# 6th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals

June 13–17, 2017

## Abstracts of Talks

### Keynote Speakers

#### Topology and fractals: measuring topological numbers with waves in quasicrystals

The diffraction pattern (Bragg or diffraction) and the spectrum of the Laplacian of Quasiperiodic tilings often exhibit fractal properties. We shall elaborate on these properties starting from familiar one-dimensional tilings (e.g. Fibonacci tilings). Then, we shall show how to relate these spectral and diffraction features to topological invariants of tilings best described using K-theory groups.

#### Power dissipation on fractal AC networks

Energy forms on graphs and on fractals can be interpreted in terms of electric linear networks by assuming that current flows between nodes (vertices) connected by resistors (edges). A resistor is called a dissipative element because there is a loss of energy when an alternating current runs through it. To the contrary, no loss is caused when the current flows through a non-dissipative element such an inductor or a capacitor. In the 60s Feynman described an infinite passive AC linear network (the infinite ladder), whose nodes were connected by inductors and capacitors, that would lead to actual power dissipation at some frequencies. Based on this idea, we study the concept of power dissipation on graphs and fractals associated to passive linear networks with non-dissipative components. We present in detail the so-called Sierpinski ladder fractal network and construct the power dissipation measure associated with continuous potentials in this network and prove it to be continuous as well as singular with respect to an appropriate Hausdorff measure defined on the fractal dust related to the network.

#### Strong shape theorems in cellular automata models on the Sierpinski gasket

It has been conjectured that on any state space, the growing clusters associated with the four cellular automata models---internal diffusion-limited aggregation (IDLA), rotor-router aggregation, divisible sandpiles, and abelian sandpiles---have the same limit shape. This conjecture is far from being proven. Even on $\mathbb{Z}^d$, while it has been proved that the limit shape is an Euclidean ball in the case of IDLA, rotor-router aggregation, and divisible sandpiles, the case of abelian sandpiles remains open.

The purpose of my talk is to explain that on the infinite graphical Sierpinski gasket ($SG$), when particles are launched from the corner vertex $o$ of $SG$, the cluster in each of the four models fills balls (centered at $o$ in the graph metric) with (asymptotically) precise rates. Thus there is a kind of "limit shape universality" on $SG$. We do not know the extent to which such phenomenon holds on other graphs, but some of our proof methods may hint at possible mechanisms underlying the universality.

This talk covers 3 papers on the aforementioned limit shape results, all written in 2017:

• IDLA: by Wilfried Huss (TU Graz), Ecaterina Sava-Huss (TU Graz), Sasha Teplyaev (UConn), and myself.
• Divisible sandpiles: by Huss and Sava-Huss.
• Rotor-router aggregation & abelian sandpiles: by Jonah Kudler-Flam (Colgate undergrad, Class of '17) and myself.

#### Multipliers of Dirichlet spaces

The aim of the talk is to introduce the notion of multiplier for a general Dirichlet space, generalizing the one studied in a Euclidean setting by Strichartz and Maz'ya. We show that bounded eigenfunctions are multipliers and various density properties of the algebra of finite energy multipliers. We conclude by showing that in a Euclidean space the natural multipliers' norm and seminorm are conformal invariants.

#### Fractal projection theorems - old and new

Marstrand’s theorems relating to the Hausdorff dimension of projections of sets date back over 60 years, but continue to motivate a great deal of research. We will review a range of developments, such as projections of specific classes of set, exceptional projections, box-dimension analogues, etc.

#### Generators of gap diffusions

The operator $d/d\mu\ d/dx$ is known to be the infinitesimal generator of a gap diffusion, the interesting case is that $\mu$ is singular, supported on a fractal. We recap some results on spectral properties and give stability estimates for the case that $\mu$ is approached by pre-fractal measures. Moreover, we explain how the gap diffusion can be obtained as a fractal transformation (in the sense of Barnsley) of Brownian motion. The talk gives a survey on joint works with Peter Arzt, Christian Seifert, Lenon Minorics and Tim Ehnes.

#### Spectral triples for noncommutative fractals

The Sierpinski gasket is studied from a functional point of view, and this provides a way to quantize it, namely to produce a self-similar noncommutative C$^*$-algebra containing the continuous functions on the gasket as a sub-algebra. The representations of this algebra are studied, it is shown that a noncommutative Dirichlet form can be defined, which restricts to the classical energy form on the gasket, and a spectral triple is proposed. Such triple reconstructs in particular the Dirichlet form via the formula $$a\to res_{s=\delta}\ tr(|D|^{-s/2}|[D,a]|^2 |D|^{-s/2}),$$ for a suitable $\delta$. Work in prograss with F.Cipriani, T.Isola e J-L.Sauvageot.

