Colloquium: Alexander Teplyaev

About this event

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.