A well-known problem in mathematics and physics consists in understanding how the geometry (or shape) of a musical instrument affects it sound. This gives rise to two related types of mathematical problems: direct spectral problems (how the shape of a drum affects its sound) and inverse spectral problems (how one can recover the shape of a drum from its sound). Here, we will consider both types of problems in the context of drums with fractal (that is, very rough) boundary. We will show, in particular, that one can “hear” the fractal dimension of the boundary (a certain measure of its roughness) and in certain cases, a fractal analog of its length. In the special case of vibrating fractal strings (the one‐dimensional situation), we will show that the corresponding inverse spectral problem is intimately connected with the Riemann Hypothesis, which is arguably the most famous open problem in mathematics and whose solution will likely unlock deep secrets about the prime numbers. In conclusion, we will briefly explain how this work eventually gave rise to a mathematical theory of complex fractal dimensions (developed by the author and his collaborators), which captures the vibrations that are intrinsic to both fractal geometries and the prime numbers.

Social and political turbulence in the United States and a number of European countries has been rising in recent years. My research, which combines analysis of historical data with the tools of complexity science, has identified the deep structural forces that work to undermine societal stability and resilience to internal and external shocks. Here I look beneath the surface of day-to-day contentious politics and social unrest, and focus on the negative social and economic trends that explain our current “Age of Discord.” I also discuss the current research by my group aiming to translate these insights from historical analysis into a computational model that would allow us to construct a series of probabilistic scenarios of social breakdown and recovery.

In this talk I will present my take on one line of research in
Computability Theory that studies logical properties of degree structures.
There are different ways in which we can compare the algorithmic complexity
between countable mathematical objects. A set A of natural numbers is
Turing reducible to a set B if there is an algorithm to determine
membership in A using B as data. A set A is enumeration reducible to B if
there is an algorithm to list the members of A using any listing of B as
data. Each reducibility gives rise to a degree structure: a partial order
in which we identify sets that have equal algorithmic complexity. The
Turing degrees and the enumeration degrees are closely related: the
Turing degrees have an isomorphic copy inside the enumeration degrees,
which we call the total degrees. We study these structures from three
interrelated aspects: how complicated is the set of true algebraic
statements is each partial order; what relations have structurally
definable presentations; what does the automorphism group for each partial
order look like?

Consider the subgraph sampling model, where we observe a random subgraph of a given (possibly non random) large graph $G_n$, by choosing vertices of $G_n$ independently at random with probability $p_n$. In this setting, we study the question of estimating the number of copies $N(H,G_n)$ of a fixed motif/small graph (think of $H$ as edges, two stars, triangles) in the big graph $G_n$. We derive necessary and sufficient conditions for the consistency and the asymptotic normality of a natural Horvitz-Thompson (HT) type estimator.



As it turns out, the asymptotic normality of the HT estimator exhibits an interesting fourth-moment phenomenon, which asserts that the HT estimator (appropriately centered and rescaled) converges in distribution to the standard normal whenever its fourth-moment converges to 3. We apply our results to several natural graph ensembles, such as sparse graphs with bounded degree, Erdős-Renyi random graphs, random regular graphs, and dense graphons.

Solvable lattice models are interesting objects that
first arose in statistical mechanics, where they can
model physical systems such as 2-dimensional ice.
"Solvability" is a special feature that allows the
models to be studied by a powerful tool, the Yang-Baxter
equation. Such models have also come up in different
contexts, such as quantum field theory, integrable
probability, and the representation theory of p-adic
groups. We will focus on the latter connection and
discuss recent work of Brubaker, Buciumas, Bump and
Gustafsson relating p-adic Whittaker functions to
new "supersymmetric" lattice models, just as similar
models appeared almost simultaneously in work of
Aggarwal, Borodin and Wheeler with motivations in
probability.

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