Marius Ionescu

Derivations and Dirichlet forms on fractals

Abstract We study derivations and Fredholm modules on metric spaces with a local regular conservative Dirichlet form. In particular, on finitely ramified fractals, we show that there is a non-trivial Fredholm module if and only if the fractal is not a tree (i.e. not simply connected). This result relates Fredholm modules and topology, refines and improves known results on p.c.f. fractals. We also discuss weakly summable Fredholm modules and the Dixmier trace in the cases of some finitely and infinitely ramified fractals (including non-self-similar fractals) if the so-called spectral dimension is less than 2. In the finitely ramified self-similar case we relate the p -summability question with estimates of the Lyapunov exponents for harmonic functions and the behavior of the pressure function.

IRREDUCIBLE REPRESENTATIONS OF GROUPOID C*-ALGEBRAS

If G is a second countable locally compact Hausdorff groupoid with Haar system, we show that every representation induced from an irreducible representation of a stability group is irreducible.

Pseudo-differential operators on fractals and other metric measure spaces.

We define and study pseudo-differential operators on a class of fractals that include the post-critically finite self-similar sets and Sierpinski carpets. Using the sub-Gaussian estimates of the heat operator we prove that our operators have kernels that decay and, in the constant coefficient case, are smooth off the diagonal. Our analysis can be extended to products of fractals. While our results are applicable to a larger class of metric measure spaces with Laplacian, we use them to study elliptic, hypoelliptic, and quasi-elliptic operators on p.c.f. fractals, answering a few open questions posed in a series of recent papers. We extend our class of operators to include the so called Hormander hypoelliptic operators and we initiate the study of wavefront sets and microlocal analysis on p.c.f. fractals.

C*-algebras associated with Mauldin-Williams graphs

A Mauldin–Williams graph

The Interplay between Student Loans and Credit Card Debt: Implications for Default in the Great Recession

M

Mauldin-Williams graphs, Morita equivalence and isomorphisms

is a generalization of an iterated function system by a directed graph. Its invariant set

Groupoid Actions on Fractafolds

K

THE RESOLVENT KERNEL FOR PCF SELF-SIMILAR FRACTALS

plays the role of the self-similar set. We associate a

Groupoid Methods in Wavelet Analysis

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REMARKS ON THE IDEAL STRUCTURE OF FELL BUNDLE C -ALGEBRAS

-algebra