We consider CFTs with large gap in the spectrum of operators and a large number of degrees of freedom (large central charge). We analytically study a Heavy-Heavy-Light-Light correlation function, where Heavy, refers to an operator with conformal dimension which scales like the central charge and Light, refers to an operator whose dimension is of order unity in the large central charge limit. In certain regimes, the correlation function can be examined analytically leading to very simple and suggestive expressions.

The classical Universal Approximation Theorem certifies that the universal approximation property holds for the class of neural networks of arbitrary width. Here we consider the natural `dual' theorem for width-bounded networks of arbitrary depth, for a broad class of activation functions. In particular we show that such a result holds for polynomial activation functions, making this genuinely different to the classical case. We will then discuss some natural extensions of this result, e.g. for nowhere differentiable activation functions, or for noncompact domains.
 

The hard Lefschetz theorem is a fundamental statement about the symmetry of the cohomology of algebraic varieties. In nearly all cases that we systematically understand it, it comes with a geometric meaning, often in form of Hodge structures and signature data for the Hodge-Riemann bilinear form.

Nevertheless, similar to the role the standard conjectures play in number theory, several intriguing combinatorial problems can be reduced to hard Lefschetz properties, though in extreme cases without much geometric meaning, lacking any existence of, for instance,  an ample cone to do Hodge theory with.

I will present a way to prove the hard Lefschetz theorem in such a situation, by introducing biased pairing and perturbation theory for intersection rings. The price we pay is that the underlying variety, in a precise sense, has itself to be sufficiently generic. For instance, we shall see that any quasismooth, but perhaps nonprojective toric variety can be "perturbed" to a toric variety with the same equivariant cohomology, and that has the hard Lefschetz property.

Finally, I will discuss how this applies to prove some interesting theorems in geometry, topology and combinatorics. In particular, we shall see a generalization of a classical result due to Descartes and Euler: We prove that if a simplicial complex embeds into euclidean 2d-space, the number of d-simplices in it can exceed the number of (d-1)-simplices by a factor of at most d+2.

Congruence monoids in the ring of integers are given by certain unions of arithmetic progressions. To each congruence monoid, there is a canonical way to associate a semigroup C*-algebra. I will explain this construction and then discuss joint work with Xin Li on K-theoretic invariants. I will also indicate how all of this generalizes to congruence monoids in the ring of integers of an arbitrary algebraic number field.

Many soft materials, such as elastomers and hydrogels, are made of long chain molecules crosslinked to form a three-dimensional network. Their mechanical properties depend on network parameters such as chain density, chain length distribution and the functionality of the crosslinks. Understanding the relationships between the topology of polymer networks and their mechanical properties has been a long-standing challenge in polymer physics.

In this work, we focus on so-called “near-ideal” networks, which are produced by the cross-coupling of star-like macromolecules with well-defined chain length. We developed a computational approach based on random discrete networks, according to which the polymer network is represented by an assembly of non-linear springs connected at junction points representing crosslinks. The positions of the crosslink points are determined from the conditions of mechanical equilibrium. Scaling relations for the elastic modulus and maximum extensibility of the network were obtained. Our scaling relations contradict some predictions of classical estimates of rubber elasticity and have implications for the interpretation of experimental data for near-ideal polymer networks.

Reference: G. Alame, L. Brassart. Relative contributions of chain density and topology to the elasticity of two-dimensional polymer networks. Soft Matter 15, 5703 (2019).

We aim to generate gene coexpression networks from gene expression data. In our networks, nodes represent genes and edges depict high positive correlation in their expression across different samples. Methods based on Pearson correlation are the most commonly used to generate gene coexpression networks. We propose the use of distance correlation as an effective alternative to Pearson correlation when constructing gene expression networks. Our methodology pipeline includes a thresholding step which allows us to discriminate which pairs of genes are coexpressed. We select the value of the threshold parameter by studying the stability of the generated network, rather than relying on exogenous biological information known a priori.

Identifying systemically important countries is crucial for global financial stability. In this work we use (multilayer) network methods to identify systemically important countries. We study the financial system as a multilayer network, where each layer represent a different type of financial investment between countries. To rank countries by their systemic importance, we implement MultiRank, as well a simplistic model of financial contagion. In this first model, we consider that each country has a capital buffer, given by the capital to assets ratio. After the default of an initial country, we model financial contagion with a simple rule: a solvent country defaults when the amount of assets lost, due to the default of other countries, is larger than its capital. Our results show that when we consider that there are various types of assets the ranking of systemically important countries changes. We make all our methods available by introducing a python library. Finally, we propose a more realistic model of financial contagion that merges multilayer network theory and the contingent claims sectoral balance sheet literature. The aim of this framework is to model the banking, private, and sovereign sector of each country and thus study financial contagion within the country and between countries. 

A fundamental problem in the context of Einstein's equations of general relativity is to understand precisely the dynamical evolution of small perturbations of stationary black hole solutions. It is expected that there is a discrete set of characteristic frequencies that play a dominant role at late time intervals and carry information about the nature of the black hole, much like the normal frequencies of a vibrating string. These frequencies are called quasi-normal frequencies or resonances and they are closely related to scattering resonances in the study of Schrödinger-type equations. I will discuss a new method of defining and studying resonances for linear wave equations on asymptotically flat black holes, developed from joint work with Claude Warnick.