Abstract.
A report on recent results and outstanding problems concerning motives associated to graphs.
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References
P. Belkale and P. Brosnan, Matroids, motives, and a conjecture of Kontsevich, Duke Math. J., 116 (2003), 147–188.
S. Bloch, H. Esnault and D. Kreimer, Motives associated to graph polynomials, to appear in Comm. Math. Phys.
D. Broadhurst and D. Kreimer, Knots and numbers in Φ4 theory to 7 loops and beyond, Internat. J. Modern Phys. C, 6 (1995), 519–524.
D. Broadhurst and D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams upto 9 loops, Phys. Lett. B, 393 (1997), 403–412.
F. Brown, Multiple zeta values and periods of moduli spaces \({\overline{M}_{0,n} (\mathbb R)}\) , thèse, to appear.
J. Conant and K. Vogtmann, On a Theorem of Kontsevich, Algebr. Geom. Topol., 3 (2003), 1167–1224.
A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. I., The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys., 210 (2000), 249–273.
A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. II., The β function, diffeomorphisms and the renormalization group, Comm. Math. Phys., 216 (2001), 215–241.
A. Connes and M. Marcolli, Quantum fields and motives, J. Geom. Phys., 56 (2006), 55–85.
P. Deligne, Local behavior of Hodge structures at infinity, preprint.
H. Esnault, V. Schechtman and E. Viehweg, Cohomology of local systems on the complement of hyperplanes, Invent. Math., 109 (1992), 557–561; Erratum: Invent. Math., 112 (1993), p. 447.
A. Goncharov and Y. Manin, Multiple zeta motives and moduli spaces \({\overline{M}_{0,n}}\), Compos. Math., 140 (2004), 1–14.
L. Illusie, Autour du théorème de monodromie locale, Astérisque, 223 (1994).
C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill, 1980.
M. Kontsevich, Feynman diagrams and low-dimensional topology, First European Congress of Mathematics, Vol. II, Paris, 1992, 97–121; Progr. Math., 120, Birkhäuser, Basel, 1994.
M. Kontsevich, Formal (non)commutative symplectic geometry, In: The Gelfand Mathematical Seminars, 1990-1992, Birkhäuser, Boston, 1993, pp. 173–187.
D. Kreimer, Lecture at IHES, 2005.
P. Orlik and H. Terao, Arrangements of hyperplanes, Grundlehren Math. Wiss., 300, Springer-Verlag, 1992.
M. Polyak, Feynman diagrams for pedestrians and mathematicians, math.GT/0406251.
M. Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci., 24 (1989), 849–995.
M. Saito, Mixed Hodge modules, Publ. Res. Inst. Math. Sci., 26 (1990), 221–333.
J. Steenbrink, Limits of Hodge structures, Invent. Math., 31 (1976), 229–257.
J, Steenbrink and S. Zucker, Variation of mixed Hodge structure. I., Invent. Math., 80 (1985), 489–542.
J. Stembridge, Counting points on varieties over finite fields related to a conjecture of Kontsevich, Ann. Comb., 2 (1998), 365–385.
C. Soulé, Régulateurs, Seminar Bourbaki, 1984-85; Astérisque, 133-134 (1986), 237–253.
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Communicated by: Takeshi Saito
This article is based on the 1st Takagi Lectures that the author delivered at Research Institute for Mathematical Sciences, Kyoto University on November 25 and 26, 2006.
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Bloch, S. Motives associated to graphs. Jpn. J. Math. 2, 165–196 (2007). https://doi.org/10.1007/s11537-007-0648-9
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DOI: https://doi.org/10.1007/s11537-007-0648-9