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.

In this work, we introduce two sets of algorithms inspired by the ideas from modern geometry. One is computational conformal geometry method, including harmonic maps, holomorphic 1-forms and Ricci flow. The other one is optimization method using affine normals.

It is well known that the nonlinear filtering problem has important applications in both military and commercial industries. The central problem of nonlinear filtering is to solve the DMZ equation in real time and memoryless manner. The purpose of this paper is to show that, under very mild conditions (which essentially say that the growth of the observation |h| is greater than the growth of the drift |f|), the DMZ equation admits a unique nonnegative weak solution u which can be approximated by a solution u_R of the DMZ equation on the ball B_R with u_R|_BR = 0. The error of this approximation is bounded by a function of R which tends to zero as R goes to infinity. The solution u_R can in turn be approximated efficiently by an algorithm depending only on solving the observation-independent Kolmogorov equation on B_R. In theory, our algorithm can solve basically all engineering problems in real time manner. Specifically

Gravitational waves are predicted by the general theory of relativity. In [6] D. Christodoulou showed that gravitational waves have a nonlinear memory. We proved in [3] that the electromagnetic field contributes at highest order to the nonlinear memory effect of gravitational waves. In the present paper, we study this electromagnetic Christodoulou memory effect and compute it for binary neutron star mergers. These are typical sources of gravitational radiation. During these processes, not only mass and momenta are radiated away in form of gravitational waves, but also very strong magnetic fields are produced and radiated away. Thus the observed effect on test masses of a laser interferometer gravitational wave detector will be enlarged by the contribution of the electromagnetic field. Therefore, the present results are important for the planned experiments. Looking at the null asymptotics of spacetimes, which are solutions of the Einstein-Maxwell (EM) equations, we derived in [3] the electromagnetic Christodoulou memory effect. Moreover, our results allow to answer astro-physical questions, as the knowledge about the amount of energy radiated away in a neutron star binary merger enables us to gain information about the source of the gravitational waves.

Based on examples from superstring/D-brane theory since the work of Douglas and Moore on resolution of singularities of a superstring target-space Y via a D-brane probe, the richness and the complexity of the stack of punctual D0-branes on a variety, and as a guiding question, we lay down a conjecture that any resolution Y of a variety Y over Y can be factored through an embedding of Y into the stack Y of punctual D0-branes of rank Y on Y for Y in Y , where Y depends on the germ of singularities of Y . We prove that this conjecture holds for the resolution Y of a reduced singular curve Y over Y . In string-theoretical language, this says that the resolution Y of a singular curve Y always arises from an appropriate D0-brane aggregation on Y and that the rank of the Chan-Paton module of the D0-branes involved can be chosen to be arbitrarily large.