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 D of the Mathematics Institute. A list of past seminars can be found here.

Spring 2026

January 13, 2026

Henna Koivusalo: On cut and project sets and substitutions

Cut and project sets are obtained by taking an irrational slice through a lattice and projecting it to a lower dimensional subspace. This usually results in a set which has no translational period, even though it retains a lot of the regularity of the lattice. As such, cut and project sets are one of the archetypical examples of point sets featuring aperiodic order. The other canonical method for generating aperiodic order is via substitutions; point sets generated from an inflation-redecoration process. Many classical examples of aperiodic order, e.g. the famous Penrose tiling, have a description from both methods. It is hence fundamental to the theory of aperiodically ordered point sets to answer the question: What property characterises those cut and project sets which have a substitution description. In this talk I will give an overview of the definition and basic properties of cut and project sets, discuss classical examples of aperiodic order, define the notion of `pattern with a substitution rule', and finally come to a new result of mine, answering the question above. This work is joint with Jamie Walton (Nottingham) and Edmund Harriss (Arkansas).

February 3, 2026

Steve Cantrell: Regularity of the pressure for quasi-morphisms on subshifts

Classical results from symbolic dynamics and thermodynamic formalism tell us about the dynamical/ergodic properties of Hölder functions on subshifts of finite type. In this talk we’ll discuss how to extend some of these results to a general class of functions called quasi-morphisms. We’ll also discuss some applications of these results to geometric problems.

February 10, 2026

Lars Olsen: The Baire Hierarchy, multifractal decomposition sets and Π_γ⁰-completeness

This talk will discuss the position of the so-called "multifractal decomposition sets" in the Baire Hierarchy. In particular, we will prove that "multifractal decomposition sets" are the building blocks from which all other Π_γ⁰-sets can be constructed; more, precisely, "multifractal decomposition sets" are Π_γ⁰-complete. As an application we find the position of the classical Eggleston-Besicovitch set in the Baire Hierarchy.

February 17, 2026

Haocong Song: Finiteness of Isometry Groups of Self-Similar Sets in R^3

This talk will give a brief introduction to the symmetry problem for self-similar sets. We resolve the problem in R^3, and I will share some of the key ideas involved in both the R^2 and R^3 cases. I will also outline the difficulties that arise when attempting to extend these methods to higher-dimensional spaces.

February 24, 2026 (2pm, Lecture Theatre D)

Kevin Hughes (Edinburgh Napier University): Estimates for the plunge region of limiting operators

A version of the uncertainty principle tells us that a nonzero function on Euclidean space cannot be compactly supported while its Fourier transform is compactly supported. This does not align with our perception of communications where signals are often treated as being localized in both physical space and frequency space. Trying to understand this, Landau, Pollack, Slepian and Widom studied the asymptotic eigenvalue distribution of time-frequency limiting operators. Their operators have eigenvalues in the interval (0,1) with the properties that 1) there are a predictable number of eigenvalues close to 1, 2) almost all eigenvalues decay quickly to 0 and 3) there is a 'plunge region' between the first two phenomena where the eigenvalues 'plunge' down towards 0. I will survey some recent work with and by A. Israel (UT Austin) and A. Mayeli (CUNY) on estimating the number of eigenvalues in the plunge region for analogous limiting operators in higher dimensions. See arXiv:1502.04404, arXiv:2301.09616, arXiv:2301.11685, arXiv:2403.13092, and arXiv:2601.21224.

February 24, 2026

Lijiang Yang: Fourier Dimension and Kakeya-Type Sets in the Plane

The Fourier dimension of a Borel set is defined through the decay rate of the Fourier transforms of measures supported on the set. In this talk, I will discuss several basic properties of the Fourier dimension and compute it for a number of examples. In particular, I will show that every Kakeya set in the plane has Fourier dimension 2. We will also study the range of possible Fourier dimensions for certain restricted Kakeya-type sets. More precisely, suppose the direction set has Hausdorff dimension at least t, and in each such direction we consider a subset of a unit line segment whose Fourier dimension is at least s. I will show that the Fourier dimension of such a set is bounded below by 2st/(s+2t) and may be as small as min{s,2t}. Joint work with Jon.

