# 7th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals

June 4–8, 2022

**photo Cornell Fractals 7 (2022)**

## Schedule of Activities

### Main Talks and Courses

The main talks and courses will all be held in Malott Hall according to the following schedule; follow the links to see pdf slides of the speakers’ talks. Light refreshments will be available in the lounge (room 532) at 8:30 a.m. and during mid-morning and mid-afternoon breaks. See also the schedule of elaborations (1:15 – 2:45 p.m. daily) below the main schedule.

Time | Room | Saturday June 4 |
Sunday June 5 |
Monday June 6 |
Tuesday June 7 |
Wednesday June 8 |
---|---|---|---|---|---|---|

refreshments |
532 | Registration check-in |
||||

9:00-9:30 a.m. | 251 | Kigami | Hera | Barnsley | Rozanova-Pierrat pdf | Akkermans |

9:35-10:05 a.m. | 251 | Kajino pdf | Samuel | Freiberg | Sava-Huss(rm 251) Suzuki(rm 207) | |

coffee break |
532 | Registration check-in |
||||

10:40-11:10 a.m. | 251 | Hewett | Shanmugalingam | Lancia | Pietruska-Pałuba | Zähle |

11:15-11:45 a.m. | 251 | Gartland | Li Chen pdf | Vélez-Santiago | Mantica pdf | Jorgensen |

lunch |
||||||

1:10-1:55 p.m. Course I |
251 | Kigami | Kigami | Shanmugalingam | Xiao | Hayes Heizmann pdf dropbox |

2:00-2:45 p.m. Course II |
251 | Freiberg | Freiberg | Teplyaev | Espigule | Rogers |

2:50-3:20 p.m. | 251 | Xiao | Treviño | Post | Peirone | Plaut |

break | ||||||

3:50-4:10 p.m. | 251 | Bessi | Capitanelli | David | 4:00pm |
Stepanenko |

4:15-4:35 p.m. | 251 | Cao, Shiping | Radunovic pdf | Lapidus | Strichartz |
Farkas |

4:40-5:00 p.m. | 251 | Mograby | Hoffer | Landry | Memorial |
Plyukhin |

5:05-5:25 p.m. | 251 | Kaleta | Menzel | Kolossváry | Cao, Jun | |

parallel talks | ||||||

3:50-4:10 p.m. | 207 | Cohen | Anurag Prasad | |||

4:15-4:35 p.m. | 207 | Rawashdeh | ||||

4:40-5:00 p.m. | 207 | Obeidat | ||||

5:05-5:25 p.m. | 207 | |||||

6:00-8:00 p.m. | 532 | Reception |

### Elaborations

Half-hour elaborations of the 30-minute main talks,

following the old tradition

(see Fractals 2),

are intended to be informal talks, discussions,

question-and-answer sessions for people particularly

interested in a given subject.

The time slots are intermingled to

allow interested people to ask questions in all sessions.

Time | Room | Saturday June 4 |
Sunday June 5 |
Monday June 6 |
Tuesday June 7 |
Wednesday June 8 |
---|---|---|---|---|---|---|

1:15-1:45 p.m. | 203 | Kajino | Xiao | Freiberg | Post | Peirone |

1:30-2:00 p.m. | 207 | Hewett | Hera | David | ||

1:45-2:15 p.m. | 203 | Gartland | Shanmugalingam | Lancia | Pietruska-Pałuba | |

2:00-2:30 p.m. | 207 | Samuel | ||||

2:15-2:45 p.m. | 203 | Kigami | Li Chen | Mantica |

### Eric Akkermans Technion

Topological Properties of Fractal Spectra

*(online)*

### Srijanani Anurag Prasad Indian Institute of Technology Tirupati

Node Insertion in Coalescence Fractal Interpolation Function

The Iterated Function System(IFS) used in the construction of Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) depends on the interpolation data. The insertion of a new point in a given set of interpolation data is called the problem of node insertion. In this talk, the effect of insertion of new point on the related IFS and the Coalescence Fractal Interpolation Function would be described.

### Michael Barnsley Australian National University

Fractal Tiling Theory

*(online)*

### Ugo Bessi Dipartimento di Matematica e Fisica, Roma Tre.

Kusuoka’s measure and Dynamical Systems.

On many fractals G one can define an expansive map F : G → G, the couple (G,F) is a dynamical system and several authors have noticed that Kusuoka’s measure is a natural object for the dynamics, i. e. a particular case of a Gibbs measure. We shall explain our construction of Kusuoka’s measure and we shall use its ergodic properties to identify Cheeger’s energy and Kusuoka’s quadratic form on several fractals.

