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Counting and dynamics in SL_2

Advances in Homogeneous Dynamics May 11, 2015 - May 15, 2015

May 12, 2015 (02:30 PM PDT - 02:55 PM PDT)
Speaker(s): Michael Magee (Institute for Advanced Study)
Location: MSRI: Simons Auditorium
  • continued fraction

  • rational approximation

  • Diophantine approximation

  • asymptotic formulas

  • counting points on a lattice

  • modular group

  • thin subgroup

  • accumulation point

  • Hausdorff dimension

  • distortion function

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC



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.

23523?type=thumb Magee. Notes 881 KB application/pdf Download
Video/Audio Files


H.264 Video 14239.mp4 133 MB video/mp4 rtsp://videos.msri.org/data/000/023/449/original/14239.mp4 Download
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