Quadrature is the term for the numerical evaluation of integrals.  It's a beautiful subject because it's so accessible, yet full of conceptual surprises and challenges.  This talk will review ten of these, with plenty of history and numerical demonstrations.  Some are old if not well known, some are new, and two are subjects of my current research.

Graded affine Hecke algebras were introduced by Lusztig for studying the representation theory of p-adic groups. In particular, some problems about extensions of representations of p-adic groups can be transferred to problems in the graded Hecke algebra setting. The study of extensions gives insight to the structure of various reducible modules. In this talk, I shall discuss some methods of computing Ext-groups for graded Hecke algebras.
The talk is based on arXiv:1410.1495, arXiv:1510.05410 and forthcoming work.

The prime spectrum of a quantum algebra has a finite stratification in terms
of a set of distinguished primes called H-primes, and we can study these
strata by passing to certain nice localizations of the algebra.  H-primes
are now starting to show up in some surprising new areas, including
combinatorics (totally nonnegative matrices) and physics, and we can borrow
techniques from these areas to answer questions about quantum algebras and
their localizations.    In particular, we can use Grassmann necklaces -- a
purely combinatorial construction -- to study the topological structure of
the prime spectrum of quantum matrices.