## Mathematics and the Cosmos

The mathematical study of the cosmos has its roots in antiquity with early attempts to describe the motions of the sun, moon, stars and planets in precise mathematical terms, allowing predictions of future positions. In modern times many of the greatest mathematical scientists turned their attention to the subject. Building on Kepler's discovery of the three basic laws of planetary motion, Newton invented the subjects of celestial mechanics and dynamics. He studied the "n-body problem" of describing the motion of a number of masses, such as the sun, planets and their moons, under the force of mutual gravitational attraction. He was able to derive improved versions of Kepler's laws, one of whose consequences yielded dramatic results just in the past decade when it was used to detect the existence of planets circling other stars. More . . .

## The Shape of the Universe

For thousands of years, people
believed that the universe revolved around the earth
and astronomers created mathematical models in order
to explain observations in the sky. Eudoxus created
a model containing rotating spheres centered about the
earth. Aristotle adopted
and described this model. While he was able to partially
explain some of the planetary motions by rotating the
spheres at different velocities, other observations,
such as differences in brightness levels, could not be
resolved. In his famous work, the *Almagest*, Claudius
Ptolemy, a 2nd century philosopher, refined and improved
the model based on the earlier work of Apollonius and Hipparchus.
In the *Ptolemaic Universe* planets now moved along
epicycles (see the related JavaSketch for
more information [8]),
which had circles attached to the spheres around the
earth, and yet this model still did not completely resolve
earlier difficulties. More
. . .

## Celestial Mechanics

Newton wrote down differential equations governing the motions of celestial bodies, that is, stars, planets, comets, and satellites. When only two bodies are involved, say the sun and a planet, Newton could solve his equations. In so doing he recovered Kepler’s laws of planetary motion and became famous. Add a third body to the mix. We get the famous three-body problem, not simply an unsolved problem, but an unsolvable problem. Poincaré and Bruns proved that the three-body problem is “unsolvable” in the precise sense that it does not admit ‘enough’ analytic integrals to integrate it. Like all the great impossibility proofs – the impossibility of trisecting an angle, of solving the general quintic, or Gödel’s incompleteness theorem, this impossibility proof adds much more to our knowledge than it subtracts. In developing his impossibilty proof he discovered what we call “chaos”, and built the foundations for a whole new field of inquiry – the qualitative theory of dynamical systems. More . . .

## Space Exploration

Starting in the 20^{th} century,
the mathematical exploration of the cosmos became inextricably
entwined with the physical exploration of space. On the
one hand, virtually all the methods of celestial mechanics
that had been developed over the centuries were transformed
into tools for the navigation of rockets, artificial
satellites and space probes. On the other hand, almost
all of those space vehicles were equipped with scientific
instruments for gathering data about the earth and other
objects in our solar system, as well as distant stars
and galaxies going back to the cosmic microwave background
radiation. Furthermore, the deviations in the paths of
satellites and probes provide direct feedback on the
gravitational field around the earth and throughout the
solar system. More .
. .

## From
Black Holes to Dark Energy: Cosmology in the 21^{st} Century

Cosmology in the 20^{th} century
was almost in its entirety the outgrowth of Einstein's
foundational paper in 1915 on general relativity. Two
years later he presented his first model of the universe
based on general relativity together with Riemann's notion
of the three-sphere. More
. . .