C1

We will be discussing a Fourier-analytic approach
to optimal matching between independent samples, with
an elementary proof of the Ajtai-Komlos-Tusnady theorem.
The talk is based on a joint work with Michel Ledoux.

The notion of emergence is at the core of many of the most challenging open scientific questions, being so much a cause of wonder as a perennial source of philosophical headaches. Two classes of emergent phenomena are usually distinguished: strong emergence, which corresponds to supervenient properties with irreducible causal power; and weak emergence, which are properties generated by the lower levels in such "complicated" ways that they can only be derived by exhaustive simulation. While weak emergence is generally accepted, a large portion of the scientific community considers causal emergence to be either impossible, logically inconsistent, or scientifically irrelevant.

In this talk we present a novel, quantitative framework that assesses emergence by studying the high-order interactions of the system's dynamics. By leveraging the Integrated Information Decomposition (ΦID) framework [1], our approach distinguishes two types of emergent phenomena: downward causation, where macroscopic variables determine the future of microscopic degrees of freedom; and causal decoupling, where macroscopic variables influence other macroscopic variables without affecting their corresponding microscopic constituents. Our framework also provides practical tools that are applicable on a range of scenarios of practical interest, enabling to test -- and possibly reject -- hypotheses about emergence in a data-driven fashion. We illustrate our findings by discussing minimal examples of emergent behaviour, and present a few case studies of systems with emergent dynamics, including Conway’s Game of Life, neural population coding, and flocking models.
[1] Mediano, Pedro AM, Fernando Rosas, Robin L. Carhart-Harris, Anil K. Seth, and Adam B. Barrett. "Beyond integrated information: A taxonomy of information dynamics phenomena." arXiv preprint arXiv:1909.02297 (2019).
 

Pick's theorem is a century-old theorem in complex analysis about interpolation with bounded analytic functions. The Kadison-Singer problem was a question about states on $C^*$-algebras originating in the work of Dirac on the mathematical description of quantum mechanics. It was solved by Marcus, Spielman and Srivastava a few years ago.

I will talk about Pick's theorem, the Kadison-Singer problem and how the two can be brought together to solve interpolation problems with infinitely many nodes. This talk is based on joint work with Alexandru Aleman, John McCarthy and Stefan Richter.

There is a classical mathematical theorem (due to Banach and Tarski) that implies the following shocking statement: An orange can be divided into finitely many pieces, these pieces can be rotated and rearranged in such a way to yield two oranges of the same size as the original one. In 1929 J.~von Neumann recognizes that one of the reasons underlying the Banach-Tarski paradox is the fact that on the unit ball there is an action of a discrete subgroup of isometries that fails to have the property of amenability ("Mittelbarkeit" in German).

In this talk we will address more recent developments in relation to the dichotomy amenability vs. existence of paradoxical decompositions in different mathematical situations like, e.g., for metric spaces, for algebras and operator algebras. We will present a result unifying all these approaches in terms of a class of C*-algebras, the so-called uniform Roe algebras.

P. Ara, K. Li, F. Lledó and J. Wu, Amenability of coarse spaces and K-algebras , Bulletin of Mathematical Sciences 8 (2018) 257-306;
P. Ara, K. Li, F. Lledó and J. Wu, Amenability and uniform Roe algebras, Journal of Mathematical Analysis and Applications 459 (2018) 686-716;

Let $X$ be a countable discrete metric space, and think of operators on $\ell^2(X)$ in terms of their $X$-by-$X$ matrix. Band operators are ones whose matrix is supported on a "band" along the main diagonal; all norm-limits of these form a C*-algebra, called uniform Roe algebra of $X$. This algebra "encodes" the large-scale (a.k.a. coarse) structure of $X$. Quasi-locality, coined by John Roe in '88, is a property of an operator on $\ell^2(X)$, designed as a condition to check whether the operator belongs to the uniform Roe algebra (without producing band operators nearby). The talk is about our attempt to make this work, and an expander-ish condition on graphs that came out of trying to find a counterexample. (Joint with: A. Tikuisis, J. Zhang, K. Li and P. Nowak.)
 

