Polynomial values in affine subspaces of finite fields

In this talk I will discuss new results about polynomial images of affine spaces  of finite fields, in particular the frequency with which they fall in another affine subspace. I will explain several improvements and  generalisations of previously known results, for example, extending such results to polynomials of degrees higher than the characteristic of the field. 

Such results have immediate consequences to intersection of orbits with affine subspaces, estimating new types of exponential sums, Waring's problem in affine subspaces, etc.