We consider the dynamical systems arising from substitution tilings. Under some hypotheses, we show that the dynamics of the substitution or inflation map on the space of tilings is topologically conjugate to a shift on a stationary inverse limit, i.e. one of R. F. Williams' generalized solenoids. The underlying space in the inverse limit construction is easily computed in most examples and frequently has the structure of a CW-complex. This allows us to compute the cohomology and K-theory of the space of tilings. This is done completely for several one- and two-dimensional tilings, including the Penrose tilings. This approach also allows computation of the zeta function for the substitution. We discuss $C^*$-algebras related to these dynamical systems and show how the above methods may be used to compute the K-theory of these.

The purpose of this paper is to apply the theory of MV polytopes to the study of components of Lusztig's nilpotent varieties. Along the way, we introduce reflection functors for modules over the non-deformed preprojective algebra of a quiver.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use PREPROJECTIVE ALGEBRAS AND MV POLYTOPES i∈I M i and of linear maps M a : M s(a) → M t(a) for each arrow a ∈ E.To each arrow a : i → j in E, we associate an arrow a * : j → i going in the opposite direction. We let H = E E * and we extend * to H by setting (a * ) * = a.

Let Gr be the affine Grassmannian for a connected complex reductive group G. Let C G be the complex vector space spanned by (equivalence classes of) Mirković-Vilonen cycles in Gr. The Beilinson-Drinfeld Grassmannian can be used to define a convolution product on MV-cycles, making C G into a commutative algebra. We show, in type A, that C G is isomorphic to C[N], the algebra of functions on the unipotent radical N of a Borel subgroup of G; then each MV-cycle defines a polynomial in C[N], which we call an MV-polynomial. We conjecture that those MVpolynomials which are cluster monomials for a Fomin-Zelevinsky cluster algebra structure on C[N] are naturally expressible as determinants, and we conjecture a formula for many of them. (Mathematics subject classification number: 14L35)This paper is dedicated to Robert MacPherson on the occasion of his 60th birthday. IntroductionSince Mirković and Vilonen first announced their discovery of a new geometric canonical basis in representation theory [MV1], the study of their algebraic varieties-here called MV-cycles-has grown quickly, with applications to geometry, representation theory, and combinatorics. Let us begin by mentioning a few of these.In the extended version of their original paper, Mirković and Vilonen [MV2] used MV-cycles to study the relation between the representation theory of a complex reductive group and the equivariant perverse sheaves on the affine Grassmannian Gr for the Langlands dual group. The MV-cycles are subvarieties of Gr giving a canonical basis for every irreducible representation. Braverman and Gaitsgory [BG] used MV-cycles to give a geometric definition of crystal graphs. Gaussent and Littelmann [GL] gave an interpretation of Littelmann's path model using MV-cycles. Kamnitzer [K] discovered the relation between the combinatorics of the MV-polytopes (moment map images of MV-cycles) and the Berenstein-Zelevinsky combinatorics used to compute Littlewood-Richardson coefficients [BZ].Essential to the Mirković-Vilonen results was the Beilinson-Drinfeld Grassmannian. This is a space which allows one to define a convolution product of MV-cycles (sometimes called a fusion product), making the vector space spanned by all MV-cycles into a commutative algebra. The purpose of this paper is to discuss this algebra and its combinatorial properties in type A. In particular, we identify it with a polynomial algebra, and conjecture a combinatorial formula for some of the polynomials.We will continue in the spirit of [AK], in which the authors computed all MV-polytopes in type A; we use mostly elementary tools and combinatorial models, and we avoid abstraction, always preferring to make concrete choices that make it easy to work with examples. There are two reasons for working in this way: (1) Our results and conjectures can be stated in this elementary language.(2) The literature about MV-cycles often gives the inaccurate impression that considerable heavy machinery and abstruse mathematics is required to understand them.Let us now discuss the main results an...

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