Seminar: Some results around Presburger constructibility, partition functions and Grothendieck rings of adeles.

Thời gian: 14:00 đến 16:00 Ngày 09/06/2023

Địa điểm: Phòng C101, Viện nghiên cứu cao cấp về Toán

Báo cáo viên: Jorge Enrique Cely García

Abstract: This talk consist of two parts.

The first one is about Presburger constructibility, partition identities and Satake inversion.

The Presburger constructible functions are those elements in the ring of constructible motivic functions (in the sense of Cluckers-Loeser) that are built from data given by the Presburger language in Z (the value group sort) and the functions and constants involving the formal symbol L. We show some results around the Presburger constructibility of certain partition functions of positive integers. Using results of Hahn et al. we show that the explicit Satake inversion that they obtain using a combinatorial approach from partition identities, can be also obtained in the ring of contructible motivic functions. We formulate questions about possible generalizations.

The second part is about model theory of adeles and Grothendieck rings.

The recent model theory of adeles developed by Derakhshana and Macintyre uses the formalism of Feferman-Vaught and among many interesting features, the theory gives a good understanding of the definable sets of the adeles of a number fields in various languages expanding the language of rings, also provides a treatment for measures in the adeles. After presenting the basics of this new theory, I will talk about Grothendieck rings of first-order structures and I will show that the Grothendieck ring of the adeles is trivial.

This is work in progress.