I will start by briefly reviewing the Tate construction and in particular, the Tate diagonal. Using these I will then illustrate Lurie’s notion of Poincaré categories by considering Poincaré structures on module categories over a ring (spectrum) in detail. In particular, I will describe the somewhat subtle genuine Poincaré structure on the category of perfect complexes of an ordinary ring, which conjecturally links the classical notion of the Grothendieck-Witt spectrum to our derived version. Finally, I will compute its associated L-groups.

Let $X$ be a K3 surface and let $Z_X(q)$ be the generating series of the topological Euler characteristics of the Hilbert scheme of points on $X$. It is known that $q/Z_X(q)$ equals the discriminant form $\Delta(\tau)$ after the change of variables $q=e^{2 \pi i \tau}$. In this talk we consider the equivariant generalization of this result, when a finite group $G$ acts on $X$ symplectically. Mukai and Xiao has shown that there are exactly 81 possibilities for such an action in terms of types of the fixed points. The analogue of $q/Z_X(q)$ in each of the 81 cases turns out to be a cusp form (after the same change of variables). Knowledge of modular forms is not assumed in the talk; I will introduce all necessary concepts. Joint work with Jim Bryan.

For a finite subgroup $G$ of $SL(2,C)$ and for $n \geq 1$,  the Hilbert scheme $X=Hilb^{[n]}(S)$ of $n$ points on the minimal resolution $S$ of the Kleinian singularity $C^2/G$ provides a crepant resolution of the symplectic quotient $C^{2n}/G_n$, where $G_n$ is the wreath product of $G$ with $S_n$. I'll explain why every projective, crepant resolution of $C^{2n}/G_n$ is a quiver variety, and why the movable cone of $X$ can be described in terms of an extended Catalan hyperplane arrangement of the root system associated to $G$ by John McKay. These results extend the algebro-geometric aspects of Kronheimer's hyperkahler description of $S$ to higher dimensions. This is joint work with Gwyn Bellamy.

The comparison of graphs is a vitally important, yet difficult task which arises across a number of diverse research areas including biological and social networks. There have been a number of approaches to define graph distance however often these are not metrics (rendering standard data-mining techniques infeasible), or are computationally infeasible for large graphs. In this work, we define a new metric based on the spectrum of the non-backtracking graph operator and show that it can not only be used to compare graphs generated through different mechanisms but can reliably compare graphs of varying size. We observe that the family of Watts-Strogatz graphs lie on a manifold in the non-backtracking spectral embedding and show how this metric can be used in a standard classification problem of empirical graphs.