Nelson Lecture 2011-12
Factoring a linear differential operator allows one to solve it in terms of solutions of simpler operators. It is well known that an ordinary linear differential operator factors as a product of irreducible operators and that in any such factorization the number of such factors is unique. This uniqueness no longer holds for partial differential operators. Nonetheless, solutions of linear partial differential operators are the most basic examples of differential algebraic groups (DAGs) and a Jordan-Hoelder type theorem for differential groups gives a kind of factorization into irreducibles where in any such factorization the number of "factors" are unique and, after a possible permutation are equivalent in a suitable sense. DAG's come up in other contexts as well, including model theory. I will give an introduction to these groups, discussing the above and giving many examples.