All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.
|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|
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.