What is your favorite isomorphism? - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-06-20T09:39:32Z http://mathoverflow.net/feeds/question/37525 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/37525/what-is-your-favorite-isomorphism What is your favorite isomorphism? Jason Orendorff 2010-09-02T18:47:09Z 2010-09-03T02:46:34Z <p>The other day I was trying to figure out how to explain why isomorphisms are important. I pulled Boyer's <a href="http://books.google.com/books?id=xwIZQwAACAAJ&amp;dq=history+of+mathematics&amp;hl=en&amp;ei=p-h_TPeRDYWclgeYu8zBDg&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=7&amp;ved=0CFkQ6AEwBg" rel="nofollow"><em>A History of Mathematics</em></a> off the bookshelf and was surprised to find that <em>isomorphism</em> isn't even listed in its index. The <a href="http://en.wikipedia.org/wiki/Isomorphism" rel="nofollow">Wikipedia article on isomorphisms</a> only gives two concrete examples.</p> <p>There are many surprising, significant, classic isomorphisms. I'll refrain from giving examples. What are your favorites?</p> <p>As usual, please limit yourself to <strong>one isomorphism per answer</strong>.</p> <p><em>(Related: <a href="http://mathoverflow.net/questions/14574/your-favorite-surprising-connections-in-mathematics" rel="nofollow">your favorite surprising connections in mathematics</a>. But this question is looking for more concrete examples, particularly those that illustrate the power of the idea.)</em></p> http://mathoverflow.net/questions/37525/what-is-your-favorite-isomorphism/37527#37527 Answer by Charles Matthews for What is your favorite isomorphism? Charles Matthews 2010-09-02T19:10:18Z 2010-09-02T19:10:18Z <p>I nominate the Chinese Remainder Theorem, in the form of an isomorphism of a ring of residues with a cartesian product ring. This isn't "profound" mathematics, but simply unpacking it (with construction of the underlying idempotents) should convince students that algebraic structure has "content". I recall a conversation about the analogue for polynomials in one variable over a finite field, in which my side was really stating that if you understand the original CRT in the correct way, this is no sweat at all.</p> http://mathoverflow.net/questions/37525/what-is-your-favorite-isomorphism/37528#37528 Answer by Tony Huynh for What is your favorite isomorphism? Tony Huynh 2010-09-02T19:29:03Z 2010-09-02T19:29:03Z <p>English and French are isomorphic. </p> <p><strong>Stronger.</strong> They are both trivial.</p> <p>See this <a href="http://www.emis.de/journals/EM/restricted/2/2.3/mestre.ps" rel="nofollow">paper</a> by Mestre, Schoof, Washington, and Zagier for a short proof. </p> http://mathoverflow.net/questions/37525/what-is-your-favorite-isomorphism/37530#37530 Answer by Steven Gubkin for What is your favorite isomorphism? Steven Gubkin 2010-09-02T19:41:23Z 2010-09-02T19:41:23Z <p>A calculator made using wire and logic gates with electrons flowing through it should be considered isomorphic to a calculator made using tubes and physical gates with water flowing through it as long as the underlying structure (the schematics for each calculator) is the same.</p> http://mathoverflow.net/questions/37525/what-is-your-favorite-isomorphism/37531#37531 Answer by Stefan Geschke for What is your favorite isomorphism? Stefan Geschke 2010-09-02T19:44:11Z 2010-09-02T19:44:11Z <p>One of my current favourites can be found on Peter Cameron's <a href="http://cameroncounts.wordpress.com/2010/07/15/the-random-graph-2/" rel="nofollow">blog</a>. Take a countable model $(M,E)$ of enough (axioms of) set theory. Symmetrize the relation $E$ to obtain a graph. This graph is the random graph (Rado's homogeneous universal countable graph).