This Presidential Young Investigator Award supports the mathematical work of Professor Frederic Bien of Princeton University. Dr. Bien's research aim is to bring to bear on several problems in mathematics and science his expertise in the representation theory of groups. Within mathematics, he will work on deciding precisely which nonlinear differential equations can be integrated by the inverse spectral transform. In theoretical computer science, Dr. Bien will work on expanding graphs, which may model ideal communication networks. He will also apply his knowledge to the field of neural networks. Group theory began in the early nineteenth century as a description of operations that might be done successively, and also undone, as a ninety-degree clockwise rotation is undone by the ninety-degree counterclockwise rotation. Lie groups were developed in the late part of that century by the path breaking Norwegian mathematician Sophus Lie, who was interested in the most fundamental questions concerning differential equations. Now, one hundred years later, group theory pervades modern science, and the utility of Lie groups in many branches of mathematics, as well as in physics, has continued undiminished.
