Counting and dynamics in SL_2
Advances in Homogeneous Dynamics May 11, 2015 - May 15, 2015
Location: SLMath: Eisenbud Auditorium
continued fraction
rational approximation
Diophantine approximation
asymptotic formulas
counting points on a lattice
modular group
thin subgroup
accumulation point
Hausdorff dimension
distortion function
11R60 - Cyclotomic function fields (class groups, Bernoulli objects, etc.)
11N32 - Primes represented by polynomials; other multiplicative structures of polynomial values
14240
In this talk I'll discuss a lattice point count for a thin semigroup inside SL_2(Z). It is important for applications I'll describe that one can perform this count uniformly throughout congruence classes.
The approach to counting is dynamical - with input from both the real place and finite primes. At the real place one brings ideas of Dolgopyat concerning oscillatory functions into play. At finite places the necessary expansion property
follows from work of Bourgain and Gamburd (at one prime) or Bourgain, Gamburd and Sarnak (at squarefree moduli).
These are underpinned by tripling estimates in SL_2(F_p) due to Helfgott. I'll try to explain in simple terms the key dynamical facts behind all of these methods.
This talk is based on joint work with Hee Oh and Dale Winter.
Magee. Notes
|
Download |
14240
H.264 Video |
14239.mp4
|
Download |
Please report video problems to itsupport@slmath.org.
See more of our Streaming videos on our main VMath Videos page.