This is the <i>Leonardo da Vinci Lecture</i> given in Milan in March 2006. It is a survey on the concept of space-time over the last 3000years: it starts with Euclidean geometry, discusses the contributions of Gauss and Riemannian geometry, presents the dynamic concept of space-time in Einsteins general relativity, describes the importance of symmetries, and ends with Calabi-Yau manifolds and their importance in todays string theories in the attempt for a unified theory of physics.

The GKZ system for the Hesse pencil of elliptic curves has more solutions than the period integrals. In this work we give different realizations and interpretations of the extra solution, in terms of oscillating integral, Eichler integral, chain integral on the elliptic curve, limit of a period of a certain compact Calabi-Yau threefold geometry, etc. We also highlight the role played by the orbifold singularity on the moduli space and its relation to the GKZ system.

Past, Present, and Future of Statistical Science was commissioned in 2013 by the Committee of Presidents of Statistical Societies (COPSS) to celebrate its 50th anniversary and the International Year of Statistics. COPSS consists of five charter member statistical societies in North America and is best known for sponsoring prestigious awards in stat

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We propose some conjectures on the generating series of (equivariant) Euler characteristics of some vector bundles constructed from the tautological bundles on Hilbert schemes of points on affine $k$-spaces. We establish the surface case of these conjectures and present some verifications of the higher dimensional cases.