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Mathematics on the Quantum Frontier

Hilbert space isn't easy to explain. We have to start by explaining vector space, which is a set of numbers in which all the numbers can be added to, or multiplied by each other to produce other members of the set. Vector space is usually used in graphs, which normally have two dimensions: one along the bottom and one up the side. Hilbert space, however, has an infinite number of dimensions.

The idea of Hilbert space is especially important in physics because a vector in Hilbert space is used to represent the state of a system in quantum mechanics. Now at the forefront of physics, quantum mechanics describes the physics of extremely small subatomic particles. Many of the rules of quantum physics are so different from the rules of "classical" physics that they seem counter-intuitive.

As holder of a Research Chair in Mathematics, Dr. George Elliott will study how algebra works for operators in Hilbert space. An operator is a rule that transforms one vector into another, and the study of operators overlaps such other areas of mathematics as geometry, topology and number theory. The theory of operator algebras involves the way in which the order of operations affects the outcome.

In particular, Dr. Elliott is following up on the surprising discovery that virtually all naturally arising "norm-closed" algebras of operators in Hilbert space can be described completely in terms of very simple data. This is called the K-theory invariant, although in some circles it is already being called the Elliott invariant.

The importance of this simple description of what are called amenable C*-algebras - a very large class of objects with very complex structure - is that these objects arise in many areas of mathematics, and also in physics (in solid state physics, and string theory, for instance).

Dr. Elliott's work could prove to be an important turning point in the history of mathematics.