Tetrahedral molecules, Colloquium, University of Texas at Arlington, September 2014
Abstract: This talk will be a completely elementary introduction to the representation theory of finite groups. We will use a classical problem from physics as motivation: explaining the vibrational frequencies of molecules. Representations enter when one tries to use the symmetries of the molecule to reduce the work. We will solve the problem for tetrahedral molecules. Along the way we will learn a little about characters and their orthogonality relations.
Is there a Heron theorem in 3D?, Informal Mathematics Research Problem Seminar, University of North Texas, September 2012
formula (possibly known to Archimedes) gives the area of a triangle
in terms of its side lengths, in a beautifully factored form.
Why is it true? Does it generalize to higher dimensions?
What should we mean by “generalize”?
Piero della Francesca (a renaissance painter) discovered a formula for the volume of a tetrahedron in terms of its side lengths. Is there a formula for the volume in terms of the face areas? What happens in higher dimensions? I will explain the limited amount I know about these questions and invite you to discover more.
Invariant differential operators, 1st EU-US Conference on automorphic forms and related topics, RWTH Aachen University, August 2012
Abstract: We will describe Helgason's method of computing algebras of invariant differential operators on homogeneous spaces, using various Jacobi group actions as examples. We will discuss in particular the factorization of invariant operators into first order covariant operators between different homogeneous vector bundles, and the computation of the central invariant operators by which the center of the universal enveloping algebra acts. This will be a survey talk, so we will do our best not to assume much prior knowledge of Lie theory.
Trigonometric identities and the Weyl character formula, MASS (Mathematics Advanced Study Semesters) Colloquium, Penn State, November 2011
Applications of representation theory to quantization, Workshop on Equivariant Quantization, University of Luxembourg, July 2011
We will describe some applications of the representation theory of
finite dimensional simple Lie algebras to quantization over Euclidean
manifolds. We will discuss the projective quantization of
differential operator modules of Vect(R^m), and in odd dimensions,
the conformal quantization of the restriction of these modules to the
subalgebra of contact vector fields.
We will show how the Casimir operators can be used to detect resonant cases, how Clebsch-Gordan rules can be used to classify the affine-invariant maps and hence to construct the quantization in the non-resonant cases, and how lowest weight vectors help to compute the quantized vector field actions.
x to the x to the x to the ..., Undergraduate Colloquium, Baylor University, November 2010
Quantizations of differential operator modules, AMS Winter Meeting, Integrability of Dynamical Systems Special Session, San Francisco, January 2010
Abstract: Suppose that one has an infinite dimensional Lie algebra of vector fields on a manifold, for example the set of all vector fields, or, if the manifold has a contact structure, the contact vector fields. Assume that this Lie algebra contains a distinguished finite dimensional semisimple maximal subalgebra, usually called its projective or conformal subalgebra.
There are various spaces of differential operators on the manifold which carry natural representations of such Lie algebras. The projective or conformal quantization of such a representation is its decomposition into irreducible representations of the subalgebra. We discuss recent results on quantizations and their applications to cohomology, geometric equivalences and symmetries of differential operator modules, and indecomposable modules. Our methods are algebraic: we consider only Euclidean manifolds.
Vect(R) & x to the x to the x to the ..., two VIGRE Student Colloquiua, Louisiana State University, November 2009
Extremal projectors, Representation Theory Seminar, Louisiana State University, November 2009
Let g be a complex finite dimensional reductive Lie algebra.
The extremal projector P(g) is an element of a certain formal
extension of the enveloping algebra U(g) which projects
representations in Category O to their highest weight vectors along
their lower weight vectors, provided that the denominator of P(g)
does not act by zero. (This denominator is a formal product in
U(h), h being the chosen Cartan subalgebra.)
In 1971 Asherova-Smirnov-Tolstoi discovered a noncommutative finite factorization of P(g), and in 1993 Zhelobenko discovered a commutative infinite product formula. We will discuss these results and some more recent formulas for the relative projector P(g,l), the projection to the highest l-subrepresentations, l being a Levi subalgebra. (The infinite commutative factorization of P(g,l) is known, but its denominator, a formal product in the center of U(l), is only known in a few simply laced cases. There are only preliminary results concerning finite factorizations of P(g,l).)
Modules of differential operators for vector field Lie algebras, AMS Sectional Meeting, Representation Theory Special Session, Waco, Texas, October 2009
Abstract: The Lie algebra Vec(R^m) of vector fields on Euclidean space contains the projective Lie algebra sl(m+1) as a maximal subalgebra. The space of differential operators on R^m is naturally a module under Vec(R^m). In this talk we will discuss the decomposition of this module under the projective subalgebra, and the use of this decomposition in analyzing the action of Vec(R^m). We will also mention some generalizations: the module of differential operators can be generalized to modules of differential operators between arbitrary tensor field modules, and in odd dimensions, Vec(R^m) can be replaced by the Lie algebra of contact vector fields, in which case the projective subalgebra is replaced by the conformal subalgebra, a copy of sp(m+1).