Related papers: A short course on geometric motivic integration
We use the theory of motivic integration in order to give a geometric explanation of the behavior of some p-adic integrals.
These are notes of a series of talks about motivic integration I gave on the M\"unster Model Theory Month. Readers are assumed to have some basic knowledge of model theory and of valued fields. The notes are closest to the Cluckers-Loeser…
The aim of this article is to develop the theory of motivic integration over Deligne-Mumford stacks and to apply it to the birational geometry of stacks.
This is an attempt at an elementary exposition, with examples, of the theory of motivic integration developed by R. Cluckers and F. Loeser, with the view towards applications in representation theory of p-adic groups.
These notes are an expanded version of an introductory lecture on contact geometry given at the 2001 Georgia Topology Conference. They are intended to present some of the "topological" aspects of three dimensional contact geometry.
The aim of these notes is to provide a reasonably short and "hands-on" introduction to the differential calculus on associative algebras over a field of characteristic zero. Following a suggestion of Ginzburg's we call the resulting theory…
A survey of work on motivic integration.
We develop the theory of motivic integration for formal schemes
Geometric mechanics is usually studied in applied mathematics and most introductory texts are hence aimed at a mathematically minded audience. The present note tries to provide the intuition of geometric mechanics and to show the relevance…
In this short survey article, we showcase a number of non-trivial geometric problems that have recently been resolved by marrying methods from functional calculus and real-variable harmonic analysis. We give a brief description of these…
We construct a theory of motivic integration for smooth rigid varieties. As an application new invariants of degenerations are obtained.
This short note is an "elementary'' introduction to the conjectural theory of motives.
This text is a slightly edited version of lecture notes for a course I gave at ETH, during the Winter term 2000-2001, to undergraduate Mathematics and Physics students. Contents: Chapter 1 - Examples of Dynamical Systems Chapter 2 -…
We give a version of geometric motivic integration that specializes to p-adic integration via point counting. This has been done before for stable sets; we extend this to more general sets. The main problem in doing this is that it requires…
This short introduction to positive geometries, targeted at a mathematical audience, is based on my talk at OPAC 2022.
We recall the question of geometric integrators in the context of Poisson geometry, and explain their construction. These Poisson integrators are tested in some mechanical examples. Their properties are illustrated numerically and they are…
We survey certain accessible aspects of Grothendieck's theory of motives in arithmetic algebraic geometry for mathematical physicists, focussing on areas that have recently found applications in quantum field theory. An appendix (by Matilde…
We provide a gentle introduction to arc spaces, motivic integration and stringy invariants. We explain the basic concepts and first results, including the p-adic number theoretic pre-history, and we provide concrete examples. The text is a…
We introduce a direct image formalism for constructible motivic functions. One deduces a very general version of motivic integration for which a change of variables theorem is proved. These constructions are generalized to the relative…
We survey our recent work on an extension of the theory of motivic integration, called arithmetic motivic integration. We developed this theory to understand how p-adic integrals of a very general type depend on p.