## 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 **Tutorial Room 1A** of the Mathematics Institute.
A historical list of seminars can be found here.

### Autumn 2022

#### September 20, 2022

**Jonathan Fraser**:

The Fourier dimension of a measure captures the rate at which its Fourier transform decays at infinity. The Hausdorff dimension of a set, on the other hand, describes how the set fills up space on small scales by studying the cost of efficient covers. Despite how different they appear at first sight, these notions are intimately connected. Following the philosophy of 'dimension interpolation', I will introduce and discuss the 'Fourier dimension spectrum', which interpolates between the two notions. Time permitting, we will encounter applications to distance sets and sumsets.

#### September 27, 2022

**Mike Todd**:

I’ll give an introduction to (exponential) decay of correlations in dynamical systems, how this can be proved and the relevant constants involved. Moving to symbolic dynamics gives a clearer perspective on the constants involved here: I’ll discuss where they come from and how they might be improved.

#### October 4, 2022

**István Kolossváry**:

Baranski carpets exhibit interesting phenomena not witnessed by systems satisfying some sort of coordinate ordering property. We demonstrate that this is also true for multifractal analysis by looking at self-affine measures on a simple Baranski carpet. Namely, the multifractal formalism fails (i.e. the Legendre transform of the `L ^{q}` spectrum is not equal to the multifractal spectrum), even though the carpet has no overlaps and its Hausdorff and box dimensions are equal. The spectrum even has a jump discontinuity in case of the natural measure.

#### October 11, 2022

**Aleksi Pyörälä**:

During recent years, the prevalence of normal numbers in natural subsets of the reals has been an active research topic in fractal geometry. The general idea is that in the absence of any special arithmetic structure, almost all numbers in a given set should be normal, in every base. In our recent joint work with Balázs Bárány, Antti Käenmäki and Meng Wu we verify this for all self-conformal sets on the line. The result is a corollary of a uniform scaling property of self-conformal measures: roughly speaking, a measure is said to be uniformly scaling if the sequence of successive magnifications of the measure equidistributes, at almost every point, for a common distribution supported on the space of measures. Dynamical properties of these distributions often give information on the geometry of the uniformly scaling measure.

#### November 1, 2022

**Amlan Banaji**:

Geodesics are important objects in geometry, representing (in some sense) the shortest paths through a space. We introduce a class of metric spaces, called multigeodesic spaces, where between any two distinct points there exist multiple distinct minimising geodesics. We will prove a simple characterisation of multigeodesic normed spaces and deduce that `L ^{1}` spaces provide an example. In general metric spaces, however, examples such as Laakso spaces show that a wider variety of behaviour is possible.

#### November 8, 2022

**Kenneth Falconer**:

I will talk about higher order transversality and applications to `L ^{q}` dimension.

#### November 22, 2022

**Alex Rutar**:

A (deterministic) substitution consists of a finite alphabet along with a set of *transformation rules*. A classical example is the *Fibonacci substitution*, which is composed of the rules `a↦ab` and `b↦a`. Random substitutions allow multiple transformation rules for each letter, along with associated probabilities. Associated with a substitution is a shift-invariant ergodic frequency measure, which quantifies the relative occurrence of finite words as subwords of the substitution. Frequency measures are an interesting class of invariant measures which witness a form of self-similarity, while exhibiting complex overlapping phenomena. In this talk, I will provide a general introduction to random substitutions as well as their dimensional properties via the `L ^{q}`-spectrum. I will also discuss a particular class of random substitutions for which the

`L`-spectrum is analytic on

^{q}`ℝ`and the complete multifractal formalism holds. This work is joint with Andrew Mitchell (University of Birmingham).

#### November 29, 2022

**Lars Olsen**:

*TBD*

#### December 6, 2022

**Natalia Jurga**:

*TBD*

#### December 8, 2022

**Mark Holland**:

*TBD*