  The principle of tannakian duality states that any neutral tannakian category
is tensorially equivalent to the category Rep_k G of finite dimensional
representations of some affine group scheme G and field k, and conversely.
  Originally motivated by an attempt to find a first-order explanation for
generic cohomology of algebraic groups, we study neutral tannakian categories
as abstract first-order structures and, in particular, ultraproducts of them.
One of the main theorems of this dissertation is that certain naturally
definable subcategories of these ultraproducts are themselves neutral tannakian
categories, hence tensorially equivalent to Comod_A for some Hopf algebra A
over a field k. We are able to give a fairly tidy description of the
representing Hopf algebras of these categories, and explicitly compute them in
several examples.
  For the second half of this dissertation we turn our attention to the
representation theories of certain unipotent algebraic groups, namely the
additive group G_a and the Heisenberg group H_1. The results we obtain for
these groups in characteristic zero are not at all new or surprising, but in
positive characteristic they perhaps are. In both cases we obtain that, for a
given dimension n, if p is large enough with respect to n, all n-dimensional
modules for these groups in characteristic p are given by commuting products of
representations, with the constituent factors resembling representations of the
same group in characteristic zero. We later use these results to extrapolate
some generic cohomology results for these particular unipotent groups.
