# Special Double Feature

### Jason Fulman Mathematics Department Dartmouth College

The Rogers-Ramanujan identities are among the most beautiful equations in mathematics. Although initially introduced in the number theoretic setting of continued fractions, these identities also appear in conformal field theory, character theory of affine Lie algebras, and in statistical mechanics. This talk, based on thesis work, will describe the appearance of these identities in two new settings: the finite general linear groups and symmetric function theory.

### Vasily Strela Mathematics Department Dartmouth College

A function phi(t) is _refinable scaling_ if it satisfies a dilation equation phi(t) = sum_k C_k phi(2t-k). A refinable function generates multiresolution analysis (MRA): a set of nested subspaces ... V_{-1} \subset V_0 \subset V_1 ... such that the union of the V_k = L^2(R) and the intersection of all the V_k is the empty set. Translates phi(t-k) form a basis of V_0. Another part of MRA is a wavelet w(t), whose translates w(t-k) generate a subspace W_0 such that V_{1} is the direct sum of V_0 and W_0. Then {w(2^jt-k) | j,k \in Z} is a basis of L^2(R).

Standard scalar MRA assumes that there is only one scaling function. We make a step forward and allow several of them: \phi_0(t), ... ,\phi_{r-1}(t). The vector Phi(t) = [phi_0(t) ... phi_{r-1}(t)^T satisfies a dilation equation with _matrix_ coefficients C_k. Associated with phi(t) is a multiwavelet w(t) = [w_0(t) ... w_{r-1}(t)]^T. Multiwavelets have new features arising from the matrix structure of the dilation equation. For example, a multi-scaling function can combine orthogonality, symmetry and high approximation order while a scalar scaling function cannot.