# The classical and quantum geometry of polyhedral singularities and their resolutions

# The classical and quantum geometry of polyhedral singularities and their resolutions

Let $G$ be a finite subgroup of $SO(3)$. Such groups admit an ADE classification: they are the cyclic groups, the dihedral groups, and the symmetries of the platonic solids. The singularity $C^3/G$ has a natural Calabi-Yau resolution $Y$ given by Nakamura's $G$-Hilbert scheme. The classical geometry of $Y$ (its cohomology) can be described in terms of the representation theory of $G$. The quantum geometry of $Y$ (its quantum cohomology) can be described in terms of $R$, the ADE root system associated to $G$. This leads to an interesting family of algebra structures on the affine root lattice of R. Other aspects of the "quantum geometry" of $Y$ and $C^3/G$ (namely their Gromov-Witten and Donaldson-Thomas theories) are also governed by the root system $R$. One nice application is an attractive formula for the number of colored boxes piled in the corner of a roomâ€”generalizing the classical formula of MacMahon for the case of uncolored boxes.