Intervals

Usage examples of spaces.

Foremost among these are the nineteenth-century mathematical insights of Georg Bernhard Riemann that firmly established the geometrical apparatus for describing curved spaces of arbitrary dimension.

Calabi-Yau spaces is intricate and subtle, we can get an idea of what they look like with a picture.

Calabi-Yau spaces that, when chosen for the curled-up dimensions required by string theory, give rise to string vibrations that are closely akin to the particles of the standard model.

Imagine, then, two Calabi-Yau spaces in which the number of holes in various dimensions differs, but in which the total number of holes is the same.

Dixon-Lerche-Vafa-Warner guess, Plesser and I pressed on to the linchpin question: Beyond the number of families of particles, do the two different Calabi-Yau spaces agree on the rest of their physical properties?

The individual spaces in a mirror pair of Calabi-Yau spaces are not literally mirror images of one another, in the sense of everyday usage.

By carefully examining a huge sample set of Calabi-Yau spaces that they had generated by computer, they found that almost all came in pairs differing precisely by the interchange of the number of even and odd holes.

Calabi-Yau spaces for purely mathematical reasons long before string theory was discovered.

They had worked out many of the detailed properties of these geometrical spaces without an inkling of a future physical application.

In essence, mirror symmetry proclaims that particular pairs of Calabi-Yau spaces, pairs that were previously thought to be completely unrelated, are now intimately connected by string theory.

By utilizing substantial contributions of the mathematicians Maxim Kontsevich, Yuri Manin, Gang Tian, Jun Li, and Alexander Givental, Yau and his collaborators Bong Lian and Kefeng Liu have finally found a rigorous mathematical proof of the formulas used to count spheres inside Calabi-Yau spaces, thereby solving problems that have puzzled mathematicians for hundreds of years.

Yau and Tian is of interest because it provides a way to produce new Calabi-Yau spaces from ones that are known.

Inspired by earlier work of Shi-Shyr Roan, a mathematician from Taiwan, he found a systematic mathematical procedure for producing pairs of Calabi-Yau spaces that are mirrors of one another.

In fact, though, the lower-dimensional drawings that we use to visualize the spaces make the transformation appear to be somewhat more complicated than it actually is.

The perfectly adapted coloring of the enclosed spaces was the coloring of her blond hair, her flawless complexion, her subdued suits.