### Jun Cao Zhejiang Univerisity of Technology

Gaussian estimates for heat kernels of higher order Schröedinger operators with potentials in generalized Schechter class

Let L = P(D) + V be a higher order Schrödinger operator on the Euclidean space with the potential V satisfying some generalized Schechter conditions. In this talk, we show that L possesses a heat kernel that satisfies some sub-Gaussian upper bound and the Hölder regularity. The Davies–Gaffney estimates for the associated semigroup and their local versions are also discussed.

### Shiping Cao Cornell University

Dirichlet forms on unconstrained Sierpinski carpets

We construct symmetric self-similar Dirichlet forms on unconstrained Sierpinski carpets, which are natural extension of planar Sierpinski carpets by allowing the small cells to live off the 1/k grids. The intersection of two cells can be a line segment of irrational length, and we also drop the non-diagonal assumption in this recurrent setting. A uniqueness theorem is also provided. Moreover, the additional freedom of unconstrained Sierpinski carpets allows us to slide the cells around. In this situation, we view unconstrained Sierpinski carpets as moving fractals. We prove that the self-similar Dirichlet forms will vary continuously in a Γ-convergence sense, and the generated diffusion processes, viewed as processes in ℝ^{2}, will converge in distribution. This is a joint work with Hua Qiu.

### Raffaela Capitanelli Univ. of Rome Sapienza

Fractional Cauchy problem on random snowflakes

We consider time-changed Brownian motions on random Koch (pre-fractal and fractal) domains where the time change is given by the inverse to a subordinator. In particular, we study the fractional Cauchy problem with Robin condition on the pre-fractal boundary obtaining asymptotic results for the corresponding fractional diffusions with Robin, Neumann and Dirichlet boundary conditions on the fractal domain.

### Nicholas Cazet UC Davis

Measure Homology

Measure homology expands on singular homology with real coefficients. Both are functors from the category of topological spaces to the category of chain complexes of real vector spaces, and the two theories agree on spaces homotopic to CW-complexes.

Measure homology uses signed measures on function spaces to give an invariant that is useful in wild (algebraic) topology. This homology theory is isometrically isomorphic to singular homology for countable CW-complexes, thus there exists a definition of simplicial volume based on signed measures.

I will discuss the measure chain complex and give recent results, including progress on understanding the topological properties influencing the zeroth homology space.

### Li Chen Louisiana State University

L^{p} Poincaré inequalities on some fractals

We study scale invariant L^{p} Poincaré inequalities on some tree-like fractals such as Vicsek set. In this setting, neither analogue of curvature nor differential structure exists, whereas the heat kernel satisfies sub-Gaussian estimate. When 1 ≤ p ≤ 2, our proof is based on recent development of the local L^{p} theory of Korevaar-Schoen-Sobolev spaces on fractals using heat kernel methods. We also discuss the range p > 2 from the viewpoint of limit approximation. This talk is based on joint work with Fabrice Baudoin.

### Nathan Cohen Fractal Antenna Systems

Diamonds in the Roughness: Fractal Hardware Applications

Mathematical and natural fractals imply the ability to exploit fractal ‘roughness’ for engineering purposes. The best known application of such ‘hardware’ use of fractals has fallen within the domain of antennas, resonators (tuned electrical circuits and filters) and metamaterials (close-spaced resonator arrays) where considerable success has been achieved in telecom and wireless. However, there are many fractal hardware applications that have an inchoate stage; have not been deeply explored in scientific publications; and have instead taken root within the patent system and tech innovation. This surprising and lengthy list will be articulated in its benefits, to encourage research efforts on these fractal innovations. Their emergence will produce a growing field of ‘fractal engineering’ in the next two decades, thereby driving further effort in the basic mathematics of fractals.

### Claire David Sorbonne Universite

Michel Lapidus UC Riverside

Fractal complex dimensions of the Weierstrass Curve

We establish a fractal tube formula for the Weierstrass curve, which gives, for small values of a strictly positive parameter ε, an explicit expression for the volume of the ε-neighborhood of the curve. For this purpose, we prove new geometric properties of the curve and of the associated function, in relation with its local Hölder and reverse Hölder continuity, with explicit estimates that had not been obtained before. We also show that the codimension 2-D_{W} is the optimal Hölder exponent for the Weierstrass function W, from which it follows that, as is well known, W is nowhere differentiable. Then the formula, that yields the expression of the ε-neighborhood, consists of a fractal power series, with underlying exponents the complex codimensions. This enables us to obtain the associated tube and distance fractal zeta functions, whose poles yield the set of complex dimensions. We prove that the nonzero complex dimensions are periodically distributed along countably many vertical lines, with the same oscillatory period. By considering the lower Minkowski content of the curve, which we prove to be strictly positive, we then show that the Weierstrass curve is Minkowski nondegenerate, as well as not Minkowski measurable, but admits a nontrivial average Minkowski content – and that, as expected, the Minkowski dimension (or box dimension) D_{W} is the complex dimension with maximal real part, and zero imaginary part.