#### Non-fixed point diffusions on p.c.f. fractals

The construction of Laplace operators on p.c.f. fractals begins with the assumption of the existence of a non-degenerate fixed point for a renormalization map on a cone of Dirichlet forms. However it may be the case that such a fixed point does not exist, and there are only degenerate fixed points for the renormalization map. Even if the fixed point does exist, there may be degenerate fixed points as well. In the 1990s the set of asymptotically one-dimensional diffusions was constructed by Hattori-Hattori-Watanabe (HHW) on some Sierpinski gaskets by exploiting these degenerate fixed points. These processes are locally anisotropic, yet globally isotropic. We extend this work by constructing such non-fixed point diffusions more generally and then answer a question raised by HHW concerning the 'ultraviolet' scaling limit of these processes.

#### Fractal diffusion problems with drift

Our talk is part of finished and ongoing projects with Joe Chen, Maria Rosaria Lancia, Paola Vernole and Alexander Teplyaev. In the first part of the talk we sketch some simple observations related to mixed diffusions on the snowflake domain that have drift terms along the fractal boundary. This eventually leads to an interesting Lipschitz analysis for such problems. In the second part we consider heat equations with certain nonlinear drift terms on resistance spaces. Combining first order calculus and standard PDE theory in a straightforward manner we deduce some existence and uniqueness statements.

#### Harmonic analysis on fractals

We report on joint work with J. Herr and E. Weber: Using specific positive definite kernels and an adapted Kaczmarz algorithm, we establish basis constructions for a family of selfsimilar fractal measures. Some of our kernels correspond to the projection of the Szego kernel onto certain subspaces of the Hardy space corresponding to the normalized Cauchy transform of $\mu$.

#### Weyl's eigenvalue asymptotics for the Laplacian on the Apollonian gasket and on circle packing limit sets of certain Kleinian groups

The purpose of this talk is to present the speaker's recent results on the construction of a "canonical" Laplacian on circle packing fractals invariant under the action of certain Kleinian groups (discrete subgroups of the group of Moebius transformations on the Riemann sphere) and on the asymptotic behavior of its eigenvalues.

The simplest example is the Apollonian gasket, which is constructed from a given ideal triangle (the closed subset of the plane enclosed by mutually tangent three circles) by repeating indefinitely the process of removing the interior of the inner tangent circles of the ideal triangles. On the Apollonian gasket, a "canonical" Laplacian (to be more precise, a "canonical" Dirichlet form) was constructed by Teplyaev (2004) as the unique one with respect to which the coordinate functions on the Apollonian gasket are harmonic. The speaker has recently discovered an explicit expression of this Dirichlet form in terms of the circle packing structure of the gasket, which immediately extends to general circle packing fractals and defines (a candidate of) a "canonical" Laplacian on such fractals.

Then the speaker has further studied this Laplacian on more general circle packing fractals. When the circle packing fractal is the limit set of a certain class of Kleinian group (the smallest non-empty closed subset of the Riemann sphere invariant under the action of the group), some explicit combinatorial structure of the fractal is known, which makes it possible to prove Weyl's asymptotic formula for the eigenvalues of this Laplacian. The asymptotic formula involves the Hausdorff dimension and measure of the fractals and is of the same form as the circle-counting asymptotic formula by Oh and Shah (Invent. Math., 2012).

#### Differential one-forms on Dirichlet spaces and Bakry-Emery estimates on metric graphs

A general framework on Dirichlet spaces is developed to prove a weak form of the Bakry-Emery estimate and study its consequences. This estimate may be satisfied in situations, like metric graphs, where generalized notions of Ricci curvature lower bounds are not available. Based on Joint work with F. Baudoin.

#### Construction of metrics on a compact metric space

I would like to consider how one can construct a metric which possesses a prescribed property. For example, if you are given a map form the collection of compact subsets of your space to non negative real numbers, is there any metric on the space under which the given map coincides with the diameter of subsets? Even for a "simple" self-similar set like the Sierpinski carpet, this problem seems not trivial (for me at least.) I am going to explain how little I have gotten for the last three years on this problem.

#### Time changes of stochastic processes associated with resistance forms

In recent years, interest in time changes of stochastic processes according to irregular measures has arisen from various sources. Fundamental examples of such time-changed processes include the so-called Fontes-Isopi-Newman (FIN) diffusion, the introduction of which was motivated by the study of the localization and aging properties of physical spin systems, and the two-dimensional Liouville Brownian motion, which is the diffusion naturally associated with planar Liouville quantum gravity. This FIN diffusion is known to be the scaling limit of the one-dimensional Bouchaud trap model, and the two-dimensional Liouville Brownian motion is conjectured to be the scaling limit of simple random walks on random planar maps. We will provide a general framework for studying such time changed processes and their discrete approximations in the case when the underlying stochastic process is strongly recurrent, in the sense that it can be described by a resistance form, as introduced by J. Kigami. In particular, this includes the case of Brownian motion on tree-like spaces and low-dimensional self-similar fractals. If time permits, we also discuss heat kernel estimates for the relevant time-changed processes. This is a joint work with D. Croydon (Warwick) and B.M. Hambly (Oxford).

