Since the introduction of motivic integration by Kontsevich in 1996 the field has been in rapid development. By now there are several related techniques which go by the name of motivic integration and many applications of these to problems in fields as various as number theory, algebraic geometry and representation theory. For a very quick introduction, see an article of K. E. Smith.

 

Fri 12.3 10-12	SII	Introduction to Motivic Integration?
Mon 15.3. 10-12	SIII	Birational Transformation rule.
Tue 16.3. 10-12	P722	Applications to birational geometry.
Thu 18.3. 10-12	P376	Various topics on Motivic Integration.

The course begins with an introductory lecture, which is meant for a general mathematician. The three subsequent lectures require good mastering of the techniques of algebraic geometry.

Introduction to Motivic Integration.

Kontsevich's proof that birational smooth Calabi-Yau's have same Hodge numbers. General setup of geometic motivic integration (for smooth varieties over the complex numbers). Very Basics of Arc spaces. Overview of some other applications of motivic integration: McKay correspondence, Igusa-Zeta functions ...

Birational Transformation rule (complete proof).

More in depth discussion of arc spaces. Concrete example of the Transformation rule in case of blowing up a smooth center (maybe just a point in A²). The (unfortunately) rather technichal proof of the Transformation rule.

Applications to birational geometry.

Characterization of singularity types in terms of the arc spaces. Log canonical threshold. Inversion of adjunction. (this follows Mustata/Ein/Yasuda). New invariants for mildly singular varieties (following Batyrev, Veys).

Various Topics on Motivic Integration.

For example:

• Motivic integration on singular varieties (what has to be whatched out for).
• Connection to p-adic integration: arithmetic motivic integration.
• Other topics.

More information: eero.hyry@helsinki.fi.