</p> http://mathoverflow.net/questions/37525/what-is-your-favorite-isomorphism/37535#37535 Answer by Dan Piponi for What is your favorite isomorphism? Dan Piponi 2010-09-02T20:23:58Z 2010-09-03T02:46:34Z <p>The set of 7-tuples of binary trees is isomorphic to the set of binary trees. For the correct <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.46.8517" rel="nofollow">definition</a> of "isomorphic" this is a surprising non-trivial result.</p> http://mathoverflow.net/questions/37525/what-is-your-favorite-isomorphism/37538#37538 Answer by Martin Bright for What is your favorite isomorphism? Martin Bright 2010-09-02T20:43:55Z 2010-09-02T20:43:55Z <p>The most striking example of an isomorphism I remember seeing as an undergraduate was when John Conway visited and gave his famous talk on rational tangles. Being able to unknot a seemingly hopelessly tangled pair of skipping ropes by manipulating rational numbers was an amazingly concrete demonstration of what it meant for structures to be isomorphic.</p> <p>For those who don't know what I'm talking about, I think there's a video online somewhere.</p> http://mathoverflow.net/questions/37525/what-is-your-favorite-isomorphism/37542#37542 Answer by Dan Ramras for What is your favorite isomorphism? Dan Ramras 2010-09-02T21:39:26Z 2010-09-02T21:39:26Z <p>I'm a fan of the isomorphism between <code>$PSL_2 (F_7)$</code> and <code>$GL_3 (F_2)$</code> (two nice descriptions of the simple group of order 168). Years ago, Richard Guy asked me if I knew an explicit map, and I didn't. But recently one was given in the Math Monthly:</p> <p>MR2572107 Brown, Ezra; Loehr, Nicholas Why is <code>${\rm PSL}(2,7)\cong{\rm GL}(3,2)$</code>? Amer. Math. Monthly 116 (2009), no. 8, 727--732.</p> <p>The paper is also available from Brown's website:</p> <p><a href="http://www.math.vt.edu/people/brown/doc.html" rel="nofollow">http://www.math.vt.edu/people/brown/doc.html</a></p> http://mathoverflow.net/questions/37525/what-is-your-favorite-isomorphism/37543#37543 Answer by Tracy Hall for What is your favorite isomorphism? Tracy Hall 2010-09-02T21:39:49Z 2010-09-02T22:11:02Z <p>The set of positive reals under multiplication is isomorphic to the set of reals under addition, which is the isomorphism underlying the operation of a slide rule. This is the only isomorphism I can think of important enough that its explicit (approximate) values used to be published in 1000-page books. The positive reals under multiplication is also a standard pedagogical example of an interesting one-dimensional abstract real vector space, where there is some content to verifying the axioms. (The other standard example is the reals with addition given by $x+y-1$ and multiplication by scalar $a$ given by $ax + 1 - a$.) </p> http://mathoverflow.net/questions/37525/what-is-your-favorite-isomorphism/37545#37545 Answer by Tracy Hall for What is your favorite isomorphism? Tracy Hall 2010-09-02T21:58:05Z 2010-09-02T21:58:05Z <p>The whole subject of non-commutative geometry arises from extending to non-commutative algebras the isomorphism that exists between commutative $\mathrm{C}^*$-algebras and locally compact Hausdorff topological spaces.</p> http://mathoverflow.net/questions/37525/what-is-your-favorite-isomorphism/37546#37546 Answer by Paul Siegel for What is your favorite isomorphism? Paul Siegel 2010-09-02T22:02:41Z 2010-09-02T22:02:41Z <p>Here is an example that Mel Hochster used to explain the notion of isomorphism to a group of talented high school students. I was one of the course assistants rather than one of the students, but I'm sure the insight was at least as valuable for me as for them.