References: There are no required textbooks for
this course, but the following references may be useful.
S. Kleiman, Algebraic cycles and the Weil conjectures , in Dix exposes
sur la cohomologie des schemas, North Holland Masson 1968.
J. Murre Lectures on motives , in Transcendental aspects of algebraic
Cycles, Lecture Notes Series vol. 313.
Y. Andre Une introduction aux motifs (motifs purs, motifs mixtes,
periodes), Panoramas et Syntheses n. 17 SMF.
M. Levine Mixed Motives , Math. Surveys and monogr. vol. 57 AMS.
U. Jannsen, S. Kleiman, J.P. Serre Eds. Motives Proceedings of Symp.
Pure Math. 55, AMS 1994.
Further reading:
S. Bloch, H. Esnault, D. Kreimer On motives associated to graph
polynomials Comm. Math. Phys. 267 (2006).
A. Connes, C. Consani, M. Marcolli Noncommutative geometry and
motives: the thermodynamics of endomotives Max-Planck Institute
Preprint series 111 (2005).
Outline of the course: The aim of this course is to provide
an introduction to the theory of motives.
Motives in algebraic geometry were introduced 40 years ago or so
with the purpose of finding a common source for several cohomological
theories and cohomological phenomena such as the notion of weight.
The goal of this theory is that of decomposing, cohomologically,
the algebraic varieties in simple fundamental blocks suitable of
being (re)combined together. Very recently, the theory of motives
has been also used to explain mathematically, the appearance of
multiple zeta values in anomalous dimensions and \beta functions
of renormalizable quantum field theories and it has also established
interesting connections with noncommutative geometry and quantum
statistical mechanics.
Prerequisites: This is the second part of 110.643. Those students
who did not enroll in 110.643 should be acquainted with the topics developed
in that course. Students should also be acquainted with the fundamental notions of
algebraic and differential geometry, commutative algebra and algebraic topology
such as: Zariski topology, algebraic varieties, sheaf theory, notions of
Weil cohomological theories: Betti, de Rham..., vector bundles, homological algebra,
spectral sequences.