### Caroline Davis Indiana University

Conformal Dimension for Rational Maps

There are many notions of dimension for fractals, probably the most well-known being the Hausdorff dimension. Certain fractals, such as Julia sets of rational maps and limit sets of Kleinian groups, naturally are defined up to some notion of quasi-symmetry (homeomorphisms that don’t distort infinitesimally small balls much). As such, for these maps it makes sense to consider a “conformal dimension” defined as the smallest possible Hausdorff dimension of a fractal quasi-symmetric to the original Julia set. In this talk, we’ll review some context about the machinery involved as well as recent results, before discussing some ongoing work related to bounding the conformal dimension for certain rational maps with Sierpinski carpet Julia set.

### Bernat Espigule Universitat de Barcelona

Smooth Variations on the Sierpinski Triangle, A Remarkable Family of Connected Self-similar Sets

In this talk I will show how to develop analysis on a recently discovered family of connected self-similar sets related to the Sierpinski triangle. Our toy models are the invariant sets Δ of three transformations of the plane, f_{0}(z) := z/2, f_{1}(z) := c – zc/2 and f_{2}(z) := 1 – z/2c. For each parameter c = a + ib there is a unique self-similar set Δ satisfying Δ = f_{0}(Δ) ∪ f_{1}(Δ) ∪ f_{2}(Δ). For c = 1/2 + i √ 3/2 the invariant set Δ is the well-known Sierpinski triangle. Changing the imaginary and real part of the parameter c = a + ib for values a ≤ 1/2 one gets analytic smooth variations on the Sierpinski triangle. We showcase what is going on for the exact algebraic curves bounding the space of parameters c where the self-similar set Δ is a simply-connected fractal dendrite. Finally, we provide the necessary tools to developed analysis on the critical fractals with exotic topological structure parameterized for the boundary’s piecewise-smooth curves.

### Abel Farkas Renyi Institute

Random measures on the paths of Markov processes

Starting with a deterministic measure ν we construct a random measure μ for almost every realization of a standard Markov process such that μ is carried by the path of the Markov process. Moreover, we further achieve that E(μ(A)) = ν for every Borel set A, i.e. the expectation of μ is ν. We obtain a formula for the expectation of double integration with respect to μ x μ that helps to prove finiteness of the energies of the random measure and so aids the geometric measure theory of random intersections. An interesting observation is that when setting ν to be the Lebesgue measure then the obtained random measure is the integral of a deterministic density function with respect to the occupation measure. Here the occupation measure counts the amount of time the Markov process spends inside a set and the deterministic density function can be expressed by the transition densities of the process.

### Uta Freiberg TU Chemnitz

The Einstein Relation on Metric Measure Spaces

We review the Einstein relation, which connects the Hausdorff, local walk and spectral dimensions on a space, in the abstract setting of a metric measure space equipped with a suitable operator. This requires some twists compared to the usual definitions from fractal geometry. The main result establishes the invariance of the three involved notions of fractal dimension under bi-Lipschitz continuous isomorphisms between mm-spaces and explains, more generally, how the transport of the analytic and stochastic structure behind the Einstein relation works. While any homeomorphism suffices for this transport of structure, non-Lipschitz maps distort the Hausdorff and the local walk dimension in different ways. To illustrate this, we take a look at Hölder regular transformations and how they influence the local walk dimension and prove some partial results concerning the Einstein relation on graphs of fractional Brownian motions. We conclude by giving a short list of further questions that may help building a general theory of the Einstein relation. The talk is based on a joint work with Fabian Burghart, Uppsala University, Sweden.

### Chris Gartland Texas A&M

Non-$L^1$-embeddability of the Wasserstein Metric over the Diamond Fractal

In a 2007 article, Naor and Schechtman – using a linear technique previously developed by Kislyakov – proved that the 1-Wasserstein space over the planar square, W_{1}([0,1]^{2}), admits no bi-Lipschitz embedding into the Banach space L^{1}. Two maps that play important roles in their proof are the Sobolev embedding W^{1,1}([0,1]^{2}) → L^{2}([0,1]^{2}) and the Fourier transform L^{2}([0,1]^{2}) ∍ f ↦ f̂ ∊ l^{2}(ℤ^{2}). In this talk, we’ll explain how to adapt their argument to the diamond fractal D_{ω} and prove that W_{1}(D_{ω}) admits no bi-Lipschitz embedding into L^{1}. As part of the adaptation, whenever f ∊ L^{2}(D_{ω},H^{2}), we interpret f̂ as the coefficients of f with respect to an eigenbasis of the Laplacian on D_{ω}. As in the planar case, analyzing the Lipschitz behavior of the eigenbasis is essential to the proof. Based on joint work with Florent Baudier and Thomas Schlumprecht.

