Abstracts

Plenary speakers

• Jiaxin Hu: We consider nonlinear parabolic (or diffusion) equations on unbounded metric measure spaces admitting a heat kernel. It is known that a solution for diffusion may cease to exist in a finite
  time as a consequence of its L^\infty-norm becoming unbounded, that is, the solution blows up.
  On the other hand, in many of these problems there exist also global solutions. Both global and
  blowing-up solutions may be very unstable and they may exhibit a rather complicated asymptotic behavior. We investigate the blow-up of solutions, the local and global existence of weak solutions, as well as the regularity of these solutions when they exists. Joint with Kenneth Falconer and Yuhua Sun.

• Maarit Järvenpää: (TBA)

• Pertti Mattila: (TBA)

• Lars Olsen: (TBA)

• Yimin Xiao: The notion of packing dimension profiles was introduced by Falconer and Howroyd (1997) for computing the packing dimension of orthogonal projections. Their definition of packing dimension profiles is based on potential-theoretical approach. Later Howroyd (2001) defined another packing dimension profile from the point of view of box-counting dimension.  In this talk, we show that packing dimension profiles are useful for determining the packing dimension of the images of stochastic processes such as Lévy processes and Gaussian random fields. These applications motive further extensions of packing dimension profiles. This talk is based on joints papers with A. Estrade, D. Khoshnevisan, R Schilling, N.-R. Shieh and D. Wu.

• Martina Zähle: (TBA)

• Abel Farkas: I will talk about how do the dimensions and measures of fractal sets behave under projections. I will give an overview of many results of the last 60 years in this area.
• Jonathan Fraser: Falconer's subadditive pressure function, introduced in the late 1980s, has proved a seminal tool in the study of self-affine fractals.  The corresponding analogue in the conformal setting is an additive pressure function, which dates back to Moran-Sinai-Bowen-Ruelle-Hutchinson, and is now very well understood.  The additive pressure is much easier to handle and displays a lot more regularity than the subadditive version, for example, it is everywhere real analytic.  In this talk I will discuss what is known about the analyticity of the subadditive pressure and, since I cannot say anything in general, I will focus on the case of upper triangular matrix products. Even in this setting the pressure fails to be analytic everywhere, but is piecewise analytic.  If time permits I will provide a crude method for bounding the number of phase transitions by identifying them with zeros of Dirichlet polynomials and challenge the audience to beat my (poor) upper bound by the end of the week.
• Xiong Jin: I will talk about the exact dimensionality and dimension conservation of random cascade measures on self-similar sets, and some generalisations of Hochman and Shmerkin’s results on the lower semi-continuity of the dimension of projections, which rely on the group extension theory of ergodic dynamical systems. This is joint work with Kenneth Falconer.
• Ronan Le Guével: We will present in this talk several constructions of the multistable processes introduced by Falconer and Lévy-Véhel in 2008. We will exhibit the links between them in the particular case of the multistable Lévy motion. We show a semi-martingale property for these processes. Hausdorff or large deviation multifractal spectrum is also obtained according to the construction.
• Jun Miao: Falconer did some fundamentral work on Lipschitz equivalence of self-similar set, which made many mathematician move into this area, and I will review the main works on self-similar sets. Finally, I will give some new results on self-affine sets.
• Vuksan Mijovic: Talk will be based on joint paper with Lars Olsen in which we introduce multifractal zeta-functions providing precise information of a very general class of multifractal spectra, including, for example, the multifractal spectra of self-conformal measures and the multifractal spectra of ergodic Birkhoff averages of continuous functions. More precisely, we prove that these and more general multifractal spectra equal the abscissae of convergence of the associated zeta functions.
• Tony Samuel: The transfer operator $\widehat{T}$ is an important tool in the study of measure preserving dynamical systems $(X, \mu, T)$ where the measure $\mu$ has infinite mass.  For instance, it allows one to obtain generalisations of Birkhoff's Ergodic Theorem.  In these generalisations, one studies the asymptotics of the partial sums $\sum_{k = 0}^{n} \widehat{T}^{k}$.  However, it turns out that asymptotics of the operators $\widehat{T}^{k}$ themselves are considerably more delicate.  In this talk we will discuss recent results concerning the asymptotic behaviour of the iterates of the transfer operators for $\alpha$-Farey maps.

