Seminar series
Date
Thu, 07 Mar 2024
Time
17:00 -
18:00
Speaker
Itay Glazer
Organisation
University of Oxford
Let f:X-->Y be a polynomial map between smooth varieties, and let mu be a smooth, compactly supported measure on X(F), where F is a local field. An interesting phenomenon is that bad singularities of f manifest themselves in poor analytic behavior of the pushforward f_*(mu) of mu by f.
I will discuss this phenomenon in two settings; the first is when f:A^n-->A^m is a polynomial map between affine spaces and mu is the Haar measure on Z_p^n, and the second is when f:G^2-->G is a word map (e.g. the commutator map (g,h)-->ghg^(-1)h^(-1)) between simple algebraic groups, and mu is a Haar measure on G(Z_p).
In these cases (and in other "real life situations"), mu and consequently f_*(mu) are constructible measures in the sense of Cluckers-Loeser motivic integration. We utilize this fact to show that the analytic behavior of f_*(mu) cannot be too bad, leading to geometric and probabilistic applications.
Based on joint works with Yotam Hendel and Raf Cluckers.