Difference between revisions of "Colloquia 2012-2013"

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(Abstracts)
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== Abstracts ==
 
== Abstracts ==
===Alex Kontorovich (Yale)===
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===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

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.