Past

TBA.

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)

Las Vergnas and Meyniel conjectured in 1981 that all the $t$-colorings of a $K_t$-minor free graph are Kempe equivalent. This conjecture can be seen as a reconfiguration counterpoint to Hadwiger's conjecture, although it neither implies it or is implied by it. We prove that for all positive $\epsilon$, for all large enough $t$, there exists a graph with no $K_{(2/3 + \epsilon)t}$ minor whose $t$-colorings are not all Kempe equivalent, thereby strongly disproving this conjecture, along with two other conjectures of the same paper.

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 aim of this talk is to present some recent developments of Geometric Measure Theory on non smooth spaces with lower Ricci Curvature bounds, mainly related to the first and second variation formula for the area, and their applications to the isoperimetric problem on non compact manifolds. The reinterpretation of some classical results in Geometric Analysis in a low regularity setting, combined with the compactness and stability theory for spaces with lower curvature bounds, leads to a series of new geometric inequalities for smooth, non compact Riemannian manifolds. The talk is based on joint works with Andrea Mondino, Gioacchino Antonelli, Enrico Pasqualetto and Marco Pozzetta.

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.

Distribution dependent equations (or McKean—Vlasov equations) have found many applications to problems in physics, biology, economics, finance and computer science. Historically, equations with either Brownian noise or zero noise have received the most attention; many well known results can be found in the monographs by A. Sznitman and F. Golse. More recently, attention has been paid to distribution dependent equations driven by random continuous noise, in particular the recent works by M. Coghi, J-D. Deuschel, P. Friz & M. Maurelli, with applications to battery modelling. Furthermore, the phenomenon of regularisation by noise has received new attention following the works of D. Davie and M. Gubinelli & R. Catellier using techniques of averaging along rough trajectories. Building on these ideas I will present recent joint work with L. Galeati and F. Harang concerning well-posedness and stability results for distribution dependent equations driven first by merely continuous noise and secondly driven by fractional Brownian motion.

Let X be a closed Riemannian manifold, and represent the algebra C(X) of continuous functions on X on the Hilbert space L^2(X) by multiplication.  Inspired by the heat kernel proof of the Atiyah-Singer index theorem, I'll explain how to describe K-homology (i.e. the dual theory to Atiyah-Hirzebruch K-theory) in terms of parametrized families of operators on L^2(X) that get more and more 'local' in X as time tends to infinity.

I'll then switch perspectives from C(X) -- the prototypical example of a commutative C*-algebra -- to noncommutative C*-algebras coming from discrete groups, and explain how the underlying large-scale geometry of the groups can give rise to approximate 'decompositions' of the C*-algebras.  I'll then explain how to use these decompositions and localization in the sense above to compute K-homology, and the connection to some conjectures in topology, geometry, and C*-algebra theory.

Donaldson-Thomas (DT) theory is an enumerative theory which produces a virtual count of stable coherent sheaves on a Calabi-Yau 3-fold. Motivic Donaldson-Thomas theory, originally introduced by Kontsevich-Soibelman, is a categorification of the DT theory. This categorification contains more refined information of the moduli space. In this talk, I will explain the role of d-critical locus structure in the definition of motivic DT invariant, following the definition by Bussi-Joyce-Meinhardt. I will also discuss results on this structure on the Hilbert schemes of zero dimensional subschemes on local toric Calabi-Yau threefolds. This is based on joint works with Sheldon Katz. The results have substantial overlap with recent work by Ricolfi-Savvas, but techniques used here are different. 

 I will present a scenario where the early universe is in a topological phase of gravity.  I will discuss a number of analogies which motivate considering gravity in such a phase. Cosmological puzzles such as the horizon problem provide a phenomenological connection to this phase and can be explained in terms of its topological nature. To obtain phenomenological estimates, a concrete realization of this scenario using Witten's four dimensional topological gravity will be used. In this model, the CMB power spectrum can be estimated by certain conformal anomaly coefficients. A qualitative prediction of this phase is the absence of tensor modes in cosmological fluctuations.

In this talk we discuss the set of generalized symmetries associated with the free graviton theory in four dimensions. These are generated by ring-like operators. As for the Maxwell field, we find a set of “electric” and a dual set of “magnetic” topological operators and compute their algebra. The associated electric and magnetic fields satisfy a set of constraints equivalent to the ones of a stress tensor of a 3d CFT. This implies that the generalized symmetry is charged under space-time symmetries, and it provides a bridge between linearized gravity and the tensor gauge theories that have been introduced recently in the context of fractonic systems in condensed matter physics.

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.

Q-systems were introduced by Longo to describe the canonical endomorphism of a finite Jones-index inclusion of infinite von Neumann factors. From our viewpoint, a Q-system is a unitary  version of a Frobenius algebra object in a tensor category or a C* 2-category. Following work of Douglass-Reutter, a Q-system is also a unitary version of a higher idempotent, and we will describe a higher unitary idempotent completion for C* 2-categories called Q-system completion. 

We will focus on the C* 2-category C*Alg with objects unital C*-algebras, 1-morphisms right Hilbert C*-correspondences, and 2-morphisms adjointable intertwiners. By adapting a subfactor reconstruction technique called realization, and using the graphical calculus available for C* 2-categories, we will show that C*Alg is Q-system complete.

This result allows for the straightforward adaptation of subfactor results to C*-algebras, characterizing finite Watatani-index extensions of unital C*-algebras equipped with a faithful conditional expectation in terms of the Q-systems in C*Alg. Q-system completion can also be used to induce new symmetries of C*-algebras from old. 

This is joint work with Quan Chen, Corey Jones and Dave Penneys (arXiv: 2105.12010).

