Edinburgh Winter 2026 

Quasisymmetric geometry of self-affine fractals 

Abstract: Quasisymmetric maps are generalisations of bi-Lipschitz maps which roughly preserve the shapes of sets but not necessarily their sizes. Consequently, unlike bi-Lipschitz maps, quasisymmetries can change the (Hausdorff, box, Assouad, etc.) dimension of a set. Quantifying how much the dimension of a set can be lowered by a quasisymmetry leads to various notions of conformal dimension, which are central invariants used in quasisymmetric classification problems. In this talk, my aim is to first motivate the study of quasisymmetric geometry, and in particular the conformal Assouad dimension, and then discuss some recent progress on quasisymmetric geometry of self-affine sets. These sets typically have a product-like local structure structure due to the non-conformal distortion of the generating affine maps, and under mild domination and irreducibilty assumptions on the matrix parts of the affine maps, this product structure leads to a dichotomy for the conformal Assouad dimension of self-affine sets: The conformal Assouad dimension is either 0 for trivial reasons, or equal to the Assouad dimension of the set. The talk is based on joint work with Alex Rutar. 

Variations on the Christ-Kiselev maximal inequality

Abstract: In 2001, M. Christ and A. Kiselev formulated an abstract maximal inequality associated to a general filtration on any measure space. Their theorem unifies many results from the literature, including work of Menchov (1927) and Paley (1931) on the almost everywhere convergence of general orthonormal systems and Zygmund's (1936) maximal Hausdorff-Young inequality.

In this talk I will present multiparameter and r-variational extensions of the Christ-Kiselev inequality. Variation semi-norm bounds are important in pointwise convergence problems, especially in  ergodic theory where there's no natural dense class of functions that pointwise convergence trivially holds. By its definition, a finite variation semi-norm implies pointwise convergence.

Abstract: TBC

Abstract: TBC