#### Magnetostatic problems in fractal domains

We will consider a magneto-static problem in a 3D fractal domain. A suitable notion of vector circulation will be given via a generalized Stokes theorem. Existence uniqueness and regularity results for the weak solution will be stated. The numerical approximation and some numerical simulations will be discussed.

#### An introduction to the theory of complex dimensions and fractal zeta functions

We will give some sample results from the new higher-dimensional theory of complex fractal dimensions developed jointly with Goran Radunovic and Darko Zubrinic in the recently published nearly 700-page research monograph (joint with these same co-authors), “Fractal Zeta Functions and Fractal Drums: Higher Dimensional Theory of Complex Dimensions” [2], published by Springer in June 2017 in the Springer Monographs in Mathematics series. We will also explain its connections with the earlier one-dimensional theory of complex dimensions developed, in particular, in the research monograph (by the speaker and M. van Frankenhuijsen) entitled "Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings" [1] (Springer Monographs in Mathematics, Springer, New York, 2013; 2nd rev. and enl. edn. of the 2006 edn.).

In particular, to an arbitrary compact subset $A$ of the $N$-dimensional Euclidean space (or, more generally, to any relative fractal drum), we will associate new distance and tube zeta functions, as well as discuss their basic properties, including their holomorphic and meromorphic extensions, and the nature and distribution of their poles (or 'complex dimensions'). We will also show that the abscissa of convergence of each of these fractal zeta functions coincides with the upper box (or Minkowski) dimension of the underlying compact set $A$, and that the associated residues are intimately related to the (possibly suitably averaged) Minkowski content of $A$. Example of classical fractals and their complex dimensions will be provided.

Finally, if time permits, we will discuss and extend to any dimension the general definition of fractality proposed by the author (and M-vF) in their earlier work [1], as the presence of nonreal complex dimensions. We will also provide examples of “hyperfractals”, for which the ‘critical line’ {${\rm Re}(s)=D$}, where $D$ is the Minkowski dimension, is not only a natural boundary for the associated fractal zeta functions, but also consist entirely of singularities of those zeta functions.

#### Neumann heat flow and gradient flow for the entropy on non-convex domains

For large classes of non-convex subsets $Y$ in $R^n$ or in Riemannian manifolds $(M,g)$ or in RCD-spaces $(X,d,m)$ we prove that the gradient flow for the Boltzmann entropy on the restricted metric measure space $(Y,d_Y,m_Y)$ exists - despite the fact that the entropy is not semiconvex - and coincides with the heat flow on Y with Neumann boundary conditions. This is joint work with K.-T. Sturm.

#### Regularity properties of Minkowski’s question mark measure

I will discuss some properties of Minkowski’s question mark measure, which originates from Minkowski’s 1904 question mark function. These properties mix in tantalizing contrast singularity (with respect to Lebesgue) and regularity (in terms of potential theory in the complex plane). In particular, I will prove regularity in the sense of Ullman, Saff, Stahl and Totik. The proof employs: an Iterated Functions System composed of Moebius maps, which yields the classical Stern-Brocot sequences, an estimate of the cardinality of large spacings in these sequences and a criterion due to Stahl and Totik. Consequences of this result will also be discussed and illustrated with numerical experiments.

#### Spectral dimension of a class of one-dimensional fractal Laplacians

The spectral dimension of the Laplacian defined by a measure has been shown to be closely related to heat kernel estimates, which under suitable conditions determine whether wave propagates with finite or infinite speed. We observe that some self-similar measures defined by finite or infinite iterated function systems with overlaps satisfy certain "bounded measure type condition", which allows us to extract useful measure-theoretic properties of iterates of the measure. We develop a technique to obtain, under this condition, a closed formula for the spectral dimension of the Laplacian. Earlier results for fractal measures with overlaps rely on Strichartz second-order identities, which are not satisfied by the measures we consider here. This is a joint work with Wei Tang and Yuanyuan Xie.

#### Szego limit theorems for pseudo-differential operators on the Sierpinski gasket

Using the existence of localized eigenfunctions for the Laplacian, we prove versions of the strong Szego limit theorem for certain classes of pseudodifferential operators defined on the Sierpinski gasket. These results can be used to establish asymptotics estimates for the spectra of these pseudo-differential operators. The talk is based on joint work with M. Ionescu and L. Rogers.

#### A P.C.F. self-similar set with no self-similar energy

A general class of finitely ramified fractals is that of P.C.F. self-similar sets. An important open problem in analysis on fractals was whether there exists a (non-degenerate) self-similar energy on every P.C.F. self-similar set (at level 1) with suitable weights. In this talk, I show an example of a P.C.F. self-similar set where there exists no (non-degenerate) self-similar energy at level 1. Note that it has been previously proved that in a general class of P.C.F. self-similar sets there exists a self-similar energy but only at a suitable sufficiently large level.