Let $\mathcal{B}$ be a (unital) commutative Banach algebra and $\Omega$ the set of non-trivial multiplicative linear functionals $\omega : \mathcal{B} \to \mathbb{C}$. Gelfand theory tells us that the kernels of these functionals are exactly the maximal ideals of $\mathcal{B}$ and, as a consequence, an element $b \in \mathcal{B}$ is invertible if and only if $\omega(b) \neq 0$ for all $\omega \in \Omega$. A generalisation to non-commutative Banach algebras is the local principle of Allan and Douglas, also known as central localisation: Let $\mathcal{B}$ be a Banach algebra, $Z$ a closed subalgebra of the center of $\mathcal{B}$ and $\Omega$ the set of maximal ideals of $Z$. For every $\omega \in \Omega$ let $\mathcal{I}_{\omega}$ be the smallest ideal of $\mathcal{B}$ which contains $\omega$. Then $b \in \mathcal{B}$ is invertible if and only if $b + \mathcal{I}_{\omega}$ is invertible in $\mathcal{B} / \mathcal{I}_{\omega}$ for every $\omega \in \Omega$.

From an operator theory point of view, one of the most important features of the local principle is the application to Calkin algebras. In that case the invertible elements are called Fredholm operators and the corresponding spectrum is called the essential spectrum. Therefore, by taking suitable subalgebras, we can obtain a characterisation of Fredholm operators. Many beautiful results in spectral theory, e.g.~formulas for the essential spectrum of Toeplitz operators, can be obtained in this way. However, the central localisation is often not sufficient to provide a satisfactory characterisation for more general operators. In this talk we therefore consider a generalisation where the ideals $\mathcal{I}_{\omega}$ do not originate from the center of the algebra. More precisely, we will start with general $L^p$-spaces and apply limit operator methods to obtain a Fredholm theory that is applicable to many different settings. In particular, we will obtain characterisations of Fredholmness and compactness in many new cases and also rediscover some classical results.

This talk is based on joint work with Christian Seifert.

 Lipschitz (or H\"older) spaces $C^\delta, \, k< \delta <k+1$, $k\in\mathbb{N}_0$, are the set of functions that are more regular than the $\mathcal{C}^k$ functions and less regular than the $\mathcal{C}^{k+1}$ functions. The classical definitions of H\"older classes involve  pointwise conditions for the functions and their derivatives.  This implies that to prove   regularity results for an operator among these spaces  we need its pointwise expression.  In many cases this can be a rather involved formula, see for example the expression of $(-\Delta)^\sigma$  in (Stinga, Torrea, Regularity Theory for the fractional harmonic oscilator, J. Funct. Anal., 2011.)

In  the 60's of last century, Stein and Taibleson, characterized bounded H\"older functions via some integral estimates of the Poisson semigroup, $e^{-y\sqrt{-\Delta}},$ and of  the Gauss semigroup, $e^{\tau{\Delta}}$. These kind of semigroup descriptions allow to obtain regularity results for fractional operators in these spaces in a more direct way.

 In this talk we shall see that we can characterize H\"older spaces adapted to other differential operators $\mathcal{L}$ by means of semigroups and that these characterizations will allow us to prove the boundedness of some fractional operators, such as $\mathcal{L}^{\pm \beta}$, Riesz transforms or Bessel potentials, avoiding the long, tedious and cumbersome computations that are needed when the pointwise expressions are handled.

Random graphs were introduced as a convenient example for demonstrating the impossibility of ‘complete disorder’ by Erdos, who also thought that these objects will never become useful in the applied areas outside of pure mathematics. In this talk, I will view random graphs as objects in the field of applied mathematics and discus how the application-driven objectives have set new directions for studying random graphs. I will focus on characterising the sizes of connected components in graphs with a given degree distribution, on the percolation-like processes on such structures, and on generalisations to the coloured graphs. These theoretical questions have interesting implications for studying resilience of networks with nontrivial structures, and for materials science where they explain kinetics-driven phase transitions. Even more surprisingly, the results reveal intricate connections between random graphs and non-linear partial differential equations indicating new possibilities for their analysis.