Research Fellows Directory
Dr Kevin McGerty
University of Oxford
My current research focuses on the quantization of certain kinds of geometric objects. This is a topic which arose from the study of quantum mechanics. In classical mechanics on can take measurements, e.g. of the temperature say, of the physical world which are mathematically encoded as``functions'' of the physical world around us. A basic feature of functions is that the algebraic operations we perform in calculating with them are commutative: if you take two measurements and multiply their value, it doesn't matter what order you do this multiplication in. However, in quantum mechanics, the physical world is given by a space of quantum states, and measurements become operators on these states. Since the multiplication of operators is inherently noncommutative -- the order you multiply them in really matters -- the quantum world is inherently noncommutative. (This noncommutativity is for example, mathematically speaking, at the heart of the famous Heisenberg uncertainly principle).
Nevertheless, by taking certain limits (which roughly correspond to ``looking from far enough away'') the quantum world collapses to the classical one, thus mathematicians view the quantum setting as a deformation of the classical one. In this sense it falls naturally into another important mathematical idea, which is to study objects in families, (a kind of mathematical analogue of trying to understand someone better by meeting their relatives).
My research focuses on the ``deformation quantization'' of a particularly interesting class of spaces known as symplectic resolutions. These spaces generalise the geometry that arises in the study of natural symmetries known as Lie groups. The theory of Lie groups plays a huge role in much of mathematics and physics, and there are many hints coming from the current interplay of mathematics and theoretical physics that these symplectic resolutions may yield exciting new contexts in which much of this theory can be extended.