Difference between revisions of "Colloquia 2012-2013"
(→Fall 2011) |
(→Abstracts) |
||
Line 106: | Line 106: | ||
== Abstracts == | == Abstracts == | ||
− | ===Alex Kontorovich (Yale)=== | + | ===Oct 14 Alex Kontorovich (Yale)=== |
''On Zaremba's Conjecture'' | ''On Zaremba's Conjecture'' | ||
It is folklore that modular multiplication is "random". This concept is useful for many applications, such as generating pseudorandom sequences, or in quasi-Monte Carlo methods for multi-dimensional numerical integration. Zaremba's theorem quantifies the quality of this "randomness" in terms of certain Diophantine properties involving continued fractions. His 40-year old conjecture predicts the ubiquity of moduli for which this Diophantine property is uniform. It is connected to Markoff and Lagrange spectra, as well as to families of "low-lying" divergent geodesics on the modular surface. We prove that a density one set satisfies Zaremba's conjecture, using recent advances such as the circle method and estimates for bilinear forms in the Affine Sieve, as well as a "congruence" analog of the renewal method in the thermodynamical formalism. This is joint work with Jean Bourgain. | It is folklore that modular multiplication is "random". This concept is useful for many applications, such as generating pseudorandom sequences, or in quasi-Monte Carlo methods for multi-dimensional numerical integration. Zaremba's theorem quantifies the quality of this "randomness" in terms of certain Diophantine properties involving continued fractions. His 40-year old conjecture predicts the ubiquity of moduli for which this Diophantine property is uniform. It is connected to Markoff and Lagrange spectra, as well as to families of "low-lying" divergent geodesics on the modular surface. We prove that a density one set satisfies Zaremba's conjecture, using recent advances such as the circle method and estimates for bilinear forms in the Affine Sieve, as well as a "congruence" analog of the renewal method in the thermodynamical formalism. This is joint work with Jean Bourgain. |
Revision as of 06:29, 13 August 2011
Contents
Mathematics Colloquium
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.
Fall 2011
date | speaker | title | host(s) | |
---|---|---|---|---|
Sep 9 | Manfred Einsiedler (ETH-Zurich) | TBA | Fish | |
Sep 16 | Richard Rimanyi (UNC-Chapel Hill) | TBA | Maxim | |
Sep 23 | Andrei Caldararu (UW-Madison) | The Hodge theorem as a derived self-intersection | (local) | |
Oct 7 | Hala Ghousseini (University of Wisconsin-Madison) | TBA | Lempp | |
Oct 14 | Alex Kontorovich (Yale) | On Zaremba's Conjecture | Shamgar | |
oct 19, Wed | Bernd Sturmfels (UC Berkeley) | Convex Algebraic Geometry | distinguished lecturer | Shamgar |
oct 20, Thu | Bernd Sturmfels (UC Berkeley) | Quartic Curves and Their Bitangents | distinguished lecturer | Shamgar |
oct 21 | Bernd Sturmfels (UC Berkeley) | Multiview Geometry | distinguished lecturer | Shamgar |
oct 28 | Peter Constantin (University of Chicago) | TBA | distinguished lecturer | |
oct 31, Mon | Peter Constantin (University of Chicago) | TBA | distinguished lecturer | |
Nov 4 | Sijue Wu (U Michigan) | TBA | Qin Li | |
Nov 11 | Henri Berestycki (EHESS and University of Chicago) | TBA | Wasow lecture | |
Nov 18 | Benjamin Recht (UW-Madison, CS Department) | TBA | Jordan | |
Dec 2 | Robert Dudley (University of California, Berkeley) | From Gliding Ants to Andean Hummingbirds: The Evolution of Animal Flight Performance | Jean-Luc | |
dec 9 | Xinwen Zhu (Harvard University) | TBA | Tonghai |
Spring 2012
date | speaker | title | host(s) |
---|---|---|---|
Feb 24 | Malabika Pramanik (University of British Columbia) | TBA | Benguria |
Abstracts
Oct 14 Alex Kontorovich (Yale)
On Zaremba's Conjecture
It is folklore that modular multiplication is "random". This concept is useful for many applications, such as generating pseudorandom sequences, or in quasi-Monte Carlo methods for multi-dimensional numerical integration. Zaremba's theorem quantifies the quality of this "randomness" in terms of certain Diophantine properties involving continued fractions. His 40-year old conjecture predicts the ubiquity of moduli for which this Diophantine property is uniform. It is connected to Markoff and Lagrange spectra, as well as to families of "low-lying" divergent geodesics on the modular surface. We prove that a density one set satisfies Zaremba's conjecture, using recent advances such as the circle method and estimates for bilinear forms in the Affine Sieve, as well as a "congruence" analog of the renewal method in the thermodynamical formalism. This is joint work with Jean Bourgain.