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About this talk

The Department of Mathematics is hosting a series of colloquia, open for all to attend. On 27 August 2026, Alexander Schied from the University of Waterloo will give a talk on 'Weierstrass Bridges'. Abstract: Weierstrass Bridges Classical Weierstrass-type functions, going back to Weierstrass, Takagi, and van der Waerden, are among the earliest and most famous examples of continuous but nowhere differentiable functions. Fractional Brownian motion, on the other hand, is a fundamental family of Gaussian processes whose trajectories range from very rough to comparatively smooth. Weierstrass bridges arise by combining these two constructions: one replaces the deterministic periodic function in a Weierstrass-type series by a fractional Brownian bridge. After introducing the necessary background, we will discuss a phase transition that appears when investigating analytic sample-path properties of Weierstrass bridges, including their moduli of continuity, Wiener-Young variation, and the Hausdorff dimension of their graphs. It turns out that this phase transition is governed by the roughness exponent, a new pathwise measure of irregularity defined through variation along refining partitions. Roughly speaking, the fine structure of Weierstrass bridges is closer to the rougher of their two components: the Weierstrass-type fractal construction or the underlying fractional Brownian motion. In the critical case, where the two roughness exponents coincide, new logarithmic phenomena appear. This talk is based on joint work with Zhenyuan Zhang. A reception will follow afterwards in the Common Room. The talk will also be streamed online via Microsoft Teams.

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