#### The homotopy critical spectrum for non-geodesic spaces

The homotopy critical spectrum (HCS) measures the size of 1-dimensional “holes” in a metric space. For compact geodesic spaces (i.e. every pair of points is joined by a path whose length realizes the distance between them), the HCS is equivalent, up to a multiplied constant, to the Sormani-Wei covering spectrum (CS). It also determines a part of the Length Spectrum (lengths of closed geodesics) and is hence related to the Laplace Spectrum. This latter observation leads to questions about isometry types of “isospectral” manifolds (having the same HCS/CS) that have been considered by de Smit-Gornet-Sutton. The homotopy critical spectrum also can be used to define strong, curvature-free finiteness theorems in Riemannian Geometry (joint work with Jay Wilkins). However, for metric spaces in general the homotopy critical spectrum behaves rather badly. We introduce a generalization of geodesic metric that allows the homotopy critical spectrum to be extended in a useful way (including some finiteness theorems) to spaces that may have no rectifiable curves at all, let alone geodesics. During the Elaborations session I will discuss the current status concerning limits of resistance metrics on finite graphs, which include many fractal metric spaces.

#### Boundary value problems for harmonic functions on domains in Sierpinski gaskets

We study boundary value problems for harmonic functions on certain domains in the level-$l$ Sierpinski gaskets $\mathcal{SG}_l$($l\geq 2$) whose boundaries are Cantor sets. We give explicit analogues of the Poisson integral formula to recover harmonic functions from their boundary values. Three types of domains, the left half domain of $\mathcal{SG}_l$ and the upper and lower domains generated by horizontal cuts of $\mathcal{SG}_l$ are considered at present. We characterize harmonic functions of finite energy and obtain their energy estimates in terms of their boundary values. This work settles several open problems raised in previous work. This is a joint work with Shiping Cao.

#### Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by $d$-sets

In the framework of the Laplacian transport, describing by a Robin boundary-valued problem in an exterior domain in $\mathbb{R}^n$, we generalize the definition of the Poincaré-Steklov operator to $d$-set boundaries, $n-1\le d<n$, and give its spectral properties to compared to a the spectrum of the interior domain and also to a truncated domain, considered as an approximation of the exterior case. The well-posedness of the Robin boundary-valued problems for the truncated and exterior domains are obtained in the general framework of $n$-sets.

#### Diffusions on fractal quantum graphs and other exotic spaces

On classical self-similar fractals, such at the Sierpinski gasket and Sierpinski carpet, there is a successful theory of diffusion processes due to Barlow, Bass, Hambly, Hino, Fukushima, Kususoka, Perkins et al. The construction of these processes usually relies on some fixed point arguments and heat kernel estimates. This talk will describe a method to construct symmetric local Dirichlet forms on some exotic spaces using algebras of functions that are in some sense `inherited' from underlying Euclidean spaces. Example of such spaces include fractal quantum graphs, snowflake domains with interior and boundary diffusions, Laakso spaces, symmetric Barlow-Evans vermiculated spaces. The presentation will emphasize some common ideas that appear in joint research with Alonso-Ruiz, Hinz, Kelleher, Lancia, Steinhurst, Vernole.

#### BVPs and trace theorems

In this talk, we firstly review the recent progress on BVPs and PDEs in singular settings. Then, we talk about our recent work on Polyharmonic BVPs on the Sierpinski Gasket and some stochastic extensions. We also report our work on $L^p$ estimates of certain extension and restriction operators on the Sierpinski Gasket.

### Other Speakers

#### A spectral triple for the stretched Sierpinski gasket

The spectral triples of noncommutative geometry are a useful tool in finding algebraic formulations for geometric concepts. Of great interest in fractal geometry are the concepts of dimension, geodesic distance, and measure. We will see how one can use spectral triples to recover these concepts for fractal sets like the stretched Sierpinski gasket. Also of interest is the use of spectral triples to recovery the energy forms from the study of analysis on fractals.

