Seminar on motivic integration, Winter 2017

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 along these lines, due to Cluckers and Loeser, combines arithmetic motivic integration with geometric motivic integration, and takes it a step further by expanding the class of functions that can be integrated. There is also an alternative construction by Hrushovki and Kazhdan.
     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.
    This seminar will be mostly focused on geometric motivic integration and its applications; we will also discuss some of the most modern unified approach, as it yields some very important results, such as an analogue of Fubini theorem for the motivic measure.

Seminar schedule:

Tuesdays, 4-5:15pm in Math 126.

  • January 10: organizational meeting and overview. At this meeting, we came up with an approximate sequence of talks for the first half of the semester: please look at the topics and sources below, and volunteer to talk! (Please note that some of these topics are independent and can be permuted).
  • January 17: Ed Belk, integration on p-adic manifolds, Weil's theorem - historical background for motivic integration.
    Sources: Section 2 of An overview of arithmetic motivic integration by J. Gordon and Y. Yaffe (which provides a brief summary of the references listed below under Item (2).
  • January 24: Nina Morishige: Jet spaces and cylindrical sets; the values of the motivic measure. We discussed that projection from n-jets to (n-1)-jets doesn't have to be surjective if the variety is singular. An example (a plane cusp) is found in W. Veys' notes , Example 2.3. Definition of the motivic measure on the arc space of X (in the case X is smooth). Definition of the function ord_Y, on the jet space, where Y is a subvariety of X. Integrability of L^{-ord_Y} function with respect to the motivic measure. Main reference: Manuel Blickle's notes.
  • January 31: Adam Gyenge: the motivic measure and the statement of the change of variables formula.
    Sources: Items (1), (3), and (4) below.
  • February 7: Julia Gordon, and overview of Quantifier elimination and rationality of Poincare series.
    Sources: Item (6) below.
  • February 14: Ed Belk, Denef-Pas language; an example of Cell Decomposition, and a rough sketch of the proof of rationality of Poincare series using cell decomposition. Sources: Item (6) below, and Sections 3.3. and 4.1.1 of this paper .
  • February 21 -- break; February 28 -- no meeting because of a conflict with Number Theory seminar.
  • March 7: Thomas Rud and Ed Belk: p-adic and motivic Igusa zeta-functions; proof of rationality using cell decomposition (sketch).
    Sources: Item (5) below.
  • March 14: Immanuel Halupczok (University of Dusseldorf) Cluckers-Loeser approach: cell decomposition and the "universal" theory of motivic integration.
    Sources: Introduction to motivic integration See also Item (7) below.
  • March 21: No Meeting!
  • March 28: Goulwen Fichou: the Milnor fibre, and the monodromy conjecture.
  • April 4: Yimu Yin, Hrushovski-Kazhdan approach to motivic integration
  • ? April 11 -- Adam and Nina, proof of Batyrev's theorem using motivic integration.
    Further topics (depending on the interests):
  • Stringy E-function and McKay correspondence.
  • Mustata's work on the invariants of singularities and jet spaces.
  • Motivic Milnor fibre, more about the monodromy conjecture.

    Topics and sources:

    1. Jet schemes; the Grothendieck group of the category of varieties, the algebra of cylindrical sets, and the measure.
    2. The p-adic numbers, Haar measues, gauge forms, and volume on p-adic manifolds.
    3. Batyrev's theorem
    4. Change of variables formula for motivic integrals. An application to Betti numbers of crepant resolutions.
    5. p-adic and motivic Poincare series. Rationality of Poincare series. Motivic Igusa zeta-functions. (This is probably 2-3 talks).
    6. Definable sets; Cell Decomposition Theorem; quantifier elimination.
    7. Constructible motivic functions, and motivic integral.
    8. Jet spaces and invariants of singularities.

    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.
    4. T. Hales, "What is motivic measure" -- this is a beautiful introduction to arithmetic motivic integration.
    5. An old (2004) but comprehensive at the time list of the litertaure on motivic integration, compiled by Manuel Blickle.