We discuss a few examples in 2+1 dimensions and 1+1 dimensions supporting a recent conjecture concerning the relation between the Planck scale and the coupling strength of a non-gravitional interaction, unlike those examples in 3+1 dimensions, we do not have to resort to exotic physics such as small black holes. However, the result concerning these low dimensional examples is a direct consequence of the 3+1 dimensional conjecture.

The action of a Lie pseudogroup $\mathcal{G}$ on a smooth manifold$M$induces a prolonged pseudogroup action on the jet spaces$J$^{$n$}of submanifolds of$M$. We prove in this paper that both the local and global freeness of the action of $\mathcal{G}$ on$J$^{$n$}persist under prolongation in the jet order$n$. Our results underlie the construction of complete moving frames and, indirectly, their applications in the identification and analysis of the various invariant objects for the prolonged pseudogroup actions.

We consider the discrete Laplacian $\Delta$ on the cubic lattice $\mathbb{Z}^d$, and deduce estimates on the group $e^{i t \Delta}$ and the resolvent $(\Delta-z)^{-1}$, weighted by $\ell^q (\mathbb{Z}^d)$-weights for suitable $q \geqslant 2$. We apply the obtained results to discrete Schrödinger operators in dimension $d \geqslant 3$ with potentials from $\ell^p (\mathbb{Z}^d)$ with suitable $p \geqslant1$.

We describe the counting of BPS states of Type II on K3 by relating the supersymmetric cycles of genus <i>g</i> to the number of rational curves with <i>g</i> double points on K3. The generating function for the number of such curves is the left-moving partition function of the bosonic string.

We study the differential polynomial rings which are defined using the special geometry of the moduli spaces of Calabi-Yau threefolds. The higher genus topological string amplitudes are expressed as polynomials in the generators of these rings, giving them a global description in the moduli space. At particular loci, the amplitudes yield the generating functions of Gromov-Witten invariants. We show that these rings are isomorphic to the rings of quasi modular forms for threefolds with duality groups for which these are known. For the other cases, they provide generalizations thereof. We furthermore study an involution which acts on the quasi modular forms. We interpret it as a duality which exchanges two distinguished expansion loci of the topological string amplitudes in the moduli space. We construct these special polynomial rings and match them with known quasi modular forms for non-compact Calabi-Yau geometries and their