#### Developments on log-harmonic mappings and some of its applications

Let $H(D)$ be the linear space of all analytic functions defined on the open unit disc $D ={z \in C : |z| < 1}$. A sense preserving log-harmonic mapping is a solution of the non-linear elliptic partial differential equation$$\frac{f_{\overline{z}}}{\overline{f}}=w(z).\frac{f_{z}}{f}$$Where $w(z)$ the second dilatation of $f$ and $w(z)\in H(D)$, $|w(z)| <1$ for every $z\in D$. It has been shown that if $f$ is non-vanishing log-harmonic mapping, then $f$ can be expressed as$$f(z)=h(z).\overline{g(z)}$$Where $h(z)$ and $g(z)$ are analytic in $D$ with the normalization $h(0)\neq0$, $g(0) = 1$. On the other hand if $f$ vanishes at $z=0$, but it is not identically zero, then $f$ admits the following representation$$f(z)=z.z^{2\beta}h(z)\overline{g(z)}$$where $Re \beta>-\frac{1}{2}$, $h(z)$ and $g(z)$ are analytic in the open disc $D$ with the normalization $h(0)\neq0$, $g(0) = 1$. In the present paper, we will give the extended idea of this class, which was first introduced by Z. Abdulhadi [1]. One of the interesting application of this extended idea is an investigation of the subclass of log-harmonic mappings for which starlike log-harmonic mappings of order alpha. By this way, we use a new method for solving the coefficient inequalities.

#### Fractal calculus from fractal (non-Diophantine) arithmetic

I will show how to formulate calculus on fractals of Cantor and Sierpiński types. This includes differentiation, integration, and harmonic analysis. For example, the resulting Fourier transform on Cantor sets is not limited to particular types of Cantor sets, but works in a general case, including the celebrated middle-third set. As opposed to more standard approaches, in the proposed formalism there is no problem with first derivatives, ana Laplacians are truly second-order.

#### A local time scaling exponent for compact metric spaces

Given a Ahlfors regular compact metric space of (local) dimension $\alpha(x)$, we define a (local) critical exponent $\beta(x)$ in several ways. The procedure used to construct $\beta$ is in many ways analogous to the procedure used to construct $\alpha,$ but is based on either mean exit times from metric balls in approximating $\epsilon$-nets or on mean exit times from metric balls using random walks defined using an Ahlfors regular measure $\mu.$ In many cases $\beta$ corresponds with the walk dimension. A local power law is placed on the scaled exit time exponent $\beta,$ and it is used to define a time scaling condition mirroring the space scaling condition of Ahlfors regularity. We show how this condition and may be used to construct a non-trivial local Dirichlet form.

#### Fractal transformed doubly reflected Brownian motions

The theory of fractal transformations has been applied to image manipulation as well as to Fourier analysis in the last few years, but fractal transformations have never been applied to stochastic processes. We present a way to compose fractal transformations and paths of a Brownian motion. An important role plays the restriction of the paths of Brownian motion to bounded intervals to enable the composition with fractal transformations whose domains are such intervals. After restricting the paths we compose with fractal transformations according to weak conditions and study the properties of the resulting processes. We show examples of such fractal transformed doubly reflected Brownian motions and give possibilities to apply the developed theory.

#### Fractal models of gas diffusion and permeation through fibrous materials

Understanding gas diffusion and permeation in nanofibrous and microfibrous materials is of importance in many applications including functional clothing, fuel cells and microfluidics. In this presentation, fractal models developed in our group for predicting the diffusivity and permeability of porous fibrous media are introduced. The predictions of these models are compared with experimental results and the effects of model parameters such as porosity, pore area and tortuosity fractal dimensions are discussed.

#### Qualitative properties of double nonlinear parabolic system with absorption

Abstract: This work is devoted to the investigation of radial symmetric solution of Emden -Fouler type system, arising as stationary solution of the nonlinear processes of the heat conductivity, the reaction diffusion, the filtration in liquid and gas in two componential medium. Such process can be described by the following degenerate type parabolic system of the equations with double nonlinearity and coupled nonlinear absorption (1) (2) where , p –numerical parameters, . Different particular cases of this problem studied by many authors (see [1-4] and literature therein). The main difficulty in the study of the properties of solutions and the numerical realization of considered problems is due to the presence degeneracy of the system. Therefore, it is necessary to consider weak solutions in a class having a physical meaning, since in the degeneracy domain the solution of problem may not exist in the classical sense. As practice of the solutions of nonlinear problems shows, one of the most effective method for investigating various qualitative properties of non-linear systems is a self-similar and approximately self-similar approach [1,4] that allows us to identify new properties of solutions such as solutions with the finite propagation velocity, blow-up solutions, spatial localization of solution. In this work in particular are established the asymptotic radial symmetric solutions with compact support, continued solution and blow-up solution. Based on established qualitative properties of solutions the numerical computations are carried out.

REFERENCES
[1] A.A Samarskii., V.A Galaktionov., S.P.Kurdyomov, A.P.Mikhailov Blow-up in quasi-linear parabolic equations. Berlin, 4, Walter de Grueter, (1995), 535.
[2] Escobedo M., Herrero M. A. Boundedness and blow up for a semi linear diffusion reaction system. J. Differential Equation 1991, 176-192
[3] Pan Zheng, Chunlai Mu, Dengming Liu Xianzhong Yao, and Shouming Zhou Blow-Up Analysis for a Quasi-linear Degenerate Parabolic Equation with Strongly Nonlinear Source Abstract and Applied analysis Volume 2012, Article ID 109546, 19 pages, doi:10.1155/2012/109546
[4] Aripov M., Sadullaeva Sh.A. Qualitative properties of solutions of a doubly nonlinear reaction-diffusion system with a source // Journal of Applied Mathematics and Physics, 2015, 3, 1090-1099 (ISSN 2327-4352) (01.00.00; USA No 6).

