How do we see things? Light is the key factor in vision. Light is a form of electromagnetic radiation. Light will strike an object and then be reflected. Upon reflection, it is directed to the eyes. The light penetrates the optical nerve, and the retina converts the light to electrical impulses which communicate with the brain.
Acoustic waves are roughly similar. For instance, when we speak, the vocal chords vibrate, and this causes the air to vibrate. These vibrations are transmitted to the ears. Inside the ear, these vibrations are converted to electrical impulses which communicate with the brain.
The key to everything is vibrations. Sound has vibrations of fairly large wave length, while light has a very tiny wave length. But there is other information, besides visual and aural, that the eyes and ears cannot extract. This includes X-rays and radio waves.
This workshop wants to understand waves. The wave length of a light wave is about 0.5 micron (a micron is one millionth of a meter). When wave lengths are much larger (as with sound), then the waves interact with the surroundings in complicated ways. There is much interference. It is the interference that makes the problems complicated, and the study interesting.
An exciting, and rather new, medical context in which some of these ideas arise is as follows. In a new application of ultrasound technology, medical technicians send several very short waves to a point in the body. At the point of convergence of the waves, a vigorous vibration results. As a result, it is possible to (completely non-invasively) "cook" cancer cells and eliminate them.
This technique works particularly well in the bodily extremities, where there is little likelihood of interference with important organs. But, in the abdomen, there is danger of damage to or interference with the rib cage and the organs inside. These "boundary conditions" give rise to the kinds of partial differential equations that this group wishes to solve.
A quite different type of application in which the very same mathematical ideas arise is in astronomy. There is of course a desire to discover new planets. But stars in the sky are very bright and interfere with our vision. There is a paradigm to send a telescope into space that is equipped witha a receptor disc that has petals. This receptor disc can filter out the bright light from the star and "see" the new planet (if it is there). This physical situation gives rise to systems of partial differential equations such as we have been describing.
The types of equations studied here are not in general elliptic, but they share many properties of elliptic equations. It is the "size" of the physical system that makes the situation complicated.
This is an AIM workshop in which pure mathematics interacts decisively with important physical systems. The workshop involved people with many different points of view: time frequency analysis, spatial frequency analysis, inverse scattering, and others. The result was a vigorous dialogue that led to many new insights.