Edinburgh Winter 2026 

Quasisymmetric geometry of self-affine fractals 

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

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.

Some boundedness results for bilinear operators with symbols in the class BS_{1,1}^m on spaces of critical smoothness. 

In this talk we will present some boundedness results for bilinear pseudodifferential operators with symbols in the class BS_{1,1}^m acting on functions in Triebel-Lizorkin spaces and certain subspaces of local bmo, involving spaces of logarithmic smoothness. 

As an application of the results obtained, we will show how in the Sobolev space of critical regularity n/2 --where the property of being a pointwise multiplicative algebra fails--, the product of two elements looses half a derivative of logarithmic order to remain in that space. More precisely, we will quantify that by showing that the product belongs to a suitable Triebel-Lizorkin space of logarithmic smoothness.

How Much of Lᵖ Is Really Commutative?

Noncommutative Lᵖ spaces associated with von Neumann algebras provide a natural operator theoretic analogue of classical Lᵖ spaces. While many familiar tools--such as interpolation and

Hölder-type inequalities--extend to this setting, some phenomena depend essentially on "commutativity". In this talk, I will present two representative results from classical Lᵖ theory: one that extends to the noncommutative framework, and one that fails.