Aschenbrenners work centers in and around model theory, but has branched into other fields of mathematics, often using results and methods originating in mathematical logic. His most spectacular achievements arising from a long-term collaboration with L. van den Dries and J. van der Hoeven, has culminated in the book Asymptotic Differential Algebra and Model Theory of Transseries.
A central point is the investigation of the differential field T of transseries, a remarkable mathematical structure also appearing in various other contexts related to asymptotic analysis, such as ordinary differential equations and o- minimal structures (and more recently, mathematical physics). Their merit is reflected by the fact that the three authors were invited for a joint talk at the next International Congress of Mathemetics in 2018, considered to be one of the most distinguished conference invitations in the field of mathematics. In a completely different direction, Aschenbrenner has collaborated with the topologist Friedl on fundamental groups of 3-manifolds. While it is well-known that every finitely presented group can be realized as the fundamental group of a compact, connected, smooth manifold of dimension 4, they proved that any fundamental group of a 3-manifolds is virtually residually p for all but finitely many primes p. Aschenbrenners results on finiteness conditions in algebra, often at the root of proving termination of algebraic algorithms, go back to his PhD thesis, which earned him the 2001 G. E. Sacks Prize for the best PhD thesis in logic worldwide. In his thesis he developed a method to obtain bounds for algorithmic questions in the ring of (multi-variate) polynomials with integer coefficients, leading to new and essentially optimal degree bounds for the ideal membership problem in this ring. Jointly with Hillar, Aschenbrenner also established a version of Hilberts Basis Theorem for ideals of infinite- dimensional polynomial rings invariant under the action of a symmetric group, allowing the development of an appropriate notion of Gröbner basis for such ideals. This has inspired a large body of work on similar topics in recent years.