01/11: Thomas Hille
Distribution of Values of Irrational Forms at Integral Points and Spherical Averages
Abstract: Let 
be a non-degenerate indefinite quadratic form in 
variables. In the mid 80’s, Margulis proved the Oppenheim conjecture, which states that if 
and is not proportional to a rational form, then the set of values of at integral points is dense in 
. In some cases, homogeneous forms of higher degree exhibit the same behavior if the number of variables is large enough in terms of the degree and if the group preserving the form is large enough, then the set of values at integral points can be studied from the point of view of homogeneous dynamics.
be a non-degenerate indefinite quadratic form in variables. In the mid 80’s, Margulis proved the Oppenheim conjecture, which states that if and is not proportional to a rational form, then the set of values of at integral points is dense in . In some cases, homogeneous forms of higher degree exhibit the same behavior if the number of variables is large enough in terms of the degree and if the group preserving the form is large enough, then the set of values at integral points can be studied from the point of view of homogeneous dynamics.In this talk we will discuss the problem of effective and quantitative distribution of values of certain forms at integral points. A central and recurrent theme revolves around (asymptotic) estimates of certain spherical averages going back to the work of Eskin, Margulis and Mozes.
This talk is based on the one hand on joint work with P. Buterus, F. Götze and G. Margulis and on the other hand with E. Fromm and H. Oh.