Analysis Seminars
This seminar series is aimed at staff, postgraduates, and final year undergraduates at the University of St Andrews; anybody else who is interested in attending is welcome to join. The intention is to provide a background in analysis, with an emphasis on the research interests of the group.
The seminar takes place on Tuesday afternoons at 15:00 in Lecture Theatre 1D of the Mathematics Institute. A historical list of seminars can be found here.
Spring 2025
February 4, 2025
Thierry de Pauw:This talk is about a generalisation of the Lebesgue density theorem. With each point x in the plane, we associate a line L(x) containing x and we ask whether, for each Borel-measurable subset A of the plane, the density ratio H1(A ∩ B(x,r) ∩ L(x))/2r converges to 1 as r ↘ 0, for a.e. x ∈ A. Here, H1 is the 1-dimensional Hausdorff measure. The answer depends on the regularity of the mapping L : x ↦ L(x). If the direction of L is constant, then the answer is positive, according to the Lebesgue density theorem in the real line combined with Fubini's theorem. For some L whose direction is merely continuous, the answer is negative due to the existence of a Nikodym set and, in fact, the above density ratio may vanish for all x ∈ A. I will report on the following progress: if the direction of L is Lipschitz then the superior limit of the above density ratio exceeds 1/4 for a.e. x ∈ A.
February 11, 2025
Ana de Orellana:We study an extension of the Stein—Tomas restriction theorem using the Fourier spectrum, a family of dimensions that live between the Fourier and Hausdorff dimensions. This leads to an Lp to L2 estimate for restriction on multifractals. Joint work with Marc Carnovale and Jonathan Fraser.
February 18, 2025
Dmytro Karvatskyi:We study the topological and fractal properties of self-similar sets on the real line, generated by an IFS with an integer similarity dimension and translations that possess a peculiar combinatorial property. A recent result obtained with O. Makarchuk on the Lebesgue measure of attractors, together with other findings, can help us uncover their topological structure. The most intriguing scenario arises when the attractor has a non-empty interior and a fractal boundary.
February 25, 2025
Pieter Allaart:We consider random subsets of self-similar sets generated by a branching random walk with exponentially decreasing steps, or equivalently, via random labelings of the edges of a full infinite M-ary tree. These random subsets can be viewed as statistically self-similar sets with extreme overlaps, hence their almost-sure box-counting and Hausdorff dimensions coincide. We show that the computation of this dimension breaks into three essentially different cases, depending on the parameters of the model. In one case, the random subset has the same dimension as the self-similar set it is part of, although it is almost surely a proper subset. We give precise formulas for the dimension in the other two cases. This is joint work with Lauritz Streck.
March 11, 2025
Antti Kaemaki:We compute the almost sure Hausdorff dimension of random self-affine sponges in Rd without imposing any separation conditions. In this context, randomness arises from the matrices in the defining semigroup, which are random yet the corresponding affine maps share a fixed point. This setup reveals a novel phenomenon: the almost sure Hausdorff dimension---even in the separated self-similar case---depends on the specific way randomness is introduced into the matrices. This talk presents ongoing research conducted with Balázs Bárány and Michal Rams.
March 18, 2025
Lijian Yang:We investigate a special class of Kakeya sets by considering the collection of midpoints of unit line segments. Given an upper bound on the dimension of this midpoint collection, we establish a lower bound on the Hausdorff dimension of the corresponding Kakeya set. As a corollary, we obtain a result that implies the Kakeya conjecture under certain conditions. Joint work with Jonathan Fraser.
March 25, 2025
Natalia Jurga:One of the fundamental features of an ergodic dynamical system (T,X,m) is that for almost every x in X, the sequence of empirical distributions along its orbit converges to the measure m. In this talk we will discuss a quantitative version of this fact and its connection to the fractal properties of the measure m. This is ongoing work with Mike Todd.
April 1, 2025
Tianyi Feng:We investigate the Assouad dimension of graphs of typical functions in Hölder-like spaces, such as the little Hölder space and the space of quasi-Lipschitz functions. Joint work with Jonathan Fraser.