#4073 Andrew Tuggle 2009-11-04T12:12:56Z 2012-01-07T17:51:49Z

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

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/4577#4577 Jose Brox 2009-11-08T02:21:45Z 2009-11-11T10:39:49Z

<p>Noncommutative Ring Theory: If it is not modules, then it is idempotents.</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/4581#4581 MLevi 2009-11-08T02:31:15Z 2009-11-08T02:31:15Z

<p>Navier-Stokes Equations: Energy estimates and more energy estimates. </p> <p>*I suppose this goes for most non-linear PDEs</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/4622#4622 Jesse 2009-11-08T07:27:11Z 2012-01-07T17:56:21Z

<p>Real Analysis: Get your hypotheses right, or suffer the counter-examples!</p> <p>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</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/5192#5192 Jon Awbrey 2009-11-12T14:10:31Z 2009-11-12T14:10:31Z

<p>The bonniest mot I can ever recall &mdash; from some graduate algebra course:</p> <ul> <li><strong><em>"Free" is just another word for nothing to do on the left.</em></strong></li> </ul>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/5235#5235 Jon Awbrey 2009-11-12T19:16:07Z 2009-11-12T19:16:07Z

<p>Another favorite of mine &hellip;</p> <ul> <li><strong><em>Redundancy is the essence of information.</em></strong></li> </ul>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/5773#5773 Jon Awbrey 2009-11-17T03:50:38Z 2009-11-17T03:50:38Z

<ul> <li><strong><em>Generating functions are the 19th Century analog of addressable memory.</em></strong></li> </ul>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/5784#5784 Csar Lozano Huerta 2009-11-17T06:12:30Z 2009-11-17T06:12:30Z

<p>One of my favorites:</p> <p>"Algebraic topology is the "art" of Not doing the integral"</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/8730#8730 Colin Tan 2009-12-13T03:45:35Z 2012-01-07T17:55:36Z

<p><em>Linear Algebra</em> is the correct generalization of dimension. (This came from Hubbard)</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/8736#8736 Dan Piponi 2009-12-13T05:16:22Z 2009-12-13T05:16:22Z

<p>"set theory is the study of well-foundedness" - A.R.D Mathias</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/8740#8740 Tim Carstens 2009-12-13T08:34:25Z 2009-12-13T08:34:25Z

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

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/23340#23340 Daniel Moskovich 2010-05-03T12:49:21Z 2010-05-03T12:49:21Z

<p>Configuration space integrals: Don't take limits- compactify!</p> <p>Dror Bar-Natan explained this punchline to me when I was just starting grad school.</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/43839#43839 Bjørn Kjos-Hanssen 2010-10-27T17:20:01Z 2010-10-27T17:20:01Z

<p>Statistics: every parameter is learnable by sampling.</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/43877#43877 darij grinberg 2010-10-27T21:15:12Z 2011-05-21T09:37:01Z

<p><strong>Representation theory of Lie groups:</strong> 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.)</p> <p><strong>Constructive logic:</strong> 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.)</p> <p><strong>Homological algebra:</strong> How badly do modules fail to behave like vector spaces?</p> <p><strong>Gröbner basis theory:</strong> polynomials in $n$ variables can be divided with rest (at least if you have some $O\left(N^{N^{N^{N}}}\right)$ of time)</p> <p><strong>Finite group classification:</strong> what works for Lie groups will surely be even simpler for finite groups, right? ;)</p> <p><strong>Algebraic group theory:</strong> 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 $\varepsilon$ ($\varepsilon^2=0$).</p> <p><strong>Semisimple algebras:</strong> The representations of a sufficiently nice algebra mirror a structure of the algebra itself, namely how it breaks into smaller algebras.</p> <p><strong>$n$-category theory:</strong> all the obvious isomorphisms, homotopies, congruences you have always been silently sweeping under the rug are coming back to have their revenge.</p> <p><strong>Modern algebraic geometry (schemes instead of varieties):</strong> let's have the beauty of geometry without its perversions.</p> <p>How many of these did I get totally wrong?</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/43878#43878 Qfwfq 2010-10-27T21:19:45Z 2010-10-27T21:19:45Z

<p>Algebraic geometry is the study of the intrinsic properties of any mathematical object which can be locally described by polynomial equations.</p> <p>Or</p> <p>Algebraic geometry is <em>not</em> about solving systems of polynomial equations, rather it's about studying the intrinsic properties thereof.</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/43883#43883 Alex R. 2010-10-27T21:44:37Z 2012-01-07T17:53:05Z

<p>Analytic Number Theory: log log log log log...</p> <p>Did I see that quote in Havil's book <a href="http://press.princeton.edu/titles/7494.html" rel="nofollow"><em>Gamma</em></a>?</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/43886#43886 Nate Eldredge 2010-10-27T22:00:55Z 2010-10-27T22:00:55Z

<p>Dirichlet forms: a symmetric Markov process is a self-adjoint operator is a closed symmetric form is a Markovian semigroup.</p> <p>(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.)</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/43887#43887 Nate Eldredge 2010-10-27T22:03:33Z 2010-10-27T22:03:33Z

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

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/43930#43930 Richard Borcherds 2010-10-28T03:07:52Z 2010-10-28T03:07:52Z

<p>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.)</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/43938#43938 Gerry Myerson 2010-10-28T04:59:28Z 2012-01-07T17:54:00Z

