We present a compact formula computing rational trigonometric sums. Such sums appeared in the work of E. Verlinde on the dimension of conformal blocks in Wess-Zumino-Witten (WZW) theory. As an application, we show that a formula of J.-M. Bismut and F. Labourie for the Riemann-Roch numbers of moduli spaces of flat connections on a Riemann surface coincides with Verlinde's expression.

We define higher-dimensional Dedekind sums that generalize the classical Dedekind-Rademacher sums as well as Zagier's sums, and we show how to compute them effectively using a generalization of the continued-fraction algorithm.

We present two applications. First, we show how to express special values of partial zeta functions associated to totally real number fields in terms of these sums via the Eisenstein cocycle introduced by R. Sczech. Hence we obtain a polynomial time algorithm for computing these special values. Second, we show how to use our techniques to compute certain special values of the Witten zeta function, and we compute some explicit examples.

In this paper we prove that braid monodromy of an affine plane curve determines the topology of a related projective plane curve.

We develop representation theory of the rational Cherednik algebra ${\rm H}\sb c$ associated to a finite Coxeter group $W$ in a vector space $\mathfrak {h}$, and a parameter "c." We use it to show that, for integral values of "c," the algebra ${\rm H}\sb c$ is simple and Morita equivalent to $\mathscr {D}(\mathfrak {h})\#W$, the cross product of $W$ with the algebra of polynomial differential operators on $\mathfrak {h}$.

O. Chalykh, M. Feigin, and A. Veselov [CV1], [FV] introduced an algebra, $Q\sb c$, of quasi-invariant polynomials on $\mathfrak {h}$, such that $\mathbb {C}[\mathfrak {h}]\sp W\subset Q\sb c\subset \mathbb {C}[\mathfrak {h}]$. We prove that the algebra $\mathscr {D}(Q\sb c)$ of differential operators on quasi-invariants is a simple algebra, Morita equivalent to $\mathscr {D}(\mathfrak {h})$. The subalgebra $\mathscr {D}(Q\sb c)\sp W\subset \mathscr {D}(Q\sb c)$ of $W$-invariant operators turns out to be isomorphic to the spherical subalgebra $\mathbf {eH}\sb c\mathbf {e}\subset {\rm H}\sb c$. We show that $\mathscr {D}(Q\sb c)$ is generated, as an algebra, by $Q\sb c$ and its "Fourier dual" $Q\sb c\sp \flat$, and that $\mathscr {D}(Q\sb c)$ is a rank-one projective $(Q\sb c\otimes Q\sb c\sp \flat)$-module (via multiplication-action on $\mathscr {D}(Q\sb c)$ on opposite sides).

We determine those $k$-tuples of conjugacy classes of matrices from which it is possible to choose matrices that have no common invariant subspace and have sum zero. This is an additive version of the Deligne-Simpson problem. We deduce the result from earlier work of ours on preprojective algebras and the moment map for representations of quivers. Our answer depends on the root system for a Kac-Moody Lie algebra.

Given $f\in \mathbf {Z}[x\sb 1,\ldots x\sb n]$, we compute the density of $x\in \mathbf {Z}\sp n$ such that $f(x)$ is squarefree, assuming the abc-conjecture. Given $f,g\in \mathbf {Z}[x\sb 1,\ldots x\sb n]$, we compute unconditionally the density of $x\in \mathbf {Z}\sp n$ such that $\gcd(f(x),g(x))=1$. Function field analogues of both results are proved unconditionally. Finally, assuming the abc-conjecture, given $f\in \mathbf {Z}[x]$, we estimate the size of the image of $f(\{1,2,\ldots n\})$ in $(\mathbf {Q}\sp \ast/\mathbf {Q}\sp {\ast 2})\cup \{0\}$.