ISQGD Distinguished Lecture Series Archive

Borel probability measure μ on $\mathbb R$ is called a spectral measure if there exists a countable set Λ ⊆ ℝ such that { e_λ(x)=e^{2π i λ x}: λ ∈ Λ } forms an orthonormal basis in L²(μ). The set Λ is called a spectrum of μ, and the pair (μ,Λ) is called a spectral pair. The study of spectral measures dates back to the famous Fuglede's spectral set conjecture. In 1998, Jorgensen and Pedersen discovered singular continuous self-similar spectral measures. An interesting phenomenon occurring in singular continuous spectral measures is the scaling spectrum: both the set Λ and its scaling tΛ for some t>0 are spectra. The scaling constant is called a spectral eigenvalue. In this talk, we will present recent results on the spectral eigenvalues for self-similar spectral measures generated by a Hadamard triple (N,B,L). The first main result is the construction of a class spectral eigenvalues for these self-similar spectral measures. The second main result shows that when #B is small relative to N, for the canonical self-similar spectral pair (μ,Λ), there are infinitely many primes p such that (μ,pΛ) is also a spectral pair. The proof is based on results concerning rational points in Cantor sets. Some of the talk will include joint with Derong Kong and Kun Li.

Polar sea ice is a multiscale composite material with complex structure on length scales ranging over many orders of magnitude. A principal challenge in sea ice modeling and computation is how to use microstructural information to find effective or homogenized behavior relevant to large-scale dynamic, thermodynamic, and ecological models. Similar challenges arise in the analysis and design of complex materials used in various scientific, engineering, industrial and medical applications. From tiny brine inclusions to ice pack dynamics on oceanic scales, and from microbes to polar bears, we will tour recent advances in multiscale modeling of sea ice, its ecosystems, and related composite media. We will encounter fractal geometry, dynamical systems, percolation, random matrix theory, Anderson localization, computational topology, anomalous diffusion, twisted bilayer graphene, and even invisibility cloaks.

I will talk about the relationship between fractals and their projections onto subspaces, in particular the relationship between their dimensions. The prototype for this was Marstrand's 1954 projection theorem which has provided impetus for an enormous amount of research which continues today.

Coded shifts are natural generalizations of sofic shifts, defined as the closure of all bi-infinite concatenations of words (generators) from a countable generating set 𝔾. In this talk, we present recent results on the computability of measures of maximal entropy (MMEs) for coded shift spaces.

Our first main result shows that if an MME assigns full measure to the set of concatenations, then it is unique, 𝔾-Bernoulli, and computable, provided that the Vere–Jones parameter of 𝔾 is computable. This yields computability of the MME for several well-known classes of shift spaces, including S-gap shifts, multiple gap shifts, and β-shifts. The same methods also establish the computability of the two ergodic MMEs of the Dyck shift.

We also examine the complementary case in which an MME assigns zero measure to the concatenation set. In this setting, MMEs are generally not unique. We show that even when the MME is unique and the Vere–Jones parameter is computable, the MME need not be computable in general. This is joint work with Tamara Kucherenko and Marco López.

Let $I$ be a compact interval and $V\supset I$ be an open interval. Assume that the elements of the finite list $\mathcal{F}=\{f_1, \dots, f_m\}$ of $C^1$ self mappings of $V$ also satisfy that $f_i(I) \subset I$ and there exists $0<c_1<c_2<1$ such that for all $1\leq i\leq m$ we have $c_1<|f'_i(x)|<c_2$, $x\in V$. Then we say that $\mathcal{F}$ is a hyperbolic IFS on $I$.

All hyperbolic IFSs have an attractor $\Lambda$, which is the unique non-empty compact set satisfying $\Lambda=\bigcup_{i=1}^{m} f_i(\Lambda)$. If the cylinders $\{f_i(\Lambda)\}_{i=1}^m$ are well separated, the so-called Open Set Condition (OSC) holds, then it is well known how to compute the Hausdorff dimension of $\Lambda$. If the OSC does not hold then we have only partial results. In my talk, I will review classical results and mention some new developments.

