Colloquium: Alexander Teplyaev

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Bridging Classical Analysis, Fractals, and Aperiodic Order

Robert Strichartz’s work helped build a bridge from classical harmonic analysis and partial differential equations to analysis on fractals. Following this perspective, I will describe how energy provides a common starting point for defining Laplacians, studying eigenfunctions, and constructing diffusion on spaces without smooth coordinates. The Sierpiński gasket and related self-similar networks illustrate how geometric scaling produces spectral gaps, localized eigenfunctions, and diffusion with scaling laws different from those of Brownian motion in Euclidean space.

I will then turn to aperiodic order, motivated by structures such as quasicrystals, which exhibit organization without periodicity. The collection of translates of an aperiodic pattern leads to a space combining Euclidean and fractal features. Studying diffusion and Laplacians on these pattern spaces reveals connections among analysis, probability, and dynamical systems. Throughout the talk, concrete examples will illustrate how geometry, self-similarity, and order influence analytic behavior, and how classical ideas extend to these unfamiliar settings. The talk will be accessible to a broad mathematical audience, including graduate students. No prior background in fractal analysis or aperiodic order will be assumed.