Area of Research
- Pure Math
I study representations of Lie algebras and Lie groups from an algebraic and combinatorial point of view. In particular, I belong to an international group of mathematicians, the Atlas of Lie Groups and Representations, attempting to solve the Unitary Dual Problem, which has been open for more than 70 years. The Unitary Dual Problem is the key unresolved component in a broad programme in abstract harmonic analysis which was introduced by I.M. Gelfand in the 1930s. Gelfand's programme is a powerful generalization of Fourier analysis which may be used to address problems in many disciplines, such as number theory, mathematical physics, and topology. A prominent example of the importance of the Unitary Dual Problem is the scarcity of differential equations which may be solved using Fourier analysis techniques: such differential equations correspond to special cases of the Unitary Dual Problem which have been solved. Evidently, a solution to the Unitary Dual Problem is desired by mathematicians and scientists from many disciplines.
Potential Research Projects
I have projects available at the Masters and Ph.D. level concerning signed Kazhdan-Lusztig polynomials, a recently introduced combinatorial object which appears when studying both the structure of standard modules and the signatures of corresponding Hermitian forms. Doctoral students with an interest in geometry can study the structure of generalized Verma modules.
- Yee, Wai Ling. Signatures of Invariant Hermitian Forms on Irreducible Highest Height Modules, Duke Mathematical Journal, 24 pages, to appear.
- Yee, Wai Ling, The Signature of the Shapovalov Form on Irreducible Verma Modules, Representation Theory 9 (2005), 638–677.
- Hare, Kathryn E. and Yee, Wai Ling, The Singularity of Orbital Measures on Compact Lie Groups, Revista Math. Iberoamericana 20 (2004), no. 2, 517–530.