'Important' applications of p-adic numbers outside of algebra (and number theory). - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T21:33:38Z http://mathoverflow.net/feeds/question/84320 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/84320/important-applications-of-p-adic-numbers-outside-of-algebra-and-number-theory 'Important' applications of p-adic numbers outside of algebra (and number theory). Mikhail Bondarko 2011-12-26T16:17:53Z 2011-12-27T18:01:12Z <p>Surely, $\mathbb{Z}_p$ and $\mathbb{Q}_p$ (and their extensions) are very important for algebra and number theory. Do they have any important applications outside of algebra (that I could easily explain to a student)? Here I do not demand the applications to be (purely) 'mathematical'; for example, I wonder whether p-adic numbers have applications to physics (outside of string theory?). Moreover, I am also interested in those applications that are partially 'algebraic', and yet important for some other parts of mathematics.</p> http://mathoverflow.net/questions/84320/important-applications-of-p-adic-numbers-outside-of-algebra-and-number-theory/84324#84324 Answer by Igor Rivin for 'Important' applications of p-adic numbers outside of algebra (and number theory). Igor Rivin 2011-12-26T17:06:39Z 2011-12-26T17:06:39Z <p>Googling "applied p-adic analysis" returns obviously interesting results.</p> http://mathoverflow.net/questions/84320/important-applications-of-p-adic-numbers-outside-of-algebra-and-number-theory/84325#84325 Answer by Anatoly Kochubei for 'Important' applications of p-adic numbers outside of algebra (and number theory). Anatoly Kochubei 2011-12-26T17:13:34Z 2011-12-26T17:13:34Z <p>See the survey paper "On p-adic mathematical physics", by B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev and I. V. Volovich: <a href="http://www.springerlink.com/content/j16m84014r878034/" rel="nofollow">http://www.springerlink.com/content/j16m84014r878034/</a></p> http://mathoverflow.net/questions/84320/important-applications-of-p-adic-numbers-outside-of-algebra-and-number-theory/84326#84326 Answer by David Feldman for 'Important' applications of p-adic numbers outside of algebra (and number theory). David Feldman 2011-12-26T17:14:52Z 2011-12-26T17:14:52Z <p>This article contains many references,</p> <p><a href="http://en.wikipedia.org/wiki/P-adic_quantum_mechanics" rel="nofollow">http://en.wikipedia.org/wiki/P-adic_quantum_mechanics</a></p> <p>though I don't know how many will count as "outside of string theory."</p> http://mathoverflow.net/questions/84320/important-applications-of-p-adic-numbers-outside-of-algebra-and-number-theory/84328#84328 Answer by Maurizio Monge for 'Important' applications of p-adic numbers outside of algebra (and number theory). Maurizio Monge 2011-12-26T17:55:19Z 2011-12-26T17:55:19Z <p>The (unsolved) Hilbert-Smith conjecture states that any locally compact group acting faithfully on a manifold has to be a Lie group: <a href="http://en.wikipedia.org/wiki/Hilbert%E2%80%93Smith_conjecture" rel="nofollow">http://en.wikipedia.org/wiki/Hilbert%E2%80%93Smith_conjecture</a></p> <p>However, it turns out that it is enough to prove this for $\mathbb{Z}_p$, and the conjecture follows proving that $\mathbb{Z}_p$ has no continuous faithful action on a manifold.</p> http://mathoverflow.net/questions/84320/important-applications-of-p-adic-numbers-outside-of-algebra-and-number-theory/84332#84332 Answer by Alain Valette for 'Important' applications of p-adic numbers outside of algebra (and number theory). Alain Valette 2011-12-26T20:06:12Z 2011-12-26T20:06:12Z <p>There is a journal devoted to p-adic numbers, p-adic analysis and applications: <a href="http://www.springer.com/mathematics/algebra/journal/12607" rel="nofollow">http://www.springer.com/mathematics/algebra/journal/12607</a></p> <p>You can also google "spin glasses + p-adic numbers"</p> http://mathoverflow.net/questions/84320/important-applications-of-p-adic-numbers-outside-of-algebra-and-number-theory/84334#84334 Answer by Joe Silverman for 'Important' applications of p-adic numbers outside of algebra (and number theory). Joe Silverman 2011-12-26T20:18:14Z 2011-12-26T20:18:14Z <p>This won't qualify as something you can explain to undergraduate students, but non-archimedean dynamics has recently seen a number of applications to classical complex dynamics. (Non-archimedean is dynamics over a field with a non-archimedean absolute value, but not specifically an extension of $\mathbb{Q}_p$.) I'll mention one beautiful example, which is a recent theorem of Matt Baker and Laura DeMarco. Let <code>$$f_c(x) = x^2+c$$</code> be the usual quadratic polynomial, and for any starting value $a$, let $O_c(a)$ be the forward orbit of $a$ for the map $f_c$. That is, <code>$$O_c(a) = \{a,f_c(a),f_c^2(a),f_c^3(a),...