If one is willing to take certain facts about motives as a black box, the class can be followed independently of the previous term 191b class on "Motives and Periods".

- Matilde Marcolli, "Feynman Motives", World Scientific, 2010.

- pdf S.Bloch, H.Esnault, D.Kreimer, "On motives associated to graph polynomials"
- pdf S.Bloch, "Motives associated to graphs"
- pdf S.Bloch, "Motives associated to graph sums"
- pdf P.Aluffi, M.Marcolli, "Algebro-geometric Feynman rules"
- pdf P.Belkale, P.Brosnan, "Matroids, motives and a conjecture of Kontsevich"
- pdf P.Aluffi, M.Marcolli, "Graph hypersurfaces and a dichotomy in the Grothendieck ring"
- pdf S.Bloch, "A note on Hodge structures associated to graphs"
- pdf O.Schnetz, "Quantum field theory over Fq"
- pdf P.Belkale, P.Brosnan, "Periods and Igusa zeta function"
- pdf P.Aluffi, M.Marcolli, "Parametric Feynman integrals and determinant hypersurfaces"
- pdf O.Ceyhan, M.Marcolli, "Feynman integrals and motives of configuration spaces"

- pdf D.J.Broadhurst, D.Kreimer, "Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops"
- pdf J.R.Stembridge, "Counting points on varieties over finite fields related to a conjecture of Kontsevich"
- pdf I.Kausz, "A modular compactification of the general linear group"
- pdf P.Aluffi, "Chern classes of graph hypersurfaces and deletion-contraction"
- pdf F.Brown, O.Schnetz, "Modular forms in quantum field theory"
- pdf F.Brown, O.Schnetz, K.Yeats, "Properties of the c2 invariants of Feynman graphs"
- pdf D.Doryn, "The c2 invariant is invariant"
- pdf F.Brown, O.Schnetz, "A K3 in phi4"
- pdf F.Brown, K.Yeats, "Spanning forest polynomials and the transcendental weight of Feynman graphs"
- pdf S.Mueller-Stach, S.Weinzierl, R.Zayadeh, "Picard-Fuchs equations for Feynman graphs"

- Tuesday April 1: Feynman integral and perturbative expansion: finite dimensional model, Feynman graphs.
- Thursday April 3: Scalar quantum field theory, partition function, Green functions, Euclidean and Lorentzian, asymptotic expansion, feynman graphs, Feynman rules, Schwinger and Feynman parameters, parametric Feynman integral.
- Tuesday April 8: Projective and affine graph hypersurfaces, Landau varieties, Feynman integral in affine and projective coordinates, differential forms on hypersurface complements (affine and projective), singularities of graph hypersurfaces, irreducibility, hypersurface complements and disjoint unions of graphs, hypersurface complements and 1PI components, planar graphs, dual graphs, Cremona transformation and graph hypersurfaces.
- Thursday April 10: Deletion contraction formulae: example of the Tutte polynomials, deletion-contraction for the graph polynomial, deletion-contraction type formula for the classes in the Grothendieck ring of the graph hypersurface complements, the role of the intersection between the hypersurfaces of deletion and contraction
- Tuesday April 15: presentation by Emad Nasrollahpoursamani on Bloch-Esnault-Kreimer
- Thursday April 17: presentation by Emad Nasrollahpoursamani (continued)
- Tuesday April 22: Operations on graphs that inductively define mixed Tate classes, splitting edges and multiplying edges, generating functions of the classes of the graph hypersurface complements, chains of polygons give mixed Tate classes.
- Thursday April 24: Belkale-Brosnan result: graph hypersurface classes additively generate a localization of the Grothendieck ring, incidence schemes, matroid representations schemes, universality of matroids
- Tuesday April 29: Bloch's motives of sums of graphs; stable birational equivalence classes of graph hypersurfaces; graph complete intersections and the parametric Feynman integral revisited for trivial external momenta
- Thursday May 1: Doryn and Schnetz's explicit counterexamples to the Kontsevich conjecture; Schnetz's quantum field theory over finite fields; dimensional regularization of parametric integrals
- Tuesday May 6: local Igusa zeta functions and periods
- Thursday May 8: parametric Feynman integrals and determinant hypersurfaces, combinatorics of graphs and embeddings, reformulation of the Feynman integral, motives of determinant hypersurfaces
- Tuesday May 13: varieties of frames, flag varieties, intersections of unions of Schubert cells, motives of varieties of frames, explicit computations of Grothendieck classes for two and three spaces.
- Thursday May 15: Feynman integrals in configuration spaces, potential theory (Riesz and Bessel potentials), Euclidean massless and massive propagators, Fourier transforms and relations between momentum space and configuration space Feynman amplitudes
- Tuesday May 20: Green functions of Laplacian and expansion in Gegenbauer orthogonal polynomials, properties of Gegenbauer polynomials and ultraspherical harmonics, expansion of the Feynman amplitude in Gegenbauer polynomials, angular and radial integrals, Feynman integral on polygons and polylogarithm function, general case and reduction to trivalent vertices, Gaunt coefficients, Racah factorization formula
- Thursday May 22: Angular integrals and summation domains, reduction to Mordell-Tornheim and Apostol-Vu summations via Euler-MacLaurin formula, case of dimension 4, matching half-edges and multiple polylog type summations for the leading term (ell=0)
- Tuesday May 27: Complexified problem, wonderful compactifications of configuration spaces, forms with logarithmic poles, pole subtraction procedure via principal value currents, via deformation to the normal cone and via Rota-Baxter algebras of meromorphic forms
- Thursday May 29: student presentation
- Tuesday June 3: student presentation
- Thursday June 5: student presentation

- May 29, Ingar Saberi: A K3 in phi4
- June 3, Siqi He: Picard-Fuchs equations for Feynman integrals
- June 5, Xiang Ni: A modular compactification of the general linear group

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