### Chris Hayes University of Connecticut

Resistance Scaling on 4N-Carpets

The 4N carpets are a class of infinitely ramified self-similar fractals with a large group of symmetries. For a 4N-carpet F, let {F_{n}}_{n ≥ 0} be the natural decreasing sequence of compact pre-fractal approximations with ⋂_{n}F_{n} = F. On each F_{n}, let ℰ(u,v) be the classical Dirichlet form and u_{n} be the unique harmonic function on F_{n} satisfying a certain mixed boundary value problem corresponding to assigning a constant potential between two specific subsets of the boundary. Using a method introduced by Barlow and Bass, we prove a resistance estimate of the following form: there is ρ = ρ(N) > 1 such that ℰ(u_{n}, u_{n}) ρ^{n} is bounded above and below by constants independent of n. Such estimates have implications for the existence and scaling properties of Brownian motion on F.

### Nico Heizmann TU Chemnitz

On the fluctuations of Internal DLA on the Sierpinski gasket graph

Internal diffusion limited aggregation (IDLA) is a random aggregation model on a graph G, whose clusters are formed by random walks started in the origin (some fixed vertex) and stopped upon visiting a previously unvisited site. On the Sierpinski gasket graph the asymptotic shape is known to be a ball in the usual graph metric. In this paper we establish bounds for the fluctuations of the cluster from its asymptotic shape.

### Kornelia Hera University of Chicago

Fubini-type theorems for Hausdorff dimension and their connection to unions of lines

It is well known that for Hausdorff dimension the naive Fubini theorem does not hold. Namely, there exist sets E in ℝ^{n} such that for all x in ℝ, the vertical section of E corresponding to x has Hausdorff dimension s, while the Hausdorff dimension of E is strictly greater than s+1. We prove weaker variants of the Fubini theorem for Hausdorff dimension such as: for any Borel set B there is a small subset G of B (in an appropriate sense) such that for B \ G the naive Fubini theorem holds, that is, the Hausdorff dimension of B \ G is equal to s + 1, where s is the essential supremum of the Hausdorff dimension of the vertical sections of B \ G, assuming that the projection of B to ℝ is 1-dimensional. Fubini-type results in a much more general setting are also shown. Among applications of our results are Fubini-type results for unions of lines. In fact we show that for small unions of lines, we do not even need to remove a subset for the naive Fubini theorem to be valid. Connections to the Kakeya problem are also investigated. Joint work with Tamás Keleti and András Máthé.

### David Hewett University College London

Singular integral equations on fractals

In this talk I will outline recent research into the analysis and numerical solution of singular integral equations on fractal domains. We will discuss questions of well-posedness, solution regularity, and error analysis of numerical approximations. We will also mention applications to acoustic wave scattering by fractal screens. This is joint work with Simon Chandler-Wilde (Reading), António Caetano (Aveiro), Andrew Gibbs (UCL) and Andrea Moiola (Pavia).

### Will Hoffer University of California, Riverside

Tube Formulae for Generalized von Koch Fractals

In this work, we discuss a new method of deducing the tube formula for the standard von Koch snowflake fractal and extend our method to produce novel tube formulae for generalized von Koch fractals constructed using specified scaling ratios and regular polygons. Namely, we create and study an approximate functional equation satisfied by the volume of an epsilon-neighborhood, and we explicitly analyzed the remainder term which is a piecewise function that changes behavior according to an underlying Cantor set.

### Palle Jorgensen University of Iowa

Fourier expansions for classes of fractals

*(online)* Starting with classical Fourier analysis, the talk focuses on classes of fractals, and their wider role in harmonic analysis.

We begin with a construction by the author and S. Pedersen of explicit orthogonal Fourier expansions, and fractals in the large, for certain affine fractals, as well as early work by Strichartz. We will cover several new directions, each one dealing with new aspect of the wider subject, including (among others) joint work with Dorin Dutkay, and with Eric Weber and john Herr (via infinite-dimensional Kaczmarz algorithms.) In our work with Weber and Herr, the orthogonality constraint for the Fourier expansions is relaxed. This is accomplished with a new infinite-dimensional Kaczmarz algorithm.

We stress how Fourier expansions for selfsimilar fractals contrast to their classical counterparts. By now the general theme of Fourier series, and harmonic analysis, on Fractals has taken off in a number of diverse directions; e.g., (i) wavelets on fractals, or frames; (ii) non-commutative analysis on graph limits, (iii) discrete approximations; to mention only three. A general question is: What kind of fractals admit what kind of Fourier series?