\}$$</code> where $f_c^n$ denotes the $n$'th iterate of $f_c$.</p> <p><strong>Theorem</strong>: Let $a$ and $b$ be complex numbers with $a^2\ne b^2$. Then <code>$$\{c\in\mathbb{C} : O_c(a) \text{ and } O_c(b) \text{ are both finite}\}$$</code> is a finite set.</p> <p>The proof is partly complex dynamics, partly equidistribution theorems (in both the complex and $p$-adic settings), and partly a reduction step in which one works in Berkovich space over a non-archimedean field. Note that the statement of the theorem is purely a statement about complex numbers, but the proof requires non-trivial methods from non-archimedean analysis.</p> http://mathoverflow.net/questions/84320/important-applications-of-p-adic-numbers-outside-of-algebra-and-number-theory/84335#84335 Answer by Joe Silverman for 'Important' applications of p-adic numbers outside of algebra (and number theory). Joe Silverman 2011-12-26T20:23:14Z 2011-12-26T20:23:14Z <p>At a somewhat speculative level, you might look at the work of Andrei Khrennikov:</p> <p><a href="http://w3.msi.vxu.se/Personer/akhmasda/home.html" rel="nofollow">http://w3.msi.vxu.se/Personer/akhmasda/home.html</a></p> <p>In particular, his home page mentions his work on:</p> <ul> <li>dynamical systems over $p$-adic fields with applications for describing the process of thinking: "the system conscious- subconscious as a p-adic dynamical processor."</li> <li>$p$-adic dynamical systems for social sciences.</li> </ul> http://mathoverflow.net/questions/84320/important-applications-of-p-adic-numbers-outside-of-algebra-and-number-theory/84337#84337 Answer by Paul Broussous for 'Important' applications of p-adic numbers outside of algebra (and number theory). Paul Broussous 2011-12-26T20:44:51Z 2011-12-26T20:44:51Z <p>I am not convinced by the applications of $p$-adic numbers (or adèles) to theoretical physics, even though I am not a physicist. I think p-adic mathematical physics has so far nothing to do with real phenomena. But of course $p$-adic analysis is useful in mathematics. $p$-adic analysis has for instance natural applications in the $p$-adic Langlands program. The basic idea of that program is to replace the field $\mathbb C$ by a $p$-adic field when considering linear representations (of a $p$-adic Lie group or a Galois group). </p> <p>There are obvious applications of $p$-adic numbers and adèles to analytic number theory via the (classical) Langlands program. These applications are not only algebraic, since they may for instance predict the analytic behaviour of $L$-functions. </p> <p>Another interesting example is the existence of a nice locally compact topology, defined by Berkovich, on $p$-adic rigid manifolds (varieties over ${\mathbb C}_p$, the completion of the algebraic closure of ${\mathbb Q}_p$). You get in such a way varieties analogous to complex varieties. You can do very similar things like dynamics, dessins d'enfant, potential theory, integration of $1$-forms, ... The following survey articles by Ducros (for french readers) are excellent:</p> <p>Géométrie analytique $p$-adique : la théorie de Berkovich, Gazette des Mathématiciens 111 (2007), 12-27.</p> <p>Espaces analytiques $p$-adiques au sens de Berkovich, exposé 958 du Séminaire Bourbaki (mars 2006).</p> <p>This is a very promissing theory. </p> http://mathoverflow.net/questions/84320/important-applications-of-p-adic-numbers-outside-of-algebra-and-number-theory/84340#84340 Answer by Sean Tilson for 'Important' applications of p-adic numbers outside of algebra (and number theory). Sean Tilson 2011-12-26T22:07:57Z 2011-12-27T18:01:12Z <p>The $p$-adics come up in homotopy theory. The main reason is because of their usefulness in the theory of formal group laws.</p> <p>They are also relevant in certain parts of algebraic geometry, they are (one of) the first examples of completions.</p> <p>References:</p> <p><a href="http://en.wikipedia.org/wiki/Formal_group#Lubin.E2.80.93Tate_formal_group_laws" rel="nofollow">http://en.wikipedia.org/wiki/Formal_group#Lubin.E2.80.93Tate_formal_group_laws</a></p> <p><a href="http://arxiv.org/abs/1005.0119" rel="nofollow">http://arxiv.org/abs/1005.0119</a></p> <p><a href="http://arxiv.org/abs/0802.0996" rel="nofollow">http://arxiv.org/abs/0802.0996</a></p> <p>The last one is supposed to tie in the others. Of course, this is all stuff that happened after Quillen's theorem and the work of many other people, such as Mike Hopkins, Jack Morava, Haynes Miller, Doug Ravenel, and Steve Wilson.</p>