Joint PU/IAS Number Theory

In joint work with Jacob Tsimerman we study when the primitive of a given algebraic function can be constructed using primitives from some given finite set of algebraic functions, their inverses, algebraic functions, and composition. When the given finite set is just ${1/x}$ this is the classical problem of "elementary integrability" (of algebraic functions). I will discuss some results, including a decision procedure for this question, and further problems and conjectures.