Topological data analysis (TDA) is a field with tools to quantify the shape of data in a manner that is concise and robust using concepts from algebraic topology. Persistent homology, one of the most popular tools in TDA, has proven useful in applications to time series data, detecting shape that changes over time and quantifying features like periodicity. In this talk, I will present two applications using tools from TDA to study time series data: the first using zigzag persistence, a generalization of persistent homology, to study bifurcations in dynamical systems and the second, using the shape of weighted, directed networks to distinguish periodic and chaotic behavior in time series data.

CRISPR is synonymous with a transformative genome editing technology that is innovating basic and applied sciences. I will report about the use of computational approaches to clarify the molecular basis and the gene-editing function of CRISPR-Cas9 and newly discovered CRISPR systems that are emerging as powerful tools for viral detection, including the SARS-CoV-2 coronavirus. We have implemented a multiscale approach, which combines classical molecular dynamics (MD) and enhanced sampling techniques, ab-initio MD, mixed Quantum Mechanics/Molecular Mechanics (QM/MM) approaches and constant pH MD (CpH MD), as well as cryo-EM fitting tools and graph theory derived analysis methods, to reveal the mechanistic basis of nucleic acid binding, catalysis, selectivity, and allostery in CRISPR systems. Using a Gaussian accelerated MD method and the Anton-2 supercluster we determined the conformational activation of CRISPR-Cas9 and the selectivity mechanism against off-target sequences. By applying network models graph theory, we have characterized a mechanism of allosteric regulation, transferring the information of DNA binding to the catalytic sites for cleavages. This mechanism is now being probed in novel Anti-CRISPR proteins, forming multi-mega Dalton complexes with the CRISPR enzymes and used for gene regulation and control. CpH MD simulations have been combined with ab-initio MD and a mixed QM/MM approach to establish the catalytic mechanism of DNA cleavage. Finally, by using multi-microsecond MD simulations we have recently probed a mechanism of DNA-induced of activation in the Cas12a enzyme, which underlies the detection of viral genetic elements, including the SARS-CoV-2 coronavirus. Overall, our outcomes contribute to the mechanistic understanding of CRISPR-based gene-editing technologies, providing information that is critical for the development of improved gene-editing tools for biomedical applications.

Neural networks have shown great success at learning function approximators between spaces X and Y, in the setting where X is a finite dimensional Euclidean space and where Y is either a finite dimensional Euclidean space (regression) or a set of finite cardinality (classification); the neural networks learn the approximator from N data pairs {x_n, y_n}. In many problems arising in the physical and engineering sciences it is desirable to generalize this setting to learn operators between spaces of functions X and Y. The talk will overview recent work in this context.

Then the talk will focus on work aimed at addressing the problem of learning operators which define the constitutive model characterizing the macroscopic behaviour of multiscale materials arising in material modeling. Mathematically this corresponds to using machine learning to determine appropriate homogenized equations, using data generated at the microscopic scale. Applications to visco-elasticity and crystal-plasticity are given.

In the '80s, Gromov proved that sequences of Riemannian manifold with a lower bound on the Ricci curvature and an upper bound on the dimension are precompact in the measured Gromov--Hausdorff topology (mGH for short). Since then, much attention has been given to the limits of such sequences, called Ricci limit spaces. A way to study these limits is to introduce a synthetic definition of Ricci curvature lower bounds and dimension upper bounds. A synthetic definition should not rely on an underlying smooth structure and should be stable when passing to the limit in the mGH topology. In this talk, I will briefly introduce CD spaces, which are a generalization of Ricci limit spaces.

Function approximation, as a goal in itself or as an ingredient in scientific computing, typically relies on having a basis. However, in many cases of interest an obvious basis is not known or is not easily found. Even if it is, alternative representations may exist with much fewer degrees of freedom, perhaps by mimicking certain features of the solution into the “basis functions" such as known singularities or phases of oscillation. Unfortunately, such expert knowledge typically doesn’t match well with the mathematical properties of a basis: it leads instead to representations which are either incomplete or overcomplete. In turn, this makes a problem potentially unsolvable or ill-conditioned. We intend to show that overcomplete representations, in spite of inherent ill-conditioning, often work wonderfully well in numerical practice. We explore a theoretical foundation for this phenomenon, use it to devise ground rules for practitioners, and illustrate how the theory and its ramifications manifest themselves in a number of applications.

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I will discuss the regularity of solutions to quasilinear systems satisfying a Legendre-Hadamard ellipticity condition. For such systems it is known that weak solutions may which fail to be C^1 in any neighbourhood, so we cannot expect a general regularity theory. However if we assume an a-priori regularity condition of the solutions we can rule out such counterexamples. Focusing on solutions to Euler-Lagrange systems, I will present an improved regularity results for solutions whose gradient satisfies a suitable BMO / VMO condition. Ideas behind the proof will be presented in the interior case, and global consequences will also be discussed.

A well-known conjecture of Ivanov states that mapping class groups of surfaces with genus at least 3 virtually do not surject onto the integers. Putman and Wieland reformulated this conjecture in terms of higher Prym representations of finite-index subgroups of mapping class groups. We show that the Putman-Wieland conjecture holds for geometrically uniform subgroups. Along the way we construct a cover S of the genus 2 surface such that the lifts of simple closed curves do not generate the rational homology of S. This is joint work with Markovic.

This talk aims to be a rigorous introduction to Social Choice Theory, a sub-branch of Game Theory with natural applications to economics, sociology and politics that tries to understand how to determine, based on the personal opinions of all individuals, the collective opinion of society. The goal is to prove the three famous and pessimistic impossibility theorems: Arrow's theorem, Gibbard's theorem and Balinski-Young's theorem. Our blunt conclusion will be that, unfortunately, there are no ideally fair social choice systems. Is there any hope yet?