Colloquia 2012-2013

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	Tentatively Scheduled	TBA	Fish
Sep 30	Alex Kontorovich (Yale)	On Zaremba's Conjecture	Shamgar
oct 19, Wed	Bernd Sturmfels (UC Berkeley)	TBA	distinguished lecturer
oct 20, Thu	Bernd Sturmfels (UC Berkeley)	TBA	distinguished lecturer
oct 21	Bernd Sturmfels (UC Berkeley)	TBA	distinguished lecturer
oct 28	Peter Constantin (University of Chicago)	TBA	distinguished lecturer
oct 31, Mon	Peter Constantin (University of Chicago)	TBA	distinguished lecturer
nov 18	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

Abstracts

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.