Introduction 1 2. Complexes and subcomplexes of curves 11 3. Projection bounds 19 4. Tight geodesics and hierarchies 25 5. Slices, resolutions, and markings 41 6. Comparison and control of hierarchies 47 7. Conjugacy bounds in the Mapping Class Group 57 References 61

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Abstract. A holomorphic 1-form on a compact Riemann surface S naturally defines a flat metric on S with cone-type singularities. We present the following surprising phenomenon: having found a geodesic segment (saddle connection) joining a pair of conical points one can find with a nonzero probability another saddle connection on S having the same direction and the same length as the initial one. A similar phenomenon is valid for the families of parallel closed geodesics.We give a complete description of all possible configurations of parallel saddle connections (and of families of parallel closed geodesics) which might be found on a generic flat surface S. We count the number of saddle connections of length less than L on a generic flat surface S; we also count the number of admissible configurations of pairs (triples,...) of saddle connections; we count the analogous numbers of configurations of families of closed geodesics. By the previous result of [EMa] these numbers have quadratic asymptotics c · (πL 2 ). Here we explicitly compute the constant c for a configuration of every type. The constant c is found from a Siegel-Veech formula.To perform this computation we elaborate the detailed description of the principal part of the boundary of the moduli space of holomorphic 1-forms and we find the numerical value of the normalized volume of the tubular neighborhood of the boundary. We use this for evaluation of integrals over the moduli space.