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

Yuyang Liu: TBC

TBC

March 17, 2026

Natalia Jurga: TBC

TBC

March 24, 2026

Fabian Despinoy: TBC

TBC

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

José Alves (Porto University): TBC

TBC

March 31, 2026

Yunlong Xu: TBC

TBC

April 7, 2026

Tianyi Feng: TBC

TBC

April 14, 2026

Luke Derry: TBC

TBC

April 21, 2026

Sascha Troscheit (Uppsala University): TBC

TBC

April 28, 2026

Kenneth Falconer: TBC

TBC