Interated Integrals and Cycles on Algebraic Manifolds -- Preface -- Contents -- 1. Iterated Integrals, Chen's Flat Connection and 1 -- 1.1 Introduction -- 1.2 Differential equations -- 1.3 Program -- 1.4 Lie algebras -- 1.5 Chen's Lie algebra and connection -- 1.6 Some work of Quillen -- 1.7 Group homology -- 1.8 The basic isomorphisms -- 1.9 Lattices in nilpotent Lie groups -- 1.10 Some Hodge theory -- 2. Iterated Integrals on Compact Riemann Surfaces -- 2.1 Introduction -- 2.2 Generalities on Riemann surfaces and iterated integrals -- 2.3 Harmonic volumes and iterated integrals -- 2.4 Use of the Jacobian -- 2.5 Variational formula for harmonic volume -- 2.6 Algebraic equivalence and homological equivalence of algebraic cycles -- 2.7 Calculations for the degree 4 Fermat Curve -- Homological versus algebraic equivalence in a Jacobian -- 2.8 Currents and Hodge theory -- De Rham's results for currents: -- 3. The Generalized Linking Pairing and the Heat Kernel -- 3.1 Introduction -- 3.2 The main theorem -- Appendix: Orientations, Fiber Integration -- Bibliography -- NOTES -- THE REFERENCES -- List of Notations -- Index.