PREFACE ix the renormalisation group physicists use to make sense of Feynman integrals which involve multivariables is beyond the scope of this book. In these lectures, we hope to modestly help clarify a few aspects of this vast pic- ture in setting some of these heuristic considerations on firm mathematical ground by providing analytic tools to describe regularisation techniques, whether those used in physics, number theory, or geometry, in a common framework. The fo- cus is set on the underlying canonical integral, discrete sum, and trace which are characterised by natural properties such as Stokes’ property, covariance, transla- tion invariance, or cyclicity. Various anomalies/discrepancies are investigated, all of which turn out to be local insofar as they can be expressed in terms of the noncommutative residue, another central figure in these lectures. We do not claim to present breakthrough results but rather a unified out- look with pedestrian proofs on results scattered in the physics and mathematics literature, which we try to bring to the forefront and to make accessible to the nonspecialist. Along the way we nevertheless prove yet unpublished original results such as • a characterisation of the noncommutative residue on classical symbols (Proposition 2.60 and Theorem 3.39) and of the canonical integral on noninteger order symbols (Theorem 2.61) in terms of their translation invariance • a characterisation of the noncommutative residue on classical symbols (Theorem 4.21) and of the canonical integral on noninteger order symbols (Theorem 3.43) in terms of their covariance • a characterisation of the noncommutative residue (Proposition 5.40) and the canonical discrete sum (Theorem 5.41) in terms of their Zd-translation invariance • a regularised Euler-Maclaurin formula on symbols (Theorem 5.29) • Taylor expansions (Theorem 4.16 part (2)) for integrals of holomorphic families extended to log-polyhomogeneous symbols (this is based on an unpublished joint work with Simon Scott) • a (local) conformal anomaly formula for the ζ-function at zero of a con- formally covariant operator in terms of noncommutative residues (Propo- sition 9.19). We hope in this way to open new perspectives on and further expand openings to concepts such as regularised integrals, sums, and traces. Far from being ex- haustive, these lectures leave out various important regularisation techniques such as Epstein-Glaser [EG], Pauli-Villars [PV], and lattice regularisation techniques, as well as other regularisation artefacts such as b-integrals [Mel] and relative de- terminants [Mu]. Regularisation procedures on manifolds with boundaries or sin- gularities are further vast topics we do not touch upon in spite of the variety of applications and extensions they offer. We also leave aside the realm of noncommu- tative geometry where zeta-type regularisation procedures are extended to abstract pseudodifferential calculus as well as the ambitious renormalisation issue, which would be needed to make sense of multiple divergent integrals, such as multiloop Feynman diagrams in physics, multiple discrete sums, such as multiple zeta val- ues in number theory, or to count lattice points on convex cones. Here we only tackle simple integrals, and discrete sums. Also, to keep this presentation down to a reasonable size, we chose not to report on regularisation methods implemented in

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