Given a finite collection of points and a finite collection of lines in the plane, the theorem of Szemeredi and Trotter gives a sharp upper bound for the number of incidences between the points and lines. What if we instead consider infinite collections of points and lines and consider some form of fractal dimension(s)? Is it possible to gain similarly shaped looking bounds? In this talk we survey the rapidly developing literature on this class of problems. Some of the talk will be based on joint work with Ciprian Demeter.

We begin by observing patterns, motion, fractals, and a variety of geometric structures in a series of chemical experiments ranging from plant pigments under the influence of atmospheric pollutants to microtornados. A common theme of these experiments is nucleation from the vapor or liquid phase of solid nanoparticles, followed by their subsequent growth. Applications of size-controlled, uniform nanoparticles are vast. Yet the chemical mechanisms governing particle formation—and how to control particle size distributions—have long remained an unsolved problem in materials chemistry.

We introduce Bayesian-enabled mechanism-enabled population balance modeling (BE-MEPBM), which allows us to determine previously inaccessible chemical mechanisms directly from measured particle size distributions. Mathematically, BE-MEPBM is based on coupled integro–partial differential equations describing nucleation, growth, and transport, and can be viewed as an infinite-dimensional dynamical system evolving in time and, in many cases, space. Determining these parameters from experimental data leads naturally to a PDE-constrained inverse problem.

By embedding the model within a Bayesian framework, we obtain uncertainty-aware reconstructions of kinetic mechanisms, posterior distributions over parameters, and quantitative assessments of identifiability and predictive reliability. Many of these experiments originated in discussions on developing engaging activities for applied mathematics courses, where chemical pattern formation serves as a concrete setting for dynamical systems, inverse problems, and uncertainty quantification. We close with reflections on this pedagogical endeavor and its role in connecting mathematical structure to visually compelling physical phenomena.

Dimension theory is an important sub-field of fractal geometry where one attempts to classify and describe fractal objects by assigning to them a “dimension,” that is, a quantity which reflects some geometric or analytic features of the object. It turns out there are many interesting and equally natural ways to define the “dimension” of a fractal, and one of the joys of the subject is in understanding how these different notions connect.

Dimension interpolation is a relatively new programme where one tries to define “dimension functions” which live in-between classical notions of dimension, with the broad aim of gleaning more nuanced information about the object and a more holistic view of the field in general. I will survey some recent developments in this area.

In dynamical systems theory, extreme events have been extensively studied and related to the statistical and asymptotical behavior of waiting (or hitting) times to small targets. In this talk, I will review some recent and less recent results in this direction, showing how power-law behavior of waiting times is linked to the local (fractal) dimension of the invariant measure, both in the autonomous and in the non-autonomous setting.

Time permitting, we will move to more refined asymptotic estimates and also discuss the limiting distribution of the waiting time for extreme events. In particular, I will present recent results in this direction for stochastic differential equations.

In a recent article, Rapaport showed that there is no dimension drop for exponentially separated analytic IFSs on the real line. We show that the set of exponentially separated IFSs in the space of analytic IFSs contains an open and dense set in the C2 topology. Moreover, we provide sufficient conditions for the IFS to be exponentially separated, thereby allowing us to construct explicit examples that are exponentially separated. The key technical tool is the introduction of the dual IFS, which we believe has significant interest in its own right. As an application, we also characterise when an analytic IFS can be conjugated to a self-similar IFS. This is joint work with István Kolossváry and Sascha Troscheit.

In recent years, quaternionic methodologies have found their way into many applications, one of which is digital signal processing. The main idea is to use the multidimensionality of the quaternions to model signals with multiple channels or images with multiple color values, and to use the underlying algebraic structure of quaternions to operate on these signals. On the other hand, fractal interpolation has been employed successfully in numerous applied settings over the last decades. This talk focuses on fractal interpolation in the quaternionic setting. Properties and main results are presented.

We present new results at the crossroads of spectral theory for operators in Hilbert space, optimization, large sparse systems, and the geometry of fractals arising from iterated function systems (IFS), together with new developments in fractal harmonic analysis. In particular, we introduce a new recursive iteration scheme that takes as input a prescribed sequence of selfadjoint projections. Applications include random Kaczmarz recursions, their limits, and associated error estimates.