Abstract: In this talk I will report some parts of a paper of Michel L. Lapidus and Jonathan J. Sarhad titled "Dirac Operators and Geodesic Metric on the Harmonic Sierpinski Gasket and Other Fractal Sets". First the Sierpinski gasket will be introduced as the unique fixed point of a certain contraction on the compact subsets of the Euclidean plane. Then, by defining the graph approximation of the Sierpinski gasket and also using a spectral triple on the unit circle, a Dirac operator and a spectral triple for the Sierpinski gasket will be constructed. Next, it will be shown that the spectral dimension of this spectral triple is bigger than or equal to 1. Connes' distance formula of noncommutative geometry provides a natural metric on this fractal. Finally, we shall see that Connes' metric is the same as the geodesic metric on the Sierpinski gasket.

Abstract: Suppose C is a category. We put a (quasi-)metric on the collection of all model structures on C, such that two model structures are more distant from one another if it takes a longer chain of Bousfield localizations and co-localizations to arrive at one from the other. We then compute the closed unit ball centered at the discrete model structure in this (quasi-)metric space.

Then we develop some methods for computing the entire (quasi-)metric space of all model structures on C! Our methods are actually categorical generalizations of constructions from classical linear algebra, namely Smith normal form. We describe categorical Smith normal form and its implications for the collection of model structures on a given category, and then we use these methods for some example computations: we explicitly compute all model structures and all their homotopy categories, their associated algebraic K-theories, and all their localizations and co-localizations, for the category of vector spaces over a field and then also for the category of modules over the tangent neighborhood of a regular closed point in a 1-dimensional normal scheme (e.g. Z/p^2-modules or k[x]/x^2-modules). Finally we do some commutative algebra and exploit our methods described above to prove the following amusing result: suppose R is a principal ideal ring whose modules admit generalized Smith normal form, that is, every indecomposable morphism of R-modules has indecomposable domain and indecomposable codomain (during the talk we will characterize exactly which principal ideal rings have this property). Then there exist exactly 5^A 11^B model structures on the category of R-modules, where A is the number of points in Spec R with reduced stalk, and B is the number of points in Spec R with non-reduced stalk.

Time allowing, we will actually say how to do the necessary commutative algebra to prove some of these results, which involves some work (for example, a cohomological solution to the compatible splitting problem: given a map from one split short exact sequence to another split short exact sequence, when does there exists a splitting of each short exact sequence which is compatible with the map?), and leads to a natural conjecture about how to continue the work into more sophisticated rings, by relating categorical Smith normal forms to edges in the Auslander-Reiten quiver; and we will sketch how one would go about giving an explicit classification of all model structures on torsion quasicoherent modules over a genus 0 nonsingular algebraic curve, if one had a proof of this conjecture.