We compute the equivariant cohomology ring of the moduli space of framed instantons over the affine plane. It is a Rees algebra associated with the center of cyclotomic degenerate affine Hecke algebras of type $A$. We also give some related results on the center of quiver Hecke algebras and the cohomology of quiver varieties.

Using a construction of Barth and Verra that realizes torsion bundles on sections of special K3 surfaces, we prove that the minimal resolution of a general paracanonical curve $C$ of odd genus $g$ and order $\ell \ge \sqrt{\frac{g+2}{2}}$ is natural, thus proving the Prym–Green conjecture. In the process, we confirm the expectation of Barth and Verra concerning the number of curves with $\ell$-torsion line bundle in a linear system on a special K3 surface.

We define an equivariant index of ${Spin}^c$-Dirac operators on possibly noncompact manifolds, acted on by compact, connected Lie groups. Our main result is that the index decomposes into irreducible representations according to the quantization commutes with reduction principle.

It is known that the $3$-manifold $SL(2,\mathbb{Z})\backslash SL(2,\mathbb{R})$ is diffeomorphic to the complement of the trefoil knot in $S^3$. E. Ghys showed that the linking number of this trefoil knot with a modular knot is given by the Rademacher symbol, which is a homogenization of the classical Dedekind symbol. The Dedekind symbol arose historically in the transformation formula of the logarithm of Dedekind’s eta function under $SL(2,\mathbb{Z})$. In this paper we give a generalization of the Dedekind symbol associated to a fixed modular knot. This symbol also arises in the transformation formula of a certain modular function. It can be computed in terms of a special value of a certain Dirichlet series and satisfies a reciprocity law. The homogenization of this symbol, which generalizes the Rademacher symbol, gives the linking number between two distinct symmetric links formed from modular knots.