In [13, 14, 15, 16], Bernd Siebert and myself have been working on a program designed to understand mirror symmetry via an algebro-geometric analogue to the Strominger-YauZaslow program [29]. The basic idea is that the controlling objects in mirror symmetry are integral affine manifolds with singularities. One can view an integral affine manifold as producing a mirror pair of manifolds, one a s...

We consider the disklikeness of the planar self-affine tile T generated by an integral expanding matrix A and a consecutive collinear digit set D = {0, v, 2v, · · · , (|q|−1)v} ⊂ Z2. Let f(x) = x2+px+q be the characteristic polynomial of A. We show that the tile T is disklike if and only if 2|p| ≤ |q+2|. Moreover, T is a hexagonal tile for all the cases except when p = 0, in which case T is a s...

We introduce a model that allows for the prediction of the permeability of self-affine rough channels (one-dimensional fracture) and two-dimensional fractures over a wide range of apertures. In the lubrication approximation, the permeability shows three different scaling regimes. For fractures with a large mean aperture or an aperture small enough to the permeability being close to disappearing...

We investigate the velocity dependence of the friction between two rigid blocks limited by a self-affine surface such as the one generated by a crack. The upper solid is subjected either to gravity or to an extemal elastic stiffness, and is driven horizontally at constant velocity, V, while the lower solid is fixed. For low velocities, the apparent friction coefficient is constant. For high vel...

We classify surjective self-maps (of degree at least two) of affine surfaces according to the log Kodaira dimension. In this paper we are interested in the following question. Question. Classify all smooth affine surfaces X/C which admit a proper morphism f : X → X with degree f > 1. In [5] and [18], a classification of smooth projective surfaces with a self-map of degree > 1 has been given. Th...

For a self similar or self a ne tile in R we study the following questions What is the boundary What is the convex hull We show that the boundary is a graph directed self a ne fractal and in the self similar case we give an algorithm to compute its dimension We give necessary and su cient conditions for the convex hull to be a polytope and we give a description of the Gauss map of the convex hull

Self-affinity and self-similarity are fundamental concepts in fractal geometry. In this paper, they are related to collage grammars syntactic devices based on hyperedge replacement that generate sets of collages. Essentially, a collage is a picture consisting of geometric parts like line segments, circles, polygons, polyhedra, etc. The overlay of all collages in a collage language yields a frac...

We present a new method to calculate the Hausdorff dimension of a certain class of fractals: boundaries of self-affine tiles. Among the interesting aspects are that even if the affine contraction underlying the iterated function system is not conjugated to a similarity we obtain an upperand and lower-bound for its Hausdorff dimension. In fact, we obtain the exact value for the dimension if the ...