Logic Seminar
James Freitag
UIC
Not Pfaffian
Abstract: In recent years, definable sets in o-minimal geometry have seen widespread application in number theory around diophantine geometry and transcendence. The diophantine applications of o-minimality center around various versions of results for counting points of bounded height originating from the work of Pila and Wilkie. In the full generality of o-minimal geometry, the results of Pila and Wilkie can not be improved (it is not even clear what this would mean), but various improvements which are vital in applications have been obtained recently.
One of the chief settings in which there is an improved counting theorem is the setting in which the functions are assumed to be Pfaffian, a setting introduced by Khovanskii.
A natural open problem is wether every differential algebraic function interpretable in an o-minimal structure is Pfaffian? We will answer this open question. Our analysis answers a number of recent open questions of Binyamini and Novikov, Bianconi, Armitage.
Tuesday September 14, 2021 at 4:00 PM in 636 SEO