### Naotaka Kajino Kyoto University

On singularity of energy measures for symmetric diffusions with full off-diagonal heat kernel estimates

This talk will present the result of a joint work with Mathav Murugan (University of British Columbia) that, for a strongly local regular symmetric Dirichlet space equipped with a geodesic metric, two-sided sub-Gaussian heat kernel bounds imply the singularity of the energy measures with respect to the reference measure.

For self-similar (scale-invariant) Dirichlet forms on self-similar sets, the singularity of the energy measures is known to hold in many cases by Kusuoka (1989, 1993), Ben-Bassat, Strichartz and Teplyaev (1999), Hino (2005), and Hino and Nakahara (2006), but these results heavily relied on the self-similarity of the space.

It was conjectured, and had remained open for the last two decades to prove, that the singularity of energy measures should follow, without assuming the self-similarity, just from two-sided sub-Gaussian heat kernel bounds of the same form as that for diffusions on typical

self-similar fractals. The main result of this talk answers this conjecture affirmatively.

The above result in fact does not cover the case where the space-time scaling relation in the heat kernel bounds deviates only very slightly from being Gaussian, and a natural conjecture seems to be that the singularity of the energy measures holds even in this case. If time

permits, a class of examples of fractals supporting this conjecture will be presented at the end of the talk.

### Kamil Kaleta Wrocław University of Science and Technology

Discrete-time and space Feynman–Kac operators with confining potentials

The talk will be devoted to a certain discrete-time counterpart of the classical Feynman–Kac semigroup with a confining potential in countable infinite spaces. For a class of long range Markov chains which satisfy the direct step property we prove sharp estimates for functions which are (sub-)harmonic in infinite sets with respect to the discrete Feynman–Kac operators. These results are compared with respective estimates for the case of a nearest-neighbour random walk which evolves on a graph of finite geometry. We also discuss applications to the decay rates of solutions to equations involving non-local discrete Schrödinger operators, including those based on fractional powers of the nearest-neighbour Laplacians and related quasi-relativistic operators. Finally, we analyse various classes of Markov chains which enjoy the direct step property. This is a joint work with W. Cygan and M. Sliwinski (Wroclaw).

### Jun Kigami Kyoto University

Conductive homogeneity of compact metric spaces

Classically, p-energy of a smooth function f on the real line is the integral of |∇f|^{p}. In this talk, we are going to explain how and when p-energy can be constructed on compact metric spaces. In particular, for p = 2, we have a non-trivial local regular Dirichlet form.

### Istvan Kolossváry University of St Andrews

On the Convergence Rate of the Chaos Game

This talk will address the question: how long does it take for the orbit of the chaos game to reach a certain density inside the attractor of a strictly contracting IFS of which we only assume that its lower dimension is positive? We show that the rate of growth of this cover time is determined by the Minkowski dimension of the push-forward of the shift invariant measure with exponential decay of correlations driving the chaos game. As an application, for Bedford-McMullen carpets, we completely characterize the family of probability vectors that minimize the Minkowski dimension of Bernoulli measures. Based on joint work with B. Bárány and N. Jurga.

### Maria Rosaria Lancia Universita’ di Roma Sapienza

Non autonomous BVPs with dynamical boundary conditions in irregular domains

Some results on Venttsel problems, possibly non autonomous, in irregular domains of fractal type are presented. Open problems will be discussed.

### Therese Landry Mathematical Sciences Research Institute

Convergence of Spectral Triples on the Sierpinski Gasket and other Fractal Curves

Many important physical processes can be described by differential equations. The solutions of such equations are often formulated in terms of operators on smooth manifolds. A natural question is to determine whether differential structures defined on fractals can be realized as a metric limit of differential structures on their approximating finite graphs. One of the fundamental tools of noncommutative geometry is Connes’ spectral triple. Because spectral triples generalize differential structure, they open up promising avenues for extending analytic methods from mathematical physics to fractal spaces. The Gromov-Hausdorff distance is an important tool of Riemannian geometry, and building on the earlier work of Rieffel, Latremoliere introduced a generalization of the Gromov-Hausdorff distance that was recently extended to spectral triples. The Sierpinski gasket can be viewed as a piecewise C^{1}-fractal curve, which is a class of fractals first formulated by Lapidus and Sarhad for their work on spectral triples that recover the geodesic distance on these spaces. In this talk, we will motivate and describe how their framework was adapted to our setting to yield approximation sequences suitable for metric approximation of spectral triples on piecewise C^{1}-fractal curves.