</p> <p>Consider the following game. I'll write down the numbers 1 through 9 on a sheet of paper, and you and I will take turns selecting numbers from the list (crossing off each number once it has been selected). The winner is the first person to have chosen exactly three numbers which add up to 15. For example if I selected 9, 6, 2 and you selected 3, 8, 1, 4 then you would win because 3 + 8 + 4 = 15.</p> <p>The interesting thing is that this game is isomorphic to tic-tac-toe. I don't know what I precisely mean by that, but I can explain why it is true. Simply consider a 3 x 3 magic square:</p> <p>4 9 2</p> <p>3 5 7</p> <p>8 1 6</p> <p>The rows, columns, and diagonals all add up to 15, and moreover every way of writing 15 as the sum of three numbers from 1 to 9 is represented. When you choose a number, draw an X over it; when I choose a number, circle it. Tic-tac-toe!</p> http://mathoverflow.net/questions/37525/what-is-your-favorite-isomorphism/37549#37549 Answer by Todd Trimble for What is your favorite isomorphism? Todd Trimble 2010-09-02T22:31:05Z 2010-09-02T22:31:05Z <p>The isomorphism between $SL_2(\mathbb{C})$ and the universal covering of the special Lorentz group $SO^+(1, 3)$ is definitely nifty in my opinion. ("Coincidences" between Lie groups are another good source of examples.) </p> http://mathoverflow.net/questions/37525/what-is-your-favorite-isomorphism/37550#37550 Answer by Vamsi for What is your favorite isomorphism? Vamsi 2010-09-02T22:31:50Z 2010-09-02T22:31:50Z <p>The De Rham Isomorphism.</p> http://mathoverflow.net/questions/37525/what-is-your-favorite-isomorphism/37552#37552 Answer by some guy on the street for What is your favorite isomorphism? some guy on the street 2010-09-02T22:36:43Z 2010-09-02T22:36:43Z <p>I should say I'm fond of the Thom isomorphism, but I still find the contents rather mysterious.</p> http://mathoverflow.net/questions/37525/what-is-your-favorite-isomorphism/37554#37554 Answer by Joel David Hamkins for What is your favorite isomorphism? Joel David Hamkins 2010-09-02T22:50:02Z 2010-09-02T22:50:02Z <p><a href="http://en.wikipedia.org/wiki/Pairing_function" rel="nofollow">The Cantor pairing function</a> is the function $p(a,b)= (a+b)(a+b+1)/2 + b$, a polynomial bijection between the pairs of natural numbers and individual numbers. Thus, it is a bijection or isomorphism of the sets $\mathbb{N}\times\mathbb{N}$ and $\mathbb{N}$. Using such a function, one may easily deduce that the set of rational numbers is countable, and more generally, that a countable union of countable sets is countable.</p> http://mathoverflow.net/questions/37525/what-is-your-favorite-isomorphism/37558#37558 Answer by Richard Borcherds for What is your favorite isomorphism? Richard Borcherds 2010-09-02T23:23:31Z 2010-09-02T23:23:31Z <p>The elliptic modular function<br> j(&tau;) = q<sup>-1</sup> + 744 +196884q + ... (q=e<sup>2&pi;i&tau;</sup>) </p> <p>This is an isomorphism from elliptic curves (such as C/(1,&tau;)) to the complex plane.</p> http://mathoverflow.net/questions/37525/what-is-your-favorite-isomorphism/37561#37561 Answer by Michael Hardy for What is your favorite isomorphism? Michael Hardy 2010-09-02T23:25:45Z 2010-09-02T23:25:45Z <p>How about the Fourier transform as an isomorphism between the Hilbert space $L^2$ of quadratically integrable complex-valued functions on the unit interval and the Hilbert space $\ell^2$ of sequences of complex numbers the sum of the squares of whose norms is finite?</p> http://mathoverflow.net/questions/37525/what-is-your-favorite-isomorphism/37562#37562 Answer by Felipe Voloch for What is your favorite isomorphism? Felipe Voloch 2010-09-03T00:13:51Z 2010-09-03T00:13:51Z <p>I like the isomorphism between a finite abelian group and its "Cartier" dual (not the bidual!) precisely because it's non-canonical. But I don't think it makes a good example for explaining isomorphism to non-mathematicians.</p>