Homogeneous dynamics provides very powerful tools to prove
equidistribution and density results, with many applications to number
theory and other fields. However, they have a weakness in that they are
not usually amenable to give effective results with rates. In my talks I
give three examples of such results where rates can be obtained. Though
connected by a common philosophical thread, the three talks will be
independent of each other, and each uses a completely different set of
tools.
First talk:
On an effective proof of the Openheim Conjecture (joint work with G. A. Margulis)
Margulis proof in mid 80’s of the longstanding Oppenheim Conjecture
concerning valves of indefinite quadratic forms at integer points using
homogeneous dynamics, and the subsequent strengthening of this result by
Dani and Margulis, were an important milestone in the development of the
subject. I will describe joint work with Margulis which
gives an effective proof of the Oppenheim conjecture, and explain how it
relates to the quantitative study of unipotent flows.