Elizabeth Gillaspy (University of Munster) Wavelets, spectral triples, and Hausdorff measure for directed graphs Sponsored by the Meyer Fund
Mar. 23, 2017 3pm (MATH 350)
Probability
Nathaniel Eldredge (University of Northern Colorado) Strong hypercontractivity on stratified Lie groups
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The space $E}^{\infty$ of infinite paths in a directed graph E is naturally a Cantor set; Julien and Savinien (building on work by Pearson and Bellissard) have studied spectral triples for such Cantor sets, and in particular shown that the Laplace-Beltrami operators ${\Delta}_{s}}_{s\in \mathbb{R}$ of these spectral triples all have the same eigenspaces. In joint work with Farsi, Julien, Kang, and Packer, we have discovered that this eigenspace decomposition agrees with a wavelet-type orthogonal decomposition of ${L}^{2}({E}^{\infty},M)$ first introduced by Marcolli and Paolucci. Moreover, the Dixmier trace measure associated to the Pearson-Bellissard spectral triple agrees with the Hausdorff measure on the Cantor set $E}^{\infty$. In my talk, I will sketch the proofs of the above results; no prior knowledge of spectral triples, wavelets, or Hausdorff measure will be assumed.
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Hypercontractivity is a property of Gaussian measure which says, roughly, that the transition semigroup of the Ornstein-Uhleneck process improves the integrability of functions as time passes, at a certain characteristic rate. It is intimately connected to another famous property, the logarithmic Sobolev inequality, which relates the integrability of a function and its derivative. If we work in a complex setting and restrict our attention to analytic functions, a stronger version of hypercontractivity turns out to hold; integrability improves faster.
I'll discuss how these topics fit together in the classical setting of Euclidean space with Gaussian measure, then introduce some recent work extending these results to the context of stratified Lie groups with hypoelliptic heat kernel measure.
This talk includes joint work with Leonard Gross and Laurent Saloff-Coste.