In this second talk we will discuss the idea of complexifying Clifford algebras and classifying them. We will give many examples of the Clifford algebra $Cl(s + t), s + t = n,$ on $R^n$ and see that these are actually matrices with entries from either $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$. Our discussions continues as we look at Clifford modules, which are representations of a Clifford algebra, and Clifford bundles. When $M$ is a Riemannian manifold with a metric $g$, the Clifford bundle of $M$ is the Clifford bundle generated by the tangent bundle $TM$.

In parallel with a classical definition due to Bieri and Eckmann, say an FP group G is an abelian duality group if $H^p(G,Z[G^{ab}])$ is zero except for a single integer $p=n$, in which case the cohomology group is torsion-free. We make an analogous definition for spaces. In contrast to the classical notion, the abelian duality property imposes some obvious constraints on the Betti numbers of abelian covers.

While related, the two notions are inequivalent: for example, surface groups of genus at least 2 are (Poincaré) duality groups, yet they are not abelian duality groups. On the other hand, using a result of Brady and Meier, we find that right-angled Artin groups are abelian duality groups if and only if they are duality groups: both properties are equivalent to the Cohen-Macaulay property for the presentation graph. Building on work of Davis, Januszkiewicz, Leary and Okun, hyperplane arrangement complements are both duality and abelian duality spaces. These results follow from a more general cohomological vanishing theorem, part of work in progress with Alex Suciu and Sergey Yuzvinsky.

One kind of representation of a spin group is obtained by restricting either a complex or real representation of the Clifford algebra to its spin group. It plays an important role in constructing generalized Dirac operators. In this talk, we will construct regular and spin representations of complex Clifford algebras. The former is a reducible representation given by Clifford multiplication on the exterior algebra and the latter is an irreducible representation on the exterior algebra of a complex polarization, also known as the polarized Fock space.

If a hexagon is inscribed in a conic, then the three points obtained by intersecting the opposite sides lie on a line. This is Pascal's theorem, first observed in 1639. By considering various pairs of sides obtained from the same six vertices, one obtains a collection of 60 such lines. This collection forms a highly intricate and symmetrical structure, usually called the 'hexagrammum mysticum'. I will explain some of the (myriad) properties of this structure, and the role of algebraic geometry and classical invariant theory in it. The pre-requisites will be kept rather modest, so as to ensure that the talk is widely accessible. 

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.

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.