March 10, 2026

Fabian Despinoy: On Measures of Maximal Dimension

The dimension of a fractal can often be understood by considering the measures that the set supports. We will examine some classical dimension results in the field, moving through similarity, non-linear and then to self-affine iterated function systems while exploring the appearances of measures of maximal dimension via thermodynamic formalism. We will then move to the real projective line and after a brief introduction into projective iterated function systems, a short summary-proof of the existence of one of these measures will be provided.

March 17, 2026

Natalia Jurga: Exceptional sets in parameter spaces of dynamical systems

Many dynamical systems arise naturally as parametrised families. One can then study the associated parameter space and ask how qualitative properties of the dynamics are reflected geometrically within it. In this talk I will discuss three examples where classification problems lead to intricate subsets of parameter space: the Mandelbrot set in complex dynamics, the Rauzy gasket arising from interval exchange transformations, and a tiling structure associated with the topological classification of a family of self-similar sets. Although these examples arise in very different settings, they exhibit striking parallels in how parameter spaces organise qualitative changes in the underlying dynamics.

March 24, 2026

Yuyang Liu: Multifractal Curvature Dimensions

Classical multifractal quantities focus on the volume of neighbourhoods(d-dimensional part). In this talk I will introduce multifractal curvature dimensions, which capture k-dimension geometry and give a hierarchy of multiracial dimensions interpolating between volume and curvature information. I will also discuss basic properties and present some surprising examples.

March 31, 2026 (2pm, Lecture Theatre D)

José Alves (Porto University): A Smooth Path Through a Rough Landscape

We address the linear response problem for Bernoulli convolutions across the entire parameter range, including both absolutely continuous and singular invariant measures. Our approach relies on a geometric representation via a fat baker map, viewed as a skew-product system with contracting fibres. We introduce a sectional transfer operator that acts on sections of probability measures on the fibres and show that it is a contraction in the Wasserstein-1 distance. This yields the existence and uniqueness of a fixed point corresponding to the physical invariant distribution. We further prove a smooth dependence of this fixed point on the parameter in a suitable Banach space, establishing the linear response uniformly across the regular and singular regimes. Our approach also provides a linear response for solenoidal attractors with intermittency. Joint work with Wael Bahsoun.

March 31, 2026

Tianyi Feng: The Assouad dimension and spectrum of Furstenberg-type sets

For s in [0,1] and t in [0,2], an (s,t)-Furstenberg set X is a subset of the plane where there exists a family of lines of Hausdorff dimension at least t, such that the intersection between every line in the family and X is of Hausdorff dimension at least s. An interesting question to consider is how small the set X can be. I will give an overview of the problem and discuss some variants, particularly concerning the Assouad dimension and spectrum.

April 14, 2026

Kenneth Falconer: Symmetry and enumeration of self-similar fractals

After a short introduction to IFS and self-similar fractals we will use a little group theory to count the number of self-similar sets in certain classes.

April 21, 2026 (2pm, Lecture Theatre B)

Markus Myllyoja (University of Oulu): Hitting probabilities of random covering sets and the Ekström–Persson conjecture

We consider the Ekström–Persson conjecture regarding the Hausdorff dimension of random covering sets formed by balls and driven by general probability measures. The conjecture relates the dimension of typical random covering sets to multifractal properties of the driving probability measure, and it has been previously shown to hold when the balls forming the random covering sets are not too large. In this talk, we focus on the remaining case where the balls are not not too large, and we show that the conjectured value is always a lower bound for the dimension. We also demonstrate that the conjecture is not true in general, and that a previously established upper bound can in fact be attained. The lower bound for the dimension is achieved by studying the so-called hitting probability problem, where the objective is to decide whether or not a given (compact) deterministic set intersects typical random covering sets. The talk is based on a joint work with Esa Järvenpää and Stéphane Seuret.

April 21, 2026

Sascha Troscheit (Uppsala University): Dimension drop and dual IFSs for analytic iterated function systems

In a recent paper, Ariel Rapaport confirmed that the exponential separation condition implies no dimension drop for analytic self-conformal sets and measures in R. In this talk, I will show that the exponential separation condition is strongly linked to the separation conditions of a `dual' IFS and use this to show that the exponential separation condition holds typically in a strong topological sense and that the dimension drop conjecture holds `typically' for analytic self-conformal IFSs. Based on joint work with Balazs Barany and Istvan Kolossvary.

April 28, 2026

Yunlong Xu: TBC

TBC