Contributed talks

• Benjamin Arras: The Rosenblatt process is a self-similar stochastic process with stationary increments which has the same covariance function as fractional Brownian motion. It is a zero quadratic variation process with continuous sample paths and it is not a semimartingale. In 2008, C. Tudor defined two stochastic integrals with respect to it using pathwise methods as well as Malliavin calculus tools. Due to the non-Gaussianity of the Rosenblatt process, Itô formula in the divergence sense seems rather difficult to obtain. In this talk, we will show how the means of Hida distribution theory can lead to an explicit Itô formula for sufficiently regular functionals of the Rosenblatt process.

• Thibaut Deheuvels: We study some questions of analysis in view of the modeling of tree-like structures, such as the human lungs. More particularly, we focus on a class of planar ramified domains $\Omega$ whose boundary contains a fractal self-similar part, $\Gamma$. We start by studying some function spaces defined for this class of domains. We first study the Sobolev regularity of the traces on the fractal part of the boundary of functions in some Sobolev spaces of the ramified domains. Then, we study the existence of Sobolev extension operators for the ramified domains we consider. In particular, we show that there exists $p*\in(1,\infty)$ such that there are $W^{1,p}$-extension operator for the ramified domains for every $1<p<p*$. The construction we propose is based on a Haar wavelet decomposition on the fractal set $\Gamma$. Finally, we compare the notion of self-similar trace on the fractal part of the boundary with more classical definitions of trace.
• Casey Donoven: The codimension formula describes the dimension of the intersection of two fractal in R^n. I will explain my recent work with Kenneth Falconer on abstracting this result into Cantor Spaces by building measures using martingales.
• Xiequan Fan: Multistable Lévy motions are extensions of the classical Lévy motions, where the stability index is allowed to vary in time. Several constructions and definitions, based on the Poisson representation, the Ferguson-Klass-LePage series representations and the characteristic functions, have been introduced quite recently by Falconer, Lévy Véhel, Le Guével and Liu. In this presentation, we give some new constructions of the independent increments multistable Lévy motion and to integrals with respect to multistable Lévy measure via sums of weighted independent random variables. These new representations allows us to prove that multistable Lévy motions are strongly localisable. Finally, we construct continuous approximations of multistable Lévy motions.
• Alexandre Richard: In this talk, we will present briefly the notion of abstract Wiener space, in order to use it in the study of a certain class of fractional Brownian fields. In particular we will present some results concerning the small ball probabilities of these fields and an application to the Hausdorff measure of a multiparameter fractional Brownian motion.

• Steffen Winter: Self-similar tilings are a useful tool in the study of geometric properties of self-similar sets. Tube formulas obtained for such tilings have been used for instance to express the Minkowski content of self-similar sets under certain assumptions in terms of the generator of an associated tiling. This allowed in particular some progress on Lapidus's conjecture concerning the Minkowski measurability of self-similar sets. In this talk we discuss the reach of the tiling approach (as well as its limitations). We will present some recent results on the existence of the Minkowski content for self-similar tilings which are based on renewal theory rather than tube formulas. We will give a necessary and sufficient condition for the Minkowski contents of a self-similar set and an associated tiling to coincide. The results extend far beyond the compatibility setting of earlier results and include e.g.\ also sets in $R^d$ of dimension less than $d-1$. The renewal theory approach is also applicable to fractal curvatures for which analogous results are obtained. In particular, the fractal curvatures of a self-similar set can be expressed in terms of the curvature data of the generator of an associated tiling.