Abstract: Consider a binary form

$ F = a_0 \, x_1^d + a_1 \, x_1^{d-1} \, x_2 + \dots + a_d \, x_2^d, \quad (a_i \in {\mathbf C}) $

of order $d$ in the variables $\{x_1,x_2\}$. Its Hessian is defined to be

$ \text{He} (F) = \frac{\partial^2 F}{\partial x_1^2} \frac{\partial^2 F}{\partial x_2^2} - \left(\frac{\partial^2 F}{\partial x_1 \partial x_2}\right)^2. $

It is classical that $F$ is the perfect $d$-th power of a linear form, if and only if $\text{He} (F)$ vanishes identically. Moreover, $\text{He}(F)$ is a covariant of $F$, in the sense that its construction commutes with a linear change of variables in $\{x_1,x_2\}$. Now assume that $d = r \, m$, and suppose we ask for a covariant whose vanishing is equivalent to $F$ being the perfect power of an order $r$ form. In 1885, Hilbert constructed such a covariant, to be denoted by $\mathcal{H}_{r,d}(F)$. In geometric terms, the variety of perfect powers of order $r$ forms defines a subvariety $X_r \subseteq {\mathbf P}^d$, and the coefficients of $\mathcal{H}_{r,d}$ give defining equations for this variety.

In this talk, I will outline a wholly different construction of this covariant, which leads to a generalisation called the G$\mathrm{\ddot{o}}$ttingen covariants. Moreover, we have the theorem that the ideal generated by the coefficients of the Hilbert covariant generates $X_r$ as a ${scheme}$, and not merely as a variety. This is joint work with Abdelmalek Abdesselam from the University of Virginia.

Abstract: In 1964, Milnor discovered flat tori in dimension 16 that are isospectral but not isometric. As amazing a result as this is, it still took about thirty years to construct isospectral plane domains that are not isometric. In this talk, I will review Sunada's method, as extended by Berard, to give an example of a pair of simply-connected nonisometric domains in the Euclidean plane that are both Dirichlet isospectral and Neumann isospectral.