Towards a Zero-one Law for Improvements to Dirichlet’s Approximation Theorem by Shucheng Yu

PROGRAM : ERGODIC THEORY AND DYNAMICAL SYSTEMS (HYBRID)

ORGANIZERS : C. S. Aravinda (TIFR-CAM, Bengaluru), Anish Ghosh (TIFR, Mumbai) and Riddhi Shah (JNU, New Delhi)
DATE : 05 December 2022 to 16 December 2022
VENUE : Ramanujan Lecture Hall and Online

The programme will have an emphasis on the many recent exciting breakthroughs in the ergodic theory of group actions on homogeneous spaces. This subject, which also goes by the name "homogeneous dynamics" has seen dramatic advances in the last few decades. Homogeneous dynamics comprises the study of group actions on homogeneous spaces of Lie groups. The dynamics of these actions are extremely rich and have surprising connections to diverse parts of mathematics. An early example of such a connection is Margulis's proof of the long standing conjecture of Oppenheim regarding values taken by quadratic forms at integer points. The programme will feature mini courses as well as research level talks exploring recent advances in homogeneous dynamics and their connections to number theory and geometry. India has a strong tradition in this area and the programme will provide an occasion to celebrate the fundamental contributions of S. G. Dani, a pioneer in the subject who turns 75 this year. Young mathematicians, especially those with an interest in these areas are encouraged to apply.

ICTS is committed to building an environment that is inclusive, non discriminatory and welcoming of diverse individuals. We especially encourage the participation of women and other under-represented groups.

0:00:00 Towards a Zero-one Law for Improvements to Dirichlet's Approximation Theorem
0:01:03 Dirichlet's Theorem on simultaneous Diophantine approximation
0:05:27 Asymptotic approximation: Khintchine-Groshev Theorem
0:11:45 Interesting cases for Psi
0:15:13 A zero-one law for Leb(DI(Psi)) when m = n = 1
0:17:47 Main result: a partial zero-one law on Leb(DI(Psi))
0:21:56 Examples
0:25:51 Uniform approximation
0:27:38 Homogeneous dynamics
0:31:09 Dani correspondence
0:37:39 Measure of limsup sets: Borel-Cantelli lemma
0:41:04 The convergence case
0:43:44 Step 1: A quantitative Hajos' theorem
0:53:10 Step 2: Effective equidistribution of expanding horoshperes
0:57:12 Divergence case: Effective doubly mixing of expanding horospheres
1:01:31 Q&A
1:04:34 Wrap Up