# Mathematical Sciences Research Institute

Home » AGRS Research Seminar Series: "Towards Analysis on Fractals: Piecewise $C^1$-Fractal Curves, Spectral Triples, and the Gromov-Hausdorff Propinquity " & "Multiplicative Chaos of the Brownian Loop Soup"

# Seminar

AGRS Research Seminar Series: "Towards Analysis on Fractals: Piecewise $C^1$-Fractal Curves, Spectral Triples, and the Gromov-Hausdorff Propinquity " & "Multiplicative Chaos of the Brownian Loop Soup" March 23, 2022 (02:00 PM PDT - 03:15 PM PDT)
Parent Program: The Analysis and Geometry of Random Spaces MSRI: Simons Auditorium, Online/Virtual
Speaker(s) Antoine Jego (University of Vienna), Therese-Marie Basa Landry (University of California, Santa Barbara; University of California, Riverside)
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Video

#### Towards Analysis on Fractals- Piecewise $C^1$-Fractal Curves, Spectral Triples, and the Gromov-Hausdorff Propinquity

Abstract/Media

To participate in this seminar, please register HERE.

This seminar is presented in 2 talks.

2 - 2:30 PM: "Towards Analysis on Fractals: Piecewise $C^1$-Fractal Curves, Spectral Triples, and the Gromov-Hausdorff Propinquity " - Therese-Marie Landry

Abstract: Many important physical processes can be described by differential equations. The solutions of such equations are often formulated in terms of operators on smooth manifolds. A natural question is to determine whether differential structures defined on fractals can be realized as a metric limit of differential structures on their approximating finite graphs. One of the fundamental tools of noncommutative geometry is Alain Connes’ spectral triple. Because spectral triples generalize differential structure, they open up promising avenues for extending analytic methods from mathematical physics to fractal spaces. The Gromov-Hausdorff distance is an important tool of Riemannian geometry and building on the earlier work of Marc Rieffel, Frederic Latremoliere introduced a generalization of the Gromov-Hausdorff distance that was recently extended to spectral triples.   The class of piecewise $C^1$-fractal curves was first characterized by Michel Lapidus and Jonathan Sarhad as a generalized setting for the spectral triple construction developed by Christensen, Ivan, and Lapidus in the context of the Sierpinski gasket.  We provide an analytic framework for the metric approximation of the Lapidus-Sarhad spectral triple on a piecewise $C^1$-fractal curve by spectral triples defined on an approximating sequence of finite graphs which exhibit properties motivated by the setting of the Sierpinski gasket.

2:45 - 3:15 PM: "Multiplicative Chaos of the Brownian Loop Soup" - Antoine Jego

Abstract: On the one hand, the 2D Gaussian free field (GFF) is a log-correlated Gaussian field whose exponential defines a random measure: the multiplicative chaos associated to the GFF, often called Liouville measure. On the other hand, the Brownian loop soup is an infinite collection of loops distributed according to a Poisson point process of intensity \theta times a loop measure. At criticality (\theta = 1/2), its occupation field is distributed like half of the GFF squared (Le Jan's isomorphism).

The purpose of this talk is to understand the infinitesimal contribution of one loop to Liouville measure in the above coupling. This work is not restricted to the critical intensity and provides the natural notion of multiplicative chaos associated to the Brownian loop soup when \theta is not equal to 1/2.

 Multiplicative Chaos of the Brownian Loop Soup 2.67 MB application/pdf Towards Analysis on Fractals: Piecewise $C^1$-Fractal Curves, Spectral Triples, and the Gromov-Hausdorff Propinquity 1.34 MB application/pdf