The Quantum McKay Correspondence for polyhedral singularities

Let G be a polyhedral group, namely a finite subgroup of SO(3). Nakamura's G-Hilbert scheme provides a preferred Calabi-Yau resolution Y of the polyhedral singularity C^3/G. The classical McKay correspondence describes the classical geometry of Y in terms of the representation theory of G. In this paper we describe the quantum geometry of Y in terms of R, an ADE root system associated to G. Namely, we give an explicit formula for the Gromov-Witten partition function of Y as a product over the positive roots of R. In terms of counts of BPS states (Gopakumar-Vafa invariants), our result can be stated as a correspondence: each positive root of R corresponds to one half of a genus zero BPS state. As an application, we use the crepant resolution conjecture to provide a full prediction for the orbifold Gromov-Witten invariants of [C^3/G].Comment: Introduction rewritten. Issue regarding non-uniqueness of conifold resolution clarified. Version to appear in Inventione

Sums of powers, and products of elements of the middle third Cantor set

Consider the set of sums of $m$'th powers of elements belonging to the Cantor middle third set $\mathscr{C}$, and the question of the number of terms required to ensure we find a large open interval in this set. Also consider the question of finding open intervals in the product of Cantor sets. A broad general framework that makes it possible to deal with the first problem was outlined in a paper by Astels. The question of finding the measure of $\mathscr{C} \cdot \mathscr{C}$ was considered recently in an article by Athreya, Reznick and Tyson. Astels' methods don't immediately apply to the second problem, and Athreya, Reznick, Tyson's methods become difficult in dealing with the first problem as $m$ becomes large. With the same elementary dynamical technique, in this paper we are able to answer both these questions in a satisfactory way. In particular, with $t_{m}=2 \cdot \lceil (\frac{3}{2})^{m-1}\rceil$ many terms, we find an open interval of measure at least $2\cdot (1- (2/3)^{m})$ in our set of sums of $t_m$ many $m'th$ powers, and the same elementary technique shows that the fourfold product of the Cantor set contains the interval $[(8/9)^{3},(8/9)]$.Comment: 9 page