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− | ===Oct 14 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 05:30, 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.