### Irfan Mahmood Mahmood University of the Punjab, Lahore

Darboux transformation and exact solitonic solutions of integrable coupled nonlinear wave equation

This talk encloses the Darboux solutions of integrable coupled nonlinear wave equation which is associated to Hirota Satsuma system with generalization of these solutions to N fold Darboux transformation in determinantal form as the ratio of Wronskians in framework of Darboux method. Here we also derive its exact solitonic solutions as one soliton solution and two soliton solution with the help of its multi-fold Darboux transformations in background of zero seed solution. This talk also includes the derivation of its equivalent zero curvature representation through its existed Lax pair which possesses traceless matrices may be assumed to be fit in AKNS scheme as it usually involves the parametric traceless matrices of order N containing field variables.

### Giorgio Mantica Universita’ degli studi dell’ Insubria

Absolute continuity vs singularity of invariant measures of IFS (overlapping and/or with infinitely many maps)

The characterisation of the invariant measure of an IFS as absolutely continuous or singular wrt Lebesgue is still elusive in the important class of overlapping IFS with finitely many maps, where only measure one results are known in specific cases, and of IFS with uncountably many maps. I will present rigorous and numerical results concerning these cases.

### Max Menzel Technical University of Chemnitz

Generalised measure geometric Krein-Feller operators as generators of fractal transformed doubly reflected Brownian motion and gap diffusions

Our aim is to calculate the generator of a semigroup associated to a so called fractal transformed doubly reflected Brownian motion. We start by reflecting a Brownian motion at 0 and 1 such that the corresponding state space is the unit interval. Understanding [0,1] as the attractor of an iterated function system we then apply the theory of fractal transformations to construct a strong Markov process such that its state space coincides with the Cantor set. After introducing the notion of a derivative w.r.t. a compactly supported non-atomic Borel measure developped by U. Freiberg we then show by an application of a generalised Taylor formula that the generator of a certain conjugated semigroup associated to the fractal transformed doubly reflected Brownian motion is essentially given by the second order derivative w.r.t. the invariant measure under natural weights on the Cantor set. Furthermore we investigate the connection of this generator to generators of gap diffusions with particular speed measure and scale that are constructed by time and space transformations of a Brownian motion.

### Gamal Mograby University of Maryland

Gaps labeling theorem for diamond self-similar graphs

Gap-labeling theorems are fascinating due to their interrelationships to distinct branches like solid-state physics, spectral analysis, and K-theory. This talk summarizes a spectral analysis for a class of fractal-type diamond graphs and provides a gap-labeling theorem for the corresponding probabilistic graph Laplacians. In the first half of the talk, we will summarize some of our results about the spectra of these Laplacians, such that they are pure-point, characterized as the closure of the Dirichlet-Neumann eigenvalues and coincide with the Julia sets of the spectral decimation functions. These results are essential to understand the Cantor set structure of the spectrum and the corresponding spectral gaps. In the second half of the talk, we provide an explicit formula for the density of states measures and show that they fulfill a weakly self-similar identity. As an application of this identity, we explicitly compute the integrated density of states on the spectral gaps and state a Gap-labeling theorem. The novelty of our result is that it provides a transparent illustration of the interplay between graph topology and spectral gaps labeling.

### Nazek Obeidat University of Vermont

New Theories and Applications of Tempered Fractional Differential Equations

### Roberto Peirone Università di Roma Tor Vergata, Dipartimento di Matematica

Anti-attracting maps and eigenforms on fractals

C. Sabot elaborated a theory concerning existence and uniqueness of eigenforms on fractals.

Later, V. Metz generalized and improved it. In this talk, I describe a way to prove the main results of Metz using anti-attracting maps on convex sets.

That is, I use some general fixed point theorems on convex sets that generalize the classical

Brouwer Theorem.

This talk is based on a paper by me appeared last year, with more recent improvements.

### Katarzyna Pietruska-Paluba University of Warsaw

Random Schrödinger operators on nested fractals driven by jump processes.

We present recent results, obtained jointly with Hubert Balsam, Kamil Kaleta and Mariusz Olszewski, concerned with properties of random Schrödinger operators on nested fractals. The driving force will be a subordinate Brownian motion with no diffusion part, and the randomness will come from potentials of Poisson or alloy type. We prove the existence of the Integrated Density of States, and then we establish the Lifschitz tail in this case. In fractal context, previously only the Poisson potential was considered, and the class of admissible processes was restricted – for example, it excluded relativistic stable processes. Most results were proven for the Sierpiński gasket. The scope of our present work is wider: we extended the class of fractals, the class of processes, as well as the class of random potentials.