#### On a recursive construction of Dirichlet form on the Sierpinski gasket

Let $\Gamma_n$ denote the $n$-th level Sierpinski graph of the Sierpinski gasket $K$. We consider, for any given conductance $(a_0, b_0, c_0)$ on $\Gamma_0$, the Dirchlet form ${\mathcal E}$ on $K$ obtained from a recursive construction of compatible sequence of conductances $(a_n, b_n, c_n)$ on $\Gamma_n, n\geq 0$. We prove that there is a dichotomy situation: either $a_0= b_0 =c_0$ and ${\mathcal E}$ is the standard Dirichlet form, or $a_0 >b_0 =c_0$ (or the two symmetric alternatives), and ${\mathcal E}$ is a non-self-similar Dirichlet form independent of $a_0, b_0$.

#### Spectral asymptotics on the Hanoi attractor

The Hanoi attractor (or Stretched Sierpinski Gasket) is an example of a non self similar fractal consisting of higher and lower dimensional parts. However it still exhibits a lot of symmetry. The existence of various symmetric resistance forms on the Hanoi attractor was shown in 2016 by Alonso-Ruiz, Freiberg and Kigami. To get the corresponding dirichlet forms (and thus the laplace operators) one has to choose a locally finite measure. All of these different choices have influence on the laplace operator. In particular we are interested in the Einstein relation and if it holds on the Hanoi attractor. The Einstein relation gives a powerful connection between the Hausdorff-, Walk- and Spectral-dimension. This talk focuses on the Spectral dimension of the Hanoi attractor which depends on the choice of the measure as well as the resistance form. The order of the first and second term of the eigenvalue counting function is obtained.

#### On the complex dimensions of nonarchimedean fractal sets

The higher dimensional theory of complex dimensions developed by Lapidus, Radunović, and Žubrinić provides a language for quantifying the oscillatory behaviour of the geometry of subsets of $\mathbb{R}^n$. In this talk, we will discuss a generalization of the theory that allows us to consider fractal sets in $p$-adic spaces. We will give several examples, and explore the geometric information that can be recovered from the complex dimensions of a set.

#### Sierpinski carpet as a Martin boundary

Denker and Sato used the Martin boundary of a certain Markov chain, to identify the Martin boundary with the Sierpinski gasket. Further they defined an operator on the Martin boundary to get a Laplacian on the gasket. We want to expand this approach to the Sierpinksi carpet and as a first step we identify the carpet with the Martin boundary.

#### Critical exponents of induced energy forms on self-similar sets

Recently we studied certain random walks on the hyperbolic graphs associated with the self-similar sets K, and showed that the discrete graph energy induces an energy form on K with a Besov-type domain. In this talk, we investigate the critical exponents in order for the induced form to be a regular Dirichlet form. We provide some criteria to determine such exponents through the effective resistance on graph, and make use of certain electrical network techniques to calculate the exponents for some concrete examples. This is a joint work with Ka-Sing Lau.

#### A noncommutative metric

The algebra of continuous functions on the Cantor Set can viewed as the inductive limit of sequences of finite dimensional C*-algebras. A noncommutative metric from which a standard ultrametric on the Cantor space can be recovered will be described.

#### Fractal zeta functions and geometric combinatorial problems

Fractal zeta functions are defined and used to study the distribution of distinct values of metric invariants determined by points in increasing families of self-similar subsets of $Z^n$.

#### Group actions, the Mattila integral and continuous sum-product problems

The Mattila integral, $${\mathcal M}(\mu)=\int {\left( \int_{S^{d-1}} {|\widehat{\mu}(r \omega)|}^2 d\omega \right)}^2 r^{d-1} dr,$$ developed by Mattila, is the main tool in the study of the Falconer distance problem. Recently this integral is interpreted by Greenleaf et al. in terms of the $L^2$-norm of the natural measure on $E-gE$, $g\in O(d)$, the orthogonal group. Following this group-theoretic viewpoint, we develop an analog of the Mattila integral associated with arbitrary groups. As an application, we prove for any $E,F,H\subset{\Bbb R}^2$, $\dim_{\mathcal{H}}(E)+\dim_{\mathcal{H}}(F)+\dim_{\mathcal{H}}(H)>4$, the set $$E\cdot(F+H)=\{x\cdot(y+z): x\in E, y\in F, z\in H\}$$ has positive Lebesgue. In particular, it implies that for any $A\subset{\Bbb R}$, $$|A(A+A)|>0$$ whenever $\dim_{\mathcal{H}}(A)>\frac{2}{3}$. We also give a very simple argument to show that on ${\Bbb R}^2$, $\dim_{\mathcal{H}}(E)>1$ is sufficient for $|E\cdot(E\pm E)|>0$, where the dimensional threshold is optimal. By taking $E=A\times A$, it follows that $$|A(A+A)+A(A+A)|>0$$ whenever $\dim_{\mathcal{H}}(A)>\frac{1}{2}$, which is also sharp. We therefore conjecture that $\frac{1}{2}$ is the best dimensional threshold for $A\subset{\Bbb R}$ to ensure $|A(A+A)|>0$.

