The period matrix of a smooth complex projective variety encodes the isomorphism between its singular homology and its algebraic De Rham cohomology. Numerical approximations with sufficient precision of the entries of the period matrix may be used to study transcendental properties of varieties. Such numbers also arise in the Birch and Swinnerton-Dyer conjecture, as well as its generalisations in higher dimensions, such as the Deligne conjecture. Approximations of the periods can be obtained from an effective description of the homology of the variety, which itself can be obtained from the monodromy representation associated to a generic fibration. We will describe these methods and show how they can be used to generate numerical evidence of the Deligne conjecture for certain Calabi-Yau threefolds.