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. Kontsevich, and 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 integration with 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 integartion 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.

Expository texts:

  1. T. Hales, "What is motivic measure" -- this is a beautiful introduction to arithmetic motivic integration.
  2. Jan Denef, "Arithmetic and geometric applications of quantifier elimination for valued fields" .
  3. Notes on jet schemes and arithmetic motivic integration on Manuel Blickle's homepage  . These excellent notes are about arc spaces and geometric motivic integration.
  4. Notes by F. Loeser.
  5. R. Cluckers and F. Loeser , exposition of the new framework of mot. integr.
  6. 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:

  1. V. Batyrev, "Birational Calabi-Yau n-folds have equal Betti numbers"
  2. A. Scholl, Classical motives.
    Motives (Seattle, WA, 1991), 163--187,
    Proc. Sympos. Pure Math., 55, Part 1,
    Amer. Math. Soc., Providence, RI, 1994

Some applications:

  1. A. Craw, An introduction to motivic integration.
  2. L. Ein, M. Mustata, Inversion of adjunction for locally complete intersection varieties.
  3. L. Ein, M. Mustata, The log canonical threshold of homogeneous affine hypersurfaces.
  4. Lawrence Ein, Robert Lazarsfeld, Mircea Mustata,Contact loci in arc spaces.
  5. Lawrence Ein, Mircea Mustata, and Takehiko Yasuda Jet schemes, log discrepancies and Inversion of Adjunction.
  6. Orbital integrals are motivic (by T.C. Hales).
  7. Good orbital integrals (by C. Cunningham and T.C. Hales).
  8. Characters of some depth-zero representations (J. Gordon) .

Motivic Integration in other contexts:

  1. E. Looijenga's lecture at Seminaire Bourbaki.
  2. F. Loeser, J. Sebag, Motivic integration on smooth rigid varieties and invariants of degenerations
  3. T. Yasuda, Motivic Integration on Deligne-Mumford stacks.
Seminar log:

Tuesday March 15, Kyu-Hwan Lee,  Measure on 2-dimentional local fields.

Tuesday 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 integration"


October 21,  November 4: Yoav Yaffe "p-adic cell decomposition Theorem".

October 7, Geoffrey Lynch, "Introduction to Chow motives".


September 30:
Elliot Lawes, "Prehistory of Motivic Integration".
                         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)


September 23:
Julia Gordon "an overview of motivic integration"  + discussion of what should be covered in the seminar.