Seminar on motivic integration, Winter 2011
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 along these lines, due to Cluckers and Loeser, combines arithmetic motivic
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.
Currently scheduled on Wednesdays, 3:30-5pm in MATX 118.
We will try to move it earlier starting next week.
(please watch e-mail announcements).
- January 5: organizational meeting and overview. At this meeting,
we came up with an approximate sequence of talks for the whole 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).
Here is a tentative schedule of the first few talks:
- January 12: NO MEETING
- January 19: Guillermo Mantilla-Soler, Jet spaces and cylindrical
sets; the values of the motivic measure.
- January 26: Robert Klinzmann, the motivic measure and the
statement of the change of variables formula.
- February 2: Lance Robson, p-adic numbers and measures.
- February 9: Andrew Morrison, p-adic and motivic Igusa zeta-functions.
- February 16: BREAK, NO MEETING.
- February 23: Andrew Morrison, the monodromy conjecture.
- March 2: Atshushi Kanazawa, Stringy E-function and McKay
- March 9: Julia Gordon, Quantifier elimination and rationality of
- March 16: Julia Gordon, cell decomposition and the "universal" theory
of motivic integration.
- March 23: Andrew Staal, Mustata's work on the invariants of
singularities and jet spaces.
- March 30, April 6: it would be good if someone returned to the
questions related to monodromy conjecture that we left unfinished...
Any other topic is fine, too.
If you'd like to give a talk, please e-mail me.
Topics and sources:
Jet schemes; the Grothendieck group of the category of varieties, the algebra of cylindrical sets, and the measure.
- The p-adic numbers, Haar measues, gauge forms, and volume on p-adic
- Batyrev's theorem
- Change of variables formula for motivic integrals. An application
to Betti numbers of crepant resolutions.
- p-adic and motivic Poincare series.
Rationality of Poincare series. Motivic Igusa zeta-functions.
(This is probably 2-3 talks).
- Definable sets; Cell Decomposition Theorem; quantifier elimination.
- Constructible motivic functions, and motivic integral.
- Jet spaces and invariants of singularities.
- 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.
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.
- T. Hales, "What
motivic measure" -- this is a beautiful introduction to arithmetic
- An old (2004) but comprehensive at the time
of the litertaure on motivic integration, compiled by Manuel Blickle.