Since the realization in 1907 by Hermann Minkowski that spacetime is a four-dimensional structure and his key invention of the light cone and the subsequent development by his former student Albert Einstein and Marcel Grossmann of the theory of gravity, the properties of wave propagation in spacetime have been under intense scrutiny.

Einstein, Nathan Rosen, and Heinrich Brinkmann first described gravitational plane waves. Later Roger Penrose showed that any spacetime in a suitable sense behaves like a plane wave in the immediate neighborhood of a light ray: his approach is now called the theory of Penrose limits.

The speaker in collaboration with his students has been trying for many years to exploit Penrose limits as a tool for understanding spacetime as a whole, hoping to get insight into the problem of linking quantum mechanics and gravity and in particular trying to see if complex numbers play an important role. Working over the past decade with Jonathan Holland, we have been assembling our ideas in book form. In the next two talks, the speaker will describe first the basic background and then some intriguing new results that we have obtained in the last eighteen months.

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