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)

In the first part of the seminar I will introduce shallow neural-networks from a statistical-mechanics perspective, focusing on simple cases and on a naive scenario where information to be learnt is structureless. Then, inspired by biological information processing, I will enrich this framework by accounting for structured datasets and by making the network able to perform challenging tasks like generalization or even "taking a nap”. Results presented are both analytical and numerical.

Identifying novel drug-target interactions (DTI) is a critical and rate limiting step in drug discovery. While deep learning models have been proposed to accelerate the identification process, we show that state-of-the-art models fail to generalize to novel (i.e., never-before-seen) structures. We first unveil the mechanisms responsible for this shortcoming, demonstrating how models rely on shortcuts that leverage the topology of the protein-ligand bipartite network, rather than learning the node features. Then, we introduce AI-Bind, a pipeline that combines network-based sampling strategies with unsupervised pre-training, allowing us to limit the annotation imbalance and improve binding predictions for novel proteins and ligands. We illustrate the value of AI-Bind by predicting drugs and natural compounds with binding affinity to SARS-CoV-2 viral proteins and the associated human proteins. We also validate these predictions via auto-docking simulations and comparison with recent experimental evidence. Overall, AI-Bind offers a powerful high-throughput approach to identify drug-target combinations, with the potential of becoming a powerful tool in drug discovery.

arXiv link: https://arxiv.org/abs/2112.13168

TBA.

This event will be hybrid and will take place in L1 and on Teams. A link will be available 30 minutes before the session begins.

Pascal Heid
Title: Adaptive iterative linearised Galerkin methods for nonlinear PDEs

Abstract: A wide variety of iterative methods for the solution of nonlinear equations exist. In many cases, such schemes can be interpreted as iterative local linearisation methods, which can be obtained by applying a suitable linear preconditioning operator to the original nonlinear equation. Based on this observation, we will derive an abstract linearisation framework which recovers some prominent iteration schemes. Subsequently, in order to cast this unified iteration procedure into a computational scheme, we will consider the discretisation by means of finite dimensional subspaces. We may then obtain an effective numerical algorithm by an instantaneous interplay of the iterative linearisation and an (optimally convergent) adaptive discretisation method. This will be demonstrated by a numerical experiment for a quasilinear elliptic PDE in divergence form.   

Ilyas Khan
Title: Geometric Analysis: Curvature and Applications

Abstract: Often, one will want to find a geometric structure on some given manifold satisfying certain properties. For example, one might want to find a minimal embedding of one manifold into another, or a metric on a manifold with constant scalar curvature, to name some well known examples of this sort of problem. In general, these problems can be seen as equivalent to solving a system of PDEs: differential relations on coordinate patches that can be assembled compatibly over the whole manifold to give a globally defined geometric equation.

In this talk, we will present the theories of minimal surfaces and mean curvature flow as representative examples of the techniques and philosophy that geometric analysis employs to solve problems in geometry of the aforementioned type. The description of the theory will be accompanied by a number of examples and applications to other fields, including physics, topology, and dynamics.