Let ${\cal E}$ be a family of elliptic curves over a base variety defined over $\mathbb C$. An additive extension ${\cal G}$ of ${\cal E}$ is a family of algebraic groups which fits into an exact sequence of group schemes $0\rightarrow {\mathbb G}_{\rm a}\rightarrow {\cal G}\rightarrow {\cal E}\rightarrow 0$. We can define the special subvarieties of ${\cal G}$ to be families of algebraic groups over the same base contained in ${\cal G}$. The relative Manin-Mumford conjecture suggests that the intersection of a curve in ${\cal G}$ with the special subvarieties of dimension 0 is contained in a finite union of special subvarieties.

To prove this we can assume that the family ${\cal E}$ is the Legendre family and then follow the strategy employed by Masser-Zannier for their proof of the relative Manin-Mumford conjecture for the fibred product of two legendre families. This has applications to classical problems such as the theory of elementary integration and Pell's equation in polynomials.

The classification of irreducible representations of pin double covers of Weyl groups was initiated by Schur (1911) for the symmetric group and was completed for the other groups by A. Morris, Read and others about 40 years ago. Recently, a new relation between these projective representations, graded Springer representations, and the geometry of the nilpotent cone has emerged. I will explain these connections and the relation with a Dirac operator for (extended) graded affine Hecke algebras.  The talk is partly based on joint work with Xuhua He.

Gaussian quadrature rules are of theoretical and practical interest because of their role in numerical integration and interpolation. For general weighting functions, their computation can be performed with the Golub-Welsch algorithm or one of its refinements. However, for the specific case of Gauss-Legendre quadrature, computation methods based on asymptotic series representations of the Legendre polynomials have recently been proposed. 
For large quadrature rules, these methods provide superior accuracy and speed at the cost of generality. We provide an overview of the progress that was made with these asymptotic methods, focusing on the ideas and asymptotic formulas that led to them. 
Finally, the limited generality will be discussed with Gauss-Jacobi quadrature rules as a prominent possibility for extension.