Given a semisimple Lie algebra and one of its involutions, it is possible to construct a coideal subalgebra B in the Êquantized enveloping algebra U which is a quantum analog of the classical enveloping algebra of the fixed Lie subalgebra. We study the space of B bi-invariants inside the associated quantized function algebra. Under the obvious restriction map, the space of bi-invariants is proved isomorphic to the Weyl group invariants of the character group ring associated to the restricted roots. The quantum Peter-Weyl decomposition and the classification of finite-dimensional spherical modules associated to U,B implies that this space of bi-invariants is a direct sum of one-dimensional eigenspaces for the action of the center of U. When the restricted root system is reduced, we show that the zonal spherical functions, i.e. representations of each eigenspace, correspond to Macdonald polynomials under a standard

Groups 32 was developed to illustrate an approach to writing mathematical research software. It has also proved useful in helping students understand group theory and learn to become better at proving theorems.

Large-scale similarity searches of genomic DNA are of fundamental importance in annotating functional elements in genomes. To perform large comparisons efficiently, BLAST and other widely used tools use seeded alignment, which compares only sequences that can be shown to share a common pattern or "seed" of matching bases. The choice of seed substantially affects the sensitivity of seeded alignment, but designing and evaluating optimal seeds is computationally challenging. In this talk I will address some of the computational and mathematical problems arising in seed design.The talk will rely on joint work with:- Ming Li, Bin Ma and John Tromp- Jeremy Buhler and Yanni Sun