Prof. Mike Zieve, Ph.D.
I will describe all rational functions which do not behave like random rational functions in certain precise senses, and give several consequences. Topics covered will include some of the following: a description of the rational functions of fixed degree n over a sufficiently large finite field for which the statistics of the induced function on the field do not resemble those of a typical degree-n function; refinements of Hilbert's irreducibility theorem about multivariate polynomials over a number field; a generalization of Mazur's theorem on rational torsion on elliptic curves to the setting of maps between arbitrary varieties; solutions of certain functional equations involving meromorphic functions; complex rational functions having orbits with infinite intersection; and progress on Hurwitz's question about the possible branching types of branched coverings of surfaces. The common theme in these applications is the use of group theory to control all "non-random" situations.
Datum: 30.04.2015, Raum: G03-106, Zeit: 17:00