<p>I'll offer two punchlines for Galois Theory. </p> <ol> <li><p>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 $F$. </p></li> <li><p>A polynomial is solvable in radicals if and only if the Galois group of its splitting field is a solvable group. </p></li> </ol>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/47549#47549 Joel Dodge 2010-11-27T23:14:01Z 2010-11-27T23:14:01Z

<p>Homotopy theory is an attempt to do homological algebra in non-abelian categories. </p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/47550#47550 Sean Tilson 2010-11-27T23:58:58Z 2010-11-27T23:58:58Z

<p>Algebraic Topology: Geometry is hard, and Algebra is easy so...</p> <p>(I am sure this applies to many other fields, and certainly algebra is hard.)</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/47553#47553 Michael Greinecker 2010-11-28T00:40:58Z 2010-11-28T00:40:58Z

<p>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.</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/65514#65514 Asaf Karagila 2011-05-20T09:50:08Z 2011-05-20T09:50:08Z

<p><strong>Set theory without choice:</strong> You have no choice, but to wonder...</p> <p><strong>Forcing:</strong> If it doesn't not fit, force it.</p> <p><strong>Large cardinals:</strong> "If you want more you have to assume more." (Dana Scott)</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/65515#65515 Thomas Bloom 2011-05-20T10:01:02Z 2011-05-20T10:01:02Z

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

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/65532#65532 Petrus 2011-05-20T12:07:50Z 2011-05-20T14:50:45Z

<p>I think this belongs on this list too:</p> <p>The <em>theory of groups</em> 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. – <a href="http://www.mathacademy.com/pr/quotes/index.asp?ACTION=TOP&amp;VAL=group%20theory" rel="nofollow">James Newman</a></p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/65542#65542 axequalsb 2011-05-20T13:51:54Z 2011-05-20T13:51:54Z

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

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/65543#65543 Piero D'Ancona 2011-05-20T14:11:15Z 2011-05-20T14:11:15Z

<p>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 :)</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/65544#65544 Kelly Davis 2011-05-20T14:40:07Z 2011-05-20T14:40:07Z

<p><strong>Four-Dimensional Smooth Manifolds</strong>: Whitney's trick gone wrong.</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/65547#65547 John Sidles 2011-05-20T14:52:02Z 2011-05-20T16:22:58Z

<p>Terry Tao, <a href="https://profiles.google.com/114134834346472219368/posts/WNYxf8DHXVi" rel="nofollow">in a post on <em>Google Buzz</em></a>, has given an overview of mathematics in the form of multiple "punch-lines" of the requested variety. </p> <p>Here are three examples from Tao's post:</p> <blockquote> <ul> <li><p>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". </p></li> <li><p>Algebra prizes structure, symmetry, and exact formulae; analysis prizes smoothness, stability, and estimates. </p></li> <li><p>Most of geometry would not be classified as either algebra or analysis, but simply as geometry.</p></li> </ul> </blockquote> <p>Definitely Tao's aphorisms are thought-provoking and inspiring ... but are they useful ? Don't ask me! :)</p> <hr> <p>Partly inspired by Tao's essay, here is a one-sentence <a href="http://faculty.washington.edu/sidles/ENC_2011/index.html#forGilKalai" rel="nofollow">definition of quantum mechanics</a> (as optimized for systems engineers) &nbsp;&hellip;</p> <blockquote> <ul> <li>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&nbsp;&hellip; $n$-particle Hilbert space.</li> </ul> </blockquote> <p>&hellip; and it turns out to be very useful (and great fun) to rewrite standard quantum physics texts like <a href="http://faculty.washington.edu/sidles/ENC_2011/index.html#Slichter" rel="nofollow">Charles Slichter's <i>Principles of Magnetic Resonance</i></a> based upon this one sentence definition. </p> <p>Joseph Landsberg's recent <em>Bull. AMS</em> review "<a href="http://www.ams.org/journals/bull/2008-45-02/S0273-0979-08-01176-2/home.html" rel="nofollow">Geometry and the complexity of matrix multiplication</a>" (2008), which has been praised in multiple MathOverflow posts, provides an overview of the broad utility&mdash;despite their unwieldy name&mdash;of stratifications of secant varieties of Segre varieties (which extends far beyond quantum physics).</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/65554#65554 Saikat Biswas 2011-05-20T15:55:10Z 2011-05-20T15:55:10Z

<p>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.</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/65555#65555 Stefan Waldmann 2011-05-20T15:55:37Z 2011-05-20T15:55:37Z

<p>Quantization: First quantization is a mystery, second quantization is a functor (Edward Nelson)</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/65640#65640 Marc Palm 2011-05-21T11:11:45Z 2011-05-21T11:11:45Z

<p>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. </p> <p>(That's the Peter Weyl Theorem!)</p> <p>Perhaps it would be a much better question, to interpret a well known theorem in one sentence!</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/71899#71899 Pietro Majer 2011-08-02T17:05:20Z 2011-08-02T17:05:20Z

<p>Morse Theory: <em>opus dynamicum maxime</em>.</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/109081#109081 fedja 2012-10-07T16:58:15Z 2012-10-07T16:58:15Z

<p>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).</p> <p>(communicated by Mark Rudelson) </p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/109087#109087 Hans 2012-10-07T18:22:29Z 2012-10-07T18:22:29Z

<p>Probability/Statistical mechanics:</p> <p>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;</p> <p>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.</p>

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/109102#109102 Reladenine Vakalwe 2012-10-07T22:00:36Z 2012-10-07T22:00:36Z

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

http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/109103#109103 David Corwin 2012-10-07T22:16:28Z 2012-10-07T22:16:28Z

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