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.

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Seminar schedule:

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 correspondence.
• March 9: Julia Gordon, Quantifier elimination and rationality of Poincare series.
• 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.

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Topics and sources:

1. Jet schemes; the Grothendieck group of the category of varieties, the algebra of cylindrical sets, and the measure.
   References:
   • A short course on geometric motivic integration by Manuel Blickle, Section 2. I think this is the main source. You can also check out:
   • A. Craw, An introduction to motivic integration, Section 2.
2. The p-adic numbers, Haar measues, gauge forms, and volume on p-adic manifolds.
   References:
   
   • A. Weil, Adeles and algebraic groups, Chapter 1. Progress in Mathematics, 23. Birkhauser, Boston, Mass., 1982.
   • J.-P. Serre, Quelques applications du th\'eor\`eme de densit\'e de Chebotarev. See Section 3.
   • R\'eduction modulo $p\sp{n}$ des sous-ensembles analytiques fermés de $Z\sp{N}\sb{p}$, by J. Oesterle.
   • V. Batyrev, "Birational Calabi-Yau n-folds have equal Betti numbers"
   • An overview of arithmetic motivic integration by J. Gordon and Y. Yaffe; Section 2 consists of a very brief summary of all the above sources.
3. Batyrev's theorem
   References:
   • V. Batyrev, "Birational Calabi-Yau n-folds have equal Betti numbers"
4. Change of variables formula for motivic integrals. An application to Betti numbers of crepant resolutions.
   References:
   • A short course on geometric motivic integration by Manuel Blickle, Section 3.
5. p-adic and motivic Poincare series. Rationality of Poincare series. Motivic Igusa zeta-functions. (This is probably 2-3 talks).
   References:
   • Notes by W. Veys
   • J. Denef and F. Loeser, On some rational generating series occurring in arithmetic geometry.
   • An introduction to p-adic and motivic zeta functions and the monodromy conjecture by Johannes Nicaise
   • Jan Denef, "Arithmetic and geometric applications of quantifier elimination for valued fields" .
6. Definable sets; Cell Decomposition Theorem; quantifier elimination.
   References:
   • Jan Denef, "Arithmetic and geometric applications of quantifier elimination for valued fields" .
   • Notes by J. Gordon and Y. Yaffe, Section 4.1.
7. Constructible motivic functions, and motivic integral.
   References:
   • An excellent expository artcile by R. Cluckers and F. Loeser.
   • Notes by J. Gordon and Y. Yaffe, Sections 3 and 4.2.
8. 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.

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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.