### Conrad Plaut University of Tennessee

Length Spectra when there is no Length

*(online)* The Length Spectrum (LS) of a Riemannian manifold is the set of lengths of closed geodesics, with various notions of multiplicity, and in the 1950s Huber showed that for compact Riemannian surfaces, LS and the Laplace Spectrum (LaS) determine one another. These results have been followed by a decades-long investigation of the relationship between various subsets of LS and LaS. The Covering Spectrum (CS) of Sormani-Wei is a subset of LS/2 defined for compact geodesic spaces and determined by the behavior of certain covering maps. We define for arbitrary compact metric spaces (including fractals) the Enhanced Covering Spectrum (ECS), which is new even for compact Riemannian manifolds, for which ECS contains 2CS and is contained in the Marked Length Spectrum (MLS). The latter is the set of shortest curves in their homotopy classes, all of which are closed geodesics, so MLS, hence ECS, is contained in LS for compact Riemannian manifolds. It is currently unknown whether LaS determines ECS in a compact Riemannian manifold. LaS does not determine either MLS (C. Gordon) or CS (de Smit, Gornet, Sutton), and this question now has an analog in fractals for which both ECS and LaS may be defined.

### Alex Plyukhin Saint Anselm College

Random walks and first-passage processes with fractally correlated traps

We consider the survival probability f(t) of a random walk with a constant hopping rate w on a host lattice of fractal dimension d and spectral dimension ds≤2, with spatially correlated traps. The traps form a sublattice with fractal dimension da<d and are characterized by the absorption rate wa which may be finite (imperfect traps) or infinite (perfect traps). Initial coordinates are chosen randomly at or within a fixed distance of a trap. For weakly absorbing traps (wa≪w), we find that f(t) can be closely approximated by a stretched exponential function over the initial stage of relaxation, with stretching exponent α=1−(d−da)/dw, where dw is the random walk dimension of the host lattice. At the end of this initial stage there occurs a crossover to power law kinetics f(t)∼t−α with the same exponent α as for the stretched exponential regime. For strong absorption wa>w, including the limit of perfect traps wa→∞, the stretched exponential regime is absent and the decay of f(t) follows, after a short transient, the aforementioned power law for all times.

### Olaf Post University of Trier

Norm resolvent concepts for spaces approximating a fractal

In this talk I will review results on norm resolvent convergence of Laplacians on fractal spaces. The approximating Laplacians act either on a sequence of discrete weighted graphs or on a sequence of shrinking manifolds. Norm resolvent convergence implies convergence of spectra and other related objects. The results are based on a joint work with Jan Simmer.

### Goran Radunovic University of Zagreb

Fractal zeta functions generated by orbits of parabolic diffeomorphisms

We study fractal zeta function (in the sense of Lapidus) generated by orbits of parabolic germs of diffeomorphisms and show that they can be meromorphically extended to the whole complex plane. We describe their set of poles (i.e. their complex dimensions) and their principal parts which can be understood as their fractal footprint. We study the fractal footprint of one orbit of a parabolic germ f and extract intrinsic information about the germ f from it, in particular, its formal class. Moreover, we relate complex dimensions to the generalized asymptotic expansion of the tube function of orbits with oscillatory”coefficients”as well as to the asymptotic expansion of their dynamically regularized tube function. Interestingly, parabolic orbits provide a first example of sets that have nontrivial Minkowski (or box) dimension and their tube function possesses higher order oscillatory terms, however, they do not posses non-real complex dimensions and are therefore not called fractal in the sense of Lapidus.

### Mahmoud Rawashdeh Jordan University of Science and Technology

Investigating the Convergence Analysis of the Nonlinear Fractional Volterra Integro-Differential Equation Using An Efficient Approach

The analysis of integral differential equations is very important for researchers in the science community. So, in the present work we explore the analysis using a newly approach goes by the Fractional Decomposition Method (FDM) applies to nonlinear fractional Volterra Integro-Differential equations. Then, we implement the method on four test problems such as; Fractional Volterra Integro-Differential Equations (FVIDE). We present exact and approximate solutions to fractional Volterra Integro-Differential equations. The fractional Caputo derivative is in discussion throughout this work. The Mathematica software was used extensively in some of the calculation terms and to sketch the graphs that related to the test problems.

### Anna Rozanova-Pierrat CentraleSupélec, Université Paris-Saclay

Fractal shape optimization with applications to linear acoustic

*(online)* We introduce new parametrized classes of shape admissible domains in ℝ^{n} and prove that they are compact with respect to the convergence in the sense of characteristic functions, the Hausdorff sense, the sense of compacts, and the weak convergence of their boundary volumes. The domains in these classes are bounded (ε,∞)-domains with possibly fractal boundaries that can have parts of any nonuniform Hausdorff dimension greater than or equal to n-1 and less than n. We prove the existence of optimal shapes in such classes for maximum energy dissipation in the framework of linear acoustics. A by-product of our proof is the result that the class of uniform domains with a fixed parameter ε is stable under Hausdorff convergence. An additional and related result is the Mosco convergence of Robin-type energy functionals on converging domains.

