Motivic integration
Encyclopedia
Motivic integration is a branch of algebraic geometry
which was invented by Maxim Kontsevich
in 1995 and was developed by Jan Denef
and François Loeser
. Since its introduction it has proved to be quite useful in various branches of algebraic geometry
, most notably birational geometry
and singularity theory
. Roughly speaking, motivic integration assigns to subsets of the arc space of an algebraic geometry a volume living in the Grothendieck ring
of algebraic varieties
. The naming 'motivic' mirrors the fact that unlike ordinary integration
, for which the values are real numbers, in motivic integration the values are geometric in nature.
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
which was invented by Maxim Kontsevich
Maxim Kontsevich
Maxim Lvovich Kontsevich is a Russian mathematician. He is a professor at the Institut des Hautes Études Scientifiques and a distinguished professor at the University of Miami...
in 1995 and was developed by Jan Denef
Jan Denef
Jan Denef is a Belgian mathematician. He is Full Professor of Mathematics at the Katholieke Universiteit Leuven.He is a specialist of model theory, number theory and algebraic geometry. He is well known for his early work on Hilbert's tenth problemand for developing the theory of motivic...
and François Loeser
François Loeser
François Loeser is a French mathematician. He is Professor of Mathematics at thePierre and Marie Curie University in Paris. From 2000 to 2010 he was Professor atÉcole Normale Supérieure....
. Since its introduction it has proved to be quite useful in various branches of algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, most notably birational geometry
Birational geometry
In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. In the case of dimension two, the birational geometry of algebraic surfaces was largely worked out by the Italian...
and singularity theory
Singularity theory
-The notion of singularity:In mathematics, singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width. What is meant by a singularity can be seen by dropping it on the floor...
. Roughly speaking, motivic integration assigns to subsets of the arc space of an algebraic geometry a volume living in the Grothendieck ring
Grothendieck group
In mathematics, the Grothendieck group construction in abstract algebra constructs an abelian group from a commutative monoid in the best possible way...
of algebraic varieties
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
. The naming 'motivic' mirrors the fact that unlike ordinary integration
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
, for which the values are real numbers, in motivic integration the values are geometric in nature.
External links
- AMS Bulletin Vol. 42 Tom HalesThomas Callister HalesThomas Callister Hales is an American mathematician. He is known for his 1998 computer-aided proof of the Kepler conjecture, a centuries-old problem in discrete geometry which states that the most space-efficient way to pack spheres is in a pyramid shape...
- What is motivic measure?
- math.AG/9911179 A.Craw
- An introduction to motivic integrationMotivic integrationMotivic integration is a branch of algebraic geometry which was invented by Maxim Kontsevich in 1995 and was developed by Jan Denef and François Loeser. Since its introduction it has proved to be quite useful in various branches of algebraic geometry, most notably birational geometry and...
- An introduction to motivic integration
- Lecture Notes François LoeserFrançois LoeserFrançois Loeser is a French mathematician. He is Professor of Mathematics at thePierre and Marie Curie University in Paris. From 2000 to 2010 he was Professor atÉcole Normale Supérieure....
- Seattle lecture notes on motivic integration
- Lecture Notes W.Veys
- Arc spaces, motivic integration and stringy invariants