Roughly speaking, a Lie group is a group which is also a manifold; the typical example is a closed subgroup of the group of invertible matrices, such as the group of orthogonal matrices or the Lorentz group (those matrices which preserve the space-time metric). Lie groups arise very naturally as groups acting in interesting ways on geometric systems, e.g., as groups of transformations preserving a Riemannian metric or preserving a system of differential equations. This is one reason Lie theory is a very interesting and active area of research at this time, with connections with many branches of mathematics as well as physics. A representation of a Lie group G is a continuous homomorphism from G into the space of continuous linear operators on a topological vector space which may be infinite dimensional. In a very broad sense the two underlying problems in representation theory are to decompose ``large'' representations into their irreducible components and to understand in great detail the irreducible ones. The former is often called noncommutative harmonic analysis, as it is the analog for a noncommutative group G of Fourier analysis for the group R or the circle group, where the irreducible representations are one-dimensional. The latter involves classification and construction of irreducible representations as well as determining certain of their properties such as unitarity.
OSU has an active group of mathematicians working on Lie groups and their representations, mainly concerned with representation theory of semi-simple Lie groups. In addition, many faculty members at OSU who work in other areas of mathematics use Lie theory in their research. Jim Cogdell, a number theorist, studies automorphic forms and uses Lie group techniques to obtain results in number theory. Besides number theory, there are strong connections with topology, algebraic geometry, differential equations, and mathematical physics.
Numerous techniques have been very useful in representation theory and many relationships between different fields of mathematics have resulted. The various techniques include ideal theory for noncommutative rings, algebraic and differential geometry (including D-modules and complex manifolds) and functional analysis. Here at OSU the faculty are mostly interested in geometric techniques.
Lie groups also provide many excellent and concrete examples of powerful theorems in various fields of mathematics, especially geometry and topology. For these reasons students are encouraged learn some Lie theory, even if they choose to specialize in a different area. Special topics courses are often offered and there is an active weekly seminar in Lie groups. Also, there are many opportunities for Master's students to complete an interesting creative component in this subject.
Faculty:
Leticia Barchini: Representation theory of semisimple Lie groups
and analysis on homogeneous spaces.
Birne Binegar: Groups of geometrical transformations and the
induced actions on spaces of functions; in particular, actions on the space
of square-integrable functions on a manifold that preserve the inner product
(finding such an action on a symplectic manifold is equivalent to finding
a ``quantization'' of the manifold.)
J. T. Chang: Representation theory for reductive Lie groups
(a typical example of a reductive group is the group of all invertible
matrices of fixed size).
Lisa Mantini: Symmetry, ranging from the motions that preserve
regular shapes in the plane to the changes of variable that preserve the
solutions to certain differential equations of mathematical physics.
Dave Witte: Algebraic aspects of the study of Lie groups, with
particular emphasis on arithmetic groups, for example questions regarding
the existence of ``almost normal'' subgroups, as well as questions
regarding how algebraic properties of the groups are reflected in their
actions.
Roger Zierau: Representation theory of reductive Lie groups
and the geometry of homogeneous spaces.