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 padic 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, numbervalued, 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, 45:15pm 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 padic 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 njets to (n1)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, DenefPas 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: padic and motivic Igusa
zetafunctions; proof of rationality using cell decomposition (sketch).
Sources: Item (5) below.
March 14: Immanuel Halupczok (University of
Dusseldorf) CluckersLoeser 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, HrushovskiKazhdan approach to motivic integration
? April 11  Adam and Nina, proof of Batyrev's theorem using motivic
integration.
Further topics (depending on the interests):
Stringy Efunction 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:

Jet schemes; the Grothendieck group of the category of varieties, the algebra of cylindrical sets, and the measure.
References:
 The padic numbers, Haar measues, gauge forms, and volume on padic
manifolds.
References:
 Batyrev's theorem
References:
 Change of variables formula for motivic integrals. An application
to Betti numbers of crepant resolutions.
References:
 padic and motivic Poincare series.
Rationality of Poincare series. Motivic Igusa zetafunctions.
(This is probably 23 talks).
References:
 Definable sets; Cell Decomposition Theorem; quantifier elimination.
References:
 Constructible motivic functions, and motivic integral.
References:
 Jet spaces and invariants of singularities.
References:
 L. Ein,
M. Mustata, Inversion of
adjunction for locally complete intersection varieties.
 L. Ein,
M. Mustata,
The log canonical
threshold of homogeneous affine hypersurfaces.
 Lawrence
Ein,
Robert Lazarsfeld, Mircea Mustata,Contact
loci in arc spaces.
 Lawrence Ein, Mircea Mustata, and Takehiko Yasuda Jet schemes, log
discrepancies and Inversion of Adjunction.
Motivic
Integration in other contexts:
 E. Looijenga's
lecture at Seminaire Bourbaki.
 F. Loeser, J.
Sebag, Motivic integration on smooth rigid varieties and invariants of
degenerations
 T. Yasuda,
Motivic Integration on DeligneMumford stacks.
 T. Hales, "What
is
motivic measure"  this is a beautiful introduction to arithmetic
motivic integration.
 An old (2004) but comprehensive at the time
list
of the litertaure on motivic integration, compiled by Manuel Blickle.