### Tony Samuel University of Birmingham

On numbers badly approximable by rationals in non-integer bases

There is a wealth of number systems, some of which are more classical than others. These number systems give different representations of real numbers, for example binary, non-integer base and continued fraction. The associated theory has led to a number of solutions to practical problems, for instance in signal processing, coding theory and modeling of quasicrystals. In this talk we will discuss various aspects of non-integer base expansions, and highlight some connections to the set of badly approximable numbers, concluding with a number of open questions.

### Ecaterina Sava-Huss University of Innsbruck, Austria

Sandpiles on the gasket: many questions and few answers

*(online)* Sandpiles are models of self-organized criticality in which, on a given state space, a configuration of particles is considered. If the number of particles at a site exceeds a given threshold, then the pile is unstable and topples by sending particles to neighboring sites, which in turn may become unstable as well. We proceed until there are no more unstable sites, and we are interested in the limit shape of the set of sites that have toppled in the process of stabilization. We investigate (divisible and abelian) sandpiles on the Sierpinski gasket by giving limit shape results, and conclude with several open questions and conjectures. The talk is based on works in collaboration with Joe Chen, Wilfried Huss, and Alexander Teplyaev.

### Nageswari Shanmugalingam University of Cincinnati

Besov functions on doubling metric spaces.

We will discuss a way of seeing Besov functions on doubling metric spaces as traces of Sobolev functions.

### Alexei Stepanenko Cardiff University

Computing eigenvalues of the Laplacian on rough domains

We prove a general Mosco convergence theorem for bounded Euclidean domains satisfying a set of mild geometric hypotheses. For bounded domains, this notion implies norm-resolvent convergence for the Dirichlet Laplacian which in turn ensures spectral convergence. A key element of the proof is the development of a novel, explicit Poincaré-type inequality. These results allow us to construct a universal algorithm capable of computing the eigenvalues of the Dirichlet Laplacian on a wide class of rough domains. Many domains with fractal boundaries, such as the Koch snowflake and certain filled Julia sets, are included among this class. Conversely, we construct a counter example showing that there does not exist a universal algorithm of the same type capable of computing the eigenvalues of the Dirichlet Laplacian on an arbitrary bounded domain.

### Kohei Suzuki Bielefeld University

BV functions and sets of finite perimeter on the configuration space.

*(online)* The configuration space U(X) over X is the set of all locally finite

point measures on X. The space U(X) describes various mathematical

objects such as infinitely many interacting particle systems, random

graphs/networks and hyperplane arrangement.

In this talk, we explore the development of geometric measure theory

regarding BV functions on U(R^n) equipped with the Poisson measure \pi.

As the space U(R^n) does not satisfy the volume doubling property nor

the Poincaré inequalities, the general theory on PI spaces does not

apply.

We construct the codimension m Poisson measure, which is formally

written as “(\infty-m)-Poisson measure” on U(R^n) as the space U(R^n) is

infinite-dimensional. We introduce three different definitions of BV

functions on U(R^n), and identify them. We introduce the notions of

perimeter measures and reduced boundaries, and show the Gauss-Green

formula and the de Giorgi identity, the latter of which states that the

perimeter measure of Caccioppoli sets can be expressed by the

codimension 1 Poisson measure restricted on the reduced boundary.

If time permits, we discuss applications to stochastic analysis of

infinite particle systems with boundaries. This talk is based on the

joint work with Elia Bruè (Institute for Advanced Study, Princeton).

### Alejandro Velez-Santiago University of Puerto Rico – Mayaguez

Fine regularity for anisotropic Robin problems with nonstandard growth structure over irregular domains

We will discuss the solvability and global regularity theory for a class of anisotropic elliptic Robin problems involving variable exponents over an extensive class of irregular regions

### Rodrigo Trevino University of Maryland

Quantitative weak mixing for tilings

*(online)* The concept of quantitative weak mixing emerged from a series of works by Bufetov and Solomyak in different dynamical settings. Qualitative weak mixing refers to lower bounds on the dimension of spectral measures associated with a dynamical system. In my talk I will consider the setting of tilings and survey some recent results, including some of my own and some in joint work with Boris Solomyak.

### Yimin Xiao Michigan State University

Fractal and Regularity Properties of Gaussian Random Fields

### Martina Zähle Friedrich Schiller University Jena, Institute of Mathematics

Mean Minkowski content of V-variable random fractals

*(online)* For general V-variable random fractals satisfying the Uniform Strong Open Set condition we show that – in contrast to the a.s. variants – the (average) mean D-dimensional Minkowski content exists, where the corresponding Minkowski dimension D in the mean sense is, in general, greater than its almost sure variant. Moreover, we indicate that the (average) mean Minkowski content agrees with the (average) mean D-dimensional surface area based content and present some integral representations. The latter are structurally the same as in different former cases of random fractals, and D agrees with the dimensions obtained there.