The aim of the seminar (which we stopped short of )
was to understand the recent
work of R. Cluckers
and F. Loeser on the theory of motivic integration, and to look at some
possible applications of this theory.
Motivic integration originated in a 1995 talk by M.
since then has developed in several directions. Historically, the
first theory that appeared (in the works of J. Denef and F. Loeser
dated 1996 -- 1998) was the theory of integration on arc spaces (what
is nowadays called "geometric motivic integration"). This theory
is designed for varieties
defined over an algebraically closed field. If the base field is not
algebraically closed, motivic integration is still possible, but
acquires a totally different
flavour. The theory of arithmetic motivic integration was developed by
J. Denef and F. Loeser in 1999, and that's when they first introduced
the machinery from logic into the construction. Arithmetic motivic
integration provides a different point of view on the classical
integration over p-adic fields.
The most recent theory combines arithmetic motivic
geometric motivic integration, and takes it a step further by expanding
the class of functions that can be integrated.
It should be noted here that the values of
motivic measure are
not numbers but geometric objects (such as, roughly speaking,
isomorphism classes of varieties, or, sometimes, Chow motives). In the
case of arithmetic motivic integration the way to get back to a
classical, number-valued, measure, is roughly by counting points on the
varieties over finite fields.
Geometric motivic integration and arithmetic motivic
already found a few striking applications
References, links, etc:
The most complete list of references on motivic integration that I know
of is at Manuel Blickle's homepage.
- T. Hales, "What
motivic measure" -- this is a beautiful introduction to arithmetic
Denef, "Arithmetic and geometric applications of quantifier elimination
for valued fields" .
- Notes on jet schemes
and arithmetic motivic integration on Manuel Blickle's homepage .
excellent notes are about arc spaces and geometric motivic integration.
by F. Loeser.
Cluckers and F. Loeser , exposition of the new framework of mot. integr.
- Willem Veys,
"Arc spaces, motivic integration and stringy invariants".
The main reference for the seminar:
R. Cluckers, F.
Loeser "Constructible motivic Functions and motivic integration"
Some background references:
- V. Batyrev,
"Birational Calabi-Yau n-folds have equal Betti numbers"
- A. Scholl, Classical motives.
Motives (Seattle, WA, 1991), 163--187,
Sympos. Pure Math., 55, Part 1,
Amer. Math. Soc., Providence, RI, 1994
- A. Craw, An
introduction to motivic integration.
- L. Ein,
M. Mustata, Inversion of
adjunction for locally complete intersection varieties.
- L. Ein,
The log canonical
threshold of homogeneous affine hypersurfaces.
Robert Lazarsfeld, Mircea Mustata,Contact
loci in arc spaces.
- Lawrence Ein, Mircea Mustata, and Takehiko Yasuda Jet schemes, log
discrepancies and Inversion of Adjunction.
integrals are motivic (by T.C. Hales).
integrals (by C. Cunningham and T.C. Hales).
some depth-zero representations (J. Gordon) .
Integration in other contexts:
- E. Looijenga's
lecture at Seminaire Bourbaki.
- F. Loeser, J.
Sebag, Motivic integration on smooth rigid varieties and invariants of
- T. Yasuda,
Motivic Integration on Deligne-Mumford stacks.
Tuesday March 15, Kyu-Hwan Lee, Measure on 2-dimentional local
March 8: Alf Dolich, expanding on
his talk about
mapping formulas to Chow motives, and doing some examples.
March 1: Julia Gordon, the transformation rule for
geometric motivic measure: example of a blow-up.
December 2: Alf Dolich, "mapping
formulas to Chow motives"
November 18 and 25. Jonathan Korman, about
geometric and arithmetic
motivic integration in the new framework.
November 11, Julia Gordon "Geometric
motivic integration from the new point of view".
October 28, And November 1: Raf Cluckers "new
framework for motivic integration"
Also by Raf Cluckers: in the
Representation theory/Number theory seminar at UofT. (room SS5017A)
"Recent developments in the theory of motivic
October 21, November 4: Yoav
Yaffe "p-adic cell decomposition Theorem".
October 7, Geoffrey Lynch,
"Introduction to Chow motives".
Elliot Lawes, "Prehistory of
Abstract: Let K denote a complete nonarchimedean local field. Let X
denote a K-analytic manifold. Weil described a canonical way to
integrate well-behaved functions on X. I will explain Weil's
integration (and many of its ingredients including "complete
nonarchimedean local field" and "K-analytic manifold").
The construction of (Geometric) Motivic Integration mimics, and is thus
motivated by, Weil's integration. I will present my explanation with
this in mind.
Despite the audience, I'll adopt a low-brow approach to this material.
Notes for Elliot's talk (to be expanded later)
Julia Gordon "an overview of motivic
integration" + discussion of what should be covered in the