{"id":5503,"date":"2018-08-02T17:23:12","date_gmt":"2018-08-02T15:23:12","guid":{"rendered":"https:\/\/wpethzprd.ethz.ch\/kowalski\/?p=5503"},"modified":"2024-02-20T13:40:31","modified_gmt":"2024-02-20T11:40:31","slug":"the-most-valuable-mathematical-restaurant-cards-in-the-world","status":"publish","type":"post","link":"https:\/\/blogs.ethz.ch\/kowalski\/2018\/08\/02\/the-most-valuable-mathematical-restaurant-cards-in-the-world\/","title":{"rendered":"The most valuable mathematical restaurant cards in the world!"},"content":{"rendered":"

Now that Akshay Venkatesh has (deservedly) received the Fields Medal, I find myself the owner of some priceless items of mathematical history: the four restaurant cards on which, some time in (probably) 2005, Akshay sketched the argument (based on Ratner theory) that proves that the Fourier coefficients of a cusp form at $latex n$ and at (say) $latex 2n$, for a non-arithmetic<\/em> group, do not correlate. In other words, if we normalize the coefficients (say $latex a(n)$) so that the mean-square is $latex 1$, then we have
\n$latex \\lim_{X\\to +\\infty} \\frac{1}{X}\\sum_{n\\leq X} a(n)\\overline{a(2n)}=0.$<\/p>\n

\"Akshay's<\/a>
Akshay’s cards<\/figcaption><\/figure>\n

(Incidentally, the great persifleur<\/em><\/a> of the world was also present that week in Bristol, if I remember correctly).<\/p>\n

The story of these cards actually starts the year before in Montr\u00e9al, where I participated in May in a workshop on Spectral Theory and Automorphic Forms<\/a>, organized by D. Jakobson<\/a> and Y. Petridis<\/a> (which, incidentally, remains one of the very best, if not the best, conference that I ever attended, as the programme<\/a> can suggest). There, Akshay talked about his beautiful proof (with Lindenstrauss) of the existence of cusp forms, and I remember that a few other speakers mentioned some of his ideas (one was A. Booker). <\/p>\n

In any case, during my own lecture, I mentioned the question. The motivation is an undeservedly little known gem of analytic number theory<\/a>: Duke and Iwaniec proved in 1990 that a similar non-correlation holds for Fourier coefficients of half-integral weight<\/em> modular forms, a fact that is of course related to the non-existence of Hecke operators in that context. Since it is known that this non-existence is also a property of non-arithmetic groups (in fact, a characteristic one, by the arithmeticity theorem of Margulis), one should expect the non-correlation to hold also for that case. This is what Akshay told me during a later coffee break. But only during our next meeting in Bristol did he explain to me how it worked. <\/p>\n

Note that this doesn’t quite give as much as Duke-Iwaniec: because the ergodic method only gives the existence of the limit, and no decay rate, we cannot currently (for instance) deduce a power-saving estimate for the sum of $latex a(p)$ over primes (which is what Duke and Iwaniec deduced from their own, quantitative, bounds; the point is that a similar estimate, for a Hecke form, would imply a zero-free strip for its $latex L$-function).<\/p>\n

For a detailed write-up of Akshay’s argument, see this short note<\/a>; if you want to go to the historic restaurant where the cards were written, here is the reverse of one of them:<\/p>\n

\"Restaurant<\/a>
Restaurant card<\/figcaption><\/figure>\n

If you want to make an offer for these invaluable objects, please refer to my lawyer.<\/p>\n","protected":false},"excerpt":{"rendered":"

Now that Akshay Venkatesh has (deservedly) received the Fields Medal, I find myself the owner of some priceless items of mathematical history: the four restaurant cards on which, some time in (probably) 2005, Akshay sketched the argument (based on Ratner theory) that proves that the Fourier coefficients of a cusp form at $latex n$ and … Continue reading The most valuable mathematical restaurant cards in the world!<\/span><\/a><\/p>\n","protected":false},"author":625,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5544,2,624],"tags":[],"class_list":["post-5503","post","type-post","status-publish","format-standard","hentry","category-mathematical-history","category-blogroll","category-science"],"_links":{"self":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/5503","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/users\/625"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/comments?post=5503"}],"version-history":[{"count":0,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/5503\/revisions"}],"wp:attachment":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/media?parent=5503"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/categories?post=5503"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/tags?post=5503"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}