Abstracts

Kenneth Falconer (St Andrews): A selective survey of projections

I first encountered John Marstrand's 1954 papers when I was a young researcher in the 1970s, and they have had an enormous influence on my research (and indeed that of many others) since then and continue to do so. I will give a selective survey of aspects of projections of sets that have particularly interested me, such as sets with prescribed projections, projection properties of packing dimensions, the exceptional set of projections, projections of 'special' sets such as self similar sets, etc.

Geoffrey Grimmett (Cambridge): The Grimmett-Marstrand theorem in percolation: then and now

The GM theorem states that the critical point of d-dimensional percolation may be approximated to any degree of accuracy by that of a sufficiently thick 2-dimensional slab. The proof is heavily geometrical, and is essentially unimproved 25 years later. I will describe the context and impact of the theorem, and will summarise its proof. The GM theorem is related to a more general conjecture concerning the manner in which a critical point is determined by the bounded balls of a graph, and in this context there has been recent progress by Martineau and Tassion. The notorious problem of continuity at the critical point remains open. It is an important open problem to prove the statement corresponding to the GM theorem for the case of a dependent model such as the random-cluster model, although there has been progress by Bodineau specific to the Ising model.

Robert Kaufman (Illinois, USA): Gap Series, Dimension, and Random Series

A sequence of integers with Hadamard gaps has the Sidon property from Fourier analysis. The main tool here is provided by Riesz products. Helson and Kahane (1965) found a variant concerning certain closed sets of measure $0$. In their theorem, the sets are necessarily $M_{0}$, a definite restriction. Using the average decay of certain Fourier-Stieltjes transforms, we obtain a similar result for each closed set of positive dimension. The random sequences found by this method grow geometrically, the minimum possible among all Sidon sequences.

Tom Kempton (St Andrews): Projection, slicing and scaling for self-affine measures

Marstrand's slicing theorem gives an almost sure value of the the dimension of a slice through a set of dimension $d$, as well as proving that the corresponding Hausdorff measure is finite. In the special case of self similar sets without rotation, we give simple conditions under which the Hausdorff measure of the slice can be shown to be positive. In fact this result is a corollary to a more general study of how to disintegrate measures on self-similar and self-affine sets via slicing. In the last part of the talk we use this more general result to study the scenery flow for self affine measures.

Henna Koivusalo (York): Projections of random covering sets

We study the almost sure properties of orthogonal projections of random covering sets to all $k$-dimensional subspaces of $\mathbb{R}^d$. Results hold for generating sequences of ball-like sets and are obtained through study of certain random Cantor sets.

François Ledrappier (Notre Dame, USA): A case of deterministic Marstrand behaviour

Marstrand Theorem and its extensions deal with projections along a random direction. We discuss the case of sets and measures on the unit tangent bundle of a surface that are invariant under the geodesic flow. Then, the fixed projection from the unit tangent bundle to the surface behaves as a random projection: it preserves as much dimension as possible and, at the limit value 2, it sends purely 2-unrectifiable sets to negligible sets.

Olga Maleva (Birmingham): Lowest fractal dimensions for universal differentiability

In a given space $X$, we are looking for as small as possible universal differentiability sets (UDS) $S$, defined by the requirement that every Lipschitz function on $X$ has a point of differentiability in $S$. We show that Euclidean spaces contain fractal universal differentiability sets of Minkowski dimension $1$. This is the lowest possible as all projections of the set of differentiability points inside UDS have positive measure. We also show that the $1$-dimensional Hausdorff measure of a UDS can never be sigma-finite. This is joint work with M. Dymond.

Pertti Mattila (Helsinki, Finland): Marstrand's line intersection theorem and related developments

In his 1954 paper Marstrand proved that an $s$-set of the plane with $s>1$ typically meets the lines, which it meets at all, in a set of Hausdorff dimension $s-1$. This has generated plenty of related questions and results. For example: what if lines are replaced by more general sets and what can be said about exceptional sets for 'typical'? And can one say something in Heisenberg groups? The talk is a survey on this topic.

Tuomas Orponen (Edinburgh): Restricted families of projections

I will start the talk by discussing the problem of 'restricted families of projections', that is, obtaining a.s. dimension conservation results for families of orthogonal projections far smaller than the full family. This part is based on joint work with K. Fässler from 2013. If time permits, I will continue by mentioning several open problems related to projections in the plane.

Michał Rams and Károly Simon (IMPAN, Poland and Budapest, Hungary): Projections of fractal percolations

An important result in geometric measure theory is Marstrand's Projection Theorem which is about the dimension of generic projections of a set/measure. An important idea in geometric measure theory is that randomness can replace genericity and usually lets us strengthen the assertions as well. Combining these two themes we obtain results on the dimension of projections of random Cantor sets (fractal percolations), which we will present. Joint work of M. Rams and K. Simon.

Julia Romanowska (Warsaw, Poland): On the dimension of the graph of the classical Weierstrass function

In my talk I will examine dimension of the graph of the famous Weierstrass non- differentiable function $$W_{\lambda,b}(x) = \sum_{n=0}^\infty \lambda^n \cos(2\pi b^n x)$$ for an integer $b \geq 2$ and $1/b < \lambda < 1$. In our recent paper, together with Balázs Bárány and Krzysztof Barański, we prove that for every $b$ there exists (explicitly given) $\lambda_b \in (1/b,1)$ such that the Hausdorff dimension of the graph of $W_{\lambda,b}$ is equal to $D = 2+ \frac{\log \lambda}{\log b}$ for every $\lambda \in (\lambda_b,1)$. We also show that the dimension is equal to $D$ for almost every $\lambda$ on some larger interval. This partially solves a well-known thirty-year-old conjecture. Furthermore, we prove that the Hausdorff dimension of the graph of the function $$f(x) = \sum_{n=0}^\infty \lambda^n\phi(b^nx) $$ for an integer $b \geq 2$ and $1/b < \lambda < 1$ is equal to $D$ for a typical $\mathbb{Z}$-periodic $C^3$ function $\phi$. In my talk I will talk about these results as well as I will introduce Ledrappier-Young theory and results of Tsujii, which were used in the proofs.

Jimmy Tseng (Bristol): Simultaneous dense and nondense orbits for commuting maps

We discuss an application of the Marstrand slicing theorem to dynamical systems. We show that, for two commuting automorphisms of the $d$-torus, many points have drastically different orbit structures for the two maps. Specifically, using measure rigidity, the Ledrappier-Young formula, and the Marstrand slicing theorem, we show that the set of points that have dense orbit under one map and nondense orbit under the other has full Hausdorff dimension. This mixed case, dense orbit under one map and nondense orbit under the other, is much more delicate than the other two possible cases. Our technique can also be applied to other settings. For example, we show the analogous result for two elements of the Cartan action on compact higher rank homogeneous spaces. This is joint work with V. Bergelson and M. Einsiedler.