In this talk, we present our study of Brown’s definition of the probabilistic zeta function of a finite lattice, and propose a natural alternative that may be better-suited for non-atomistic lattices. The probabilistic zeta function admits a general Dirichlet series expression, which need not be ordinary. We investigate properties of the function and compute it on several examples of finite lattices, establishing connections with well-known identities. Furthermore, we investigate when the series is an ordinary Dirichlet series. Since this is the case for coset lattices, we call such lattices coset-like. In this regard, we focus on partition lattices and d-divisible partition lattices and show that they typically fail to be coset-like. We do this by using the prime number theorem, establishing a connection with number theory.

We'll sketch how the $K$-rational solutions of a system $X$ of multivariate polynomials can be viewed as the solutions of a "classical mechanics" problem on an associated affine space.

When $X$ has a suitable topology, e.g. if its $\mathbb{C}$-solutions form a Riemann surface of genus $>1$, we'll observe some of the advantages of this new point of view such as a relatively computable algorithm for effective finiteness (with some stipulations). This is joint work with Minhyong Kim.
 

How can one integrate singular functions over fractals? And why would one want to do this? In this talk I will present a general approach to numerical quadrature on the compact attractor of an iterated function system of contracting similarities, where integration is with respect to the relevant Hausdorff measure. For certain singular integrands of logarithmic or algebraic type the self-similarity of the integration domain can be exploited to express the singular integral exactly in terms of regular integrals that can be approximated using standard techniques. As an application we show how this approach, combined with a singularity-subtraction technique, can be used to accurately evaluate the singular double integrals that arise in Hausdorff-measure Galerkin boundary element methods for acoustic wave scattering by fractal screens. This is joint work with Andrew Gibbs (UCL) and Andrea Moiola (Pavia).

The placenta is the essential organ of maternal-fetal interactions, where nutrient, oxygen, and waste exchange occur. In recent studies, differences in the morphology of the placental chorionic surface vascular network (PCSVN) have been associated with developmental disorders such as autism. This suggests that the PCSVN could potentially serve as a biomarker for the early diagnosis and treatment of autism. Studying PCSVN features in large cohorts requires a reliable and automated mechanism to extract the vascular networks. In this talk, we present a method for PCSVN extraction. Our algorithm builds upon a directional multiscale mathematical framework based on a combination of shearlets and Laplacian eigenmaps and can isolate vessels with high success in high-contrast images such as those produced in CT scans. 

In this talk we will show the application of (multilayer) network science to a wide spectrum of problems related to the ongoing COVID-19 pandemic, ranging from the molecular to the societal scale. Specifically, we will discuss our recent results about how network analysis: i) has been successfully applied to virus-host protein-protein interactions to unravel the systemic nature of SARS-CoV-2 infection; ii) has been used to gain insights about the potential role of non-compliant behavior in spreading of COVID-19; iii) has been crucial to assess the infodemic risk related to the simultaneous circulation of reliable and unreliable information about COVID-19.

References:

Assessing the risks of "infodemics" in response to COVID-19 epidemics
R. Gallotti, F. Valle, N. Castaldo, P. Sacco, M. De Domenico, Nature Human Behavior 4, 1285-1293 (2020)

CovMulNet19, Integrating Proteins, Diseases, Drugs, and Symptoms: A Network Medicine Approach to COVID-19
N. Verstraete, G. Jurman, G. Bertagnolli, A. Ghavasieh, V. Pancaldi, M. De Domenico, Network and Systems Medicine 3, 130 (2020)

Multiscale statistical physics of the pan-viral interactome unravels the systemic nature of SARS-CoV-2 infections
A. Ghavasieh, S. Bontorin, O. Artime, N. Verstraete, M. De Domenico, Communications Physics 4, 83 (2021)

Individual risk perception and empirical social structures shape the dynamics of infectious disease outbreaks
V. D'Andrea, R. Gallotti, N. Castaldo, M. De Domenico, To appear in PLOS Computational Biology (2022)