#### p-adic fractal strings and complex fractal dimensions

The theory of p-adic fractal strings extends certain geometric aspects of the theory of real fractal strings (as developed by Michel Lapidus and Machiel van Franknenhuijsen in Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings) to the nonarchimedean field of p-adic numbers. We show that the Minkowski dimension of a p-adic fractal string coincides with the abscissa of convergence of the associated geometric zeta function. We also show that all self-similar p-adic fractal strings are lattice, and hence non-Minkowski measurable. Finally, we give a general explicit formula for the volume of the tubular neighborhood of a p-adic fractal strings and illustrate the general theory with the some concrete examples such as the 3-adic Cantor string and p-adic Euler string.

#### The viscous Burgers Equation on the Sierpinski gasket

In this talk, we are going to consider the viscous Burgers Equation on the Sierpinski gasket and discuss existence of weak solutions by using the Cole-Hopf Transformation. Starting with a regular symmetric Dirichlet form, we develop a suitable notation of abstract derivations which we need to impress the convection term. Further, we transfer the Cole-Hopf Transformation to metric spaces.

#### A family of self-avoiding walks on the Sierpinski gasket

We construct a multi-paramter family of self-avoiding walks on the Sierpinski gasket. This new parameter model contains a previous one-parameter family of self-avoiding processes such as the loop-erased self-repelling walk(LESRW), a family of self-avoiding walks interpolating the loop-erased random walk and a 'standard' self-avoiding walk. LESRW has a scaling limit and the limit process is self-avoiding regardless of the value of the parameter, however the limit processes of this new model can have infinitely many self-intersections by changing these parameters even if it is not the Peano-curve. In addition, the limit processes can be self-avoiding whose Hausdorff dimension of the paths takes any value greater than 1 and lower than the dimension of the state space. We also study an exponent which governs short time behavior and a law of the iterated logarithm. In this talk, I will describe a construction of this multi-parameter model and give a framework for understanding the path properties of the limit processes.

#### A projective Fraisse presentation of the Menger curve

Fraisse constructions have been used extensively in model theory for analyzing automorphism groups of countable discrete structures. These techniques were dualized by T.Irwin and S.Solecki and applied in the study of a certain compact space known as the pseudo-arc. In this talk we will provide a projective Fraisse presentation for the Menger curve and illustrate how various homogeneity properties of this space reduce to standard Fraisse theory and basic combinatorics. This is a joint work with Slawomir Solecki.

#### Riemann-Liouville fractional calculus of coalescence hidden-variable fractal interpolation functions

In this talk, I will first show that Riemann-Liouville fractional integral of a Fractal Interpolation Function (FIF) defined on any interval [a,b] is also a FIF. Then, it is extended to show that Riemann-Liouville fractional integral of a Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) defined on any interval [a,b] is also a CHFIF. Using these, I shall derive conditions for existence of Riemann-Liouville fractional derivative of FIF and CHFIF.

#### Log-Minkowski measurability and complex dimensions

In the past several years the well developed theory of complex dimension for fractal strings has been generalized to the higher-dimensional setting of bounded subsets of the Euclidean space and, more generally, relative fractal drums (RFDs in short). These complex dimensions are defined as the poles of the associated distance (or tube) fractal zeta function and are closely related (under suitable hypotheses) to the geometry of the given set or RFD. More precisely, the complex dimensions of a given set or RFD appear as the co-exponents in the asymptotic power expansion of its tube function (defined as the volume of its delta-neighborhood), when delta is close to zero. We show here some results about the situation when the tube function asymptotics of a given set (or RFD) is not of power type. More precisely, we give sufficient conditions on the distance (or tube) fractal zeta function which imply that the leading term in the tube function asymptotics is monotonic and of power-log type. In that case, we call the coefficient of the leading term the Log-Minkowski content of the given set or RFD. We illustrate our results by interesting examples some of which arise naturally from dynamical systems.

