Abstract

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The space X of Penrose tilings of the plane has a natural topology on it. Two 

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tilings are equivalent if one can be obtained from the other by a translation. 

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The quotient topological space X/~ is bad: every point in it is dense. The 

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doctrine of non-commutative geometry is to refrain from passing to the quotient 

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and construct a non-commutative algebra that encodes some of the data lost in 

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passing to X/~. In this example (see Connes book for details) the relevant 

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non-commutative algebra is a direct limit of products of matrix algebras. We 

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will obtain this non-commutative algebra by treatng the free algebra on two 

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variables x and y modulo the relation y^2=0 as the homogeneous coordinate ring 

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of a non-commutative curve. The category of quasi-coherent sheaves on this 

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non-commutative curve is equivalent to the module category over a simple von 

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Neumann regular ring. That von Neumann regular ring is the same as the direct 

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limit  algebra that Connes associates to X/~. We will discuss algebraic analogues 

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of various topological features of X/~. For example, the non-vanishing of 

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extension groups between simple modules is analogous to the fact that every point 

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in X/~ is dense (which is analogous to the fact that any finite region of one 

<br>
Penrose tiling appears infinitely often in every other tiling).