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 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.