#### A criterion for box-counting measurability

Motivated by the seminal work of Lapidus and van Frankenhuijsen on the Minkowski measurability of bounded open subsets of the real line, this talk will cover a criterion for the box-counting measurability (an analog of Minkowski measurability) of bounded subsets of Euclidean space. The criterion is stated in terms of complex dimensions, i.e., the poles of a suitably defined zeta function, which encode a wealth of geometric information on the set in question.

#### A counterexample to the “hot spots” conjecture on nested fractals

Although the "hot spots" conjecture was proved to be false on some classical domains, the problem still generates a lot of interests on identifying the domains that the conjecture hold. The question can also be asked on fractal sets that admit Laplacians. It is known that the conjecture holds on the Sierpinski gasket and its variants. In this talk, we will show that the "hot spots" conjecture fails on the hexagasket, a typical nested fractal set. The technique we use is the spectral decimation method of eigenvalues of Laplacian on fractals.

#### The Sierpinski gasket as Martin boundary: the non-isotropic case

In the recent article "Martin boundary and exit space on the Sierpinski gasket", released 2012, Ka-Sing Lau and Sze-Man Ngai construct an isotropic Markov chain on the symbolic space which represents the Sierpinski gasket (SG). They show that the Martin boundary of this chain is homeomorphic to the Sierpinski gasket, whereas the minimal Martin boundary is equal to the boundary $V_0$ of the SG. Moreover, the induced harmonic functions on the Sierpinski gasket coincide with the canonical harmonic functions on the Sierpinski gasket. In this talk we will see how these results can be generalised to a non-isotropic Markov chain.

#### Approximation of fractals by discrete graphs: norm convergence of resolvents and spectra

We show a norm convergence result for the Laplacian on a class of pcf fractals with arbitary Borel regular probability measure which can be approximated by a sequence of finite-dimensional graph Laplacians with corresponding discrete probability measures. As a consequence other functions of the Laplacians (heat operator, spectral projections etc.) converge as well in operator norm. One also deduces convergence of the spectrum and the eigenfunctions in energy norm.

#### Generating functions for the Bernstein basis functions and their application to propbality distributions and fractals

In this talk, we study generating functions for the Bernstein basis functions. By using this functions we investigate some fundamental properties of the Bernstein basis functions. We give relations between this generating functions, the Erlang distribution, and the Poisson distribution. By using these relations, we derive some identities including the Bernstein basis functions and combinatorial sums. Finally, by applying the Bernstein polynomials, we give some remarks and observations on the elements of fractal geometry and its application in computer graphics and geometric modeling.

#### A trace theorem for the Laplacian on the Sierpinski gasket

As our world grows increasingly irregular, it is imperative that we reformulate our mathematical models to accurately describe this inherent natural roughness. The classical "smooth" analysis on manifolds is inadequate for a comprehensive study of the dynamic fractal phenomena that permeate nature. In the last two decades, a theory of analysis on fractals has been developed that centers on the construction of a Laplacian on "rough" sets such as the Sierpinski gasket SG (also called the Sierpinski triangle). This project characterizes the trace of the domain of this Laplacian operator (dom $\Delta$) to the boundary of the gasket. In particular, if SG is embedded in $\mathbb{R}^2$ such that its base coincides with the unit interval $I$, and $u \in$ dom $\Delta \cap C(SG)$, we apply a biharmonic spline approximation scheme and a novel convergence condition for the Laplacian to elucidate the regularity properties of $u|_I$.

#### Dirichlet form on a homogeneous hierarchical gasket

In this paper we define the Dirichlet form and its related harmonic functions on a homogeneous hierarchical gasket by the method of averages, and show that they enjoy the same properties as those defined in the pointwise sense.

#### Exact Hausdorff and packing measure of random attractors

Several standard models for randomising iterated function systems have been considered in the literature. Most notably random recursive and random homogeneous attractors. While the appropriate gauge function for Hausdorff and packing measure of the random recursive model has been found, the random homogeneous case has so far not been considered. In this talk we establish simple bounds on the class of appropriate gauge function and show that the random homogeneous model is much less well-behaved than its random recursive analogue.

#### Degenerate harmonic structures on fractal graphs

Harmonic functions on finitely ramified self-similar sets can be understood by studying them on a sequence of finite graphs approximating the fractal. The Dirichlet problem for the Laplacian on the self-similar set can then be reduced to the Dirichlet problem on a finite graph. We will study the invertibility of the harmonic extension matrices based on the vertex connectivity of the fractal graphs and prove that the harmonic structure of the two dimensional Sierpinski gaskets of level-k is always non-degenerate.

#### The LLL algorithm and the complex roots of a nonlattice Dirichlet polynomial

In this talk, we introduce a lattice basis reduction algorithm known as the LLL algorithm. This algorithm can be used to obtain simultaneous Diophantine approximations, which we use to approximate the complex roots of a nonlattice Dirichlet polynomial. We will also note the quasiperiodic structure of such a Dirichlet polynomial, which can be seen by obtaining many good approximations.