Virtual

The SI model is the most basic of all compartmental models used to describe the spreading of information through a population. In this talk we will present a mathematical formalism to solve the SI model on generic networks. Our methods rely on a tensor product formulation of the dynamical spreading process, inspired by many-body quantum systems. Here we will focus on time-dependent expectation values for the state of individual nodes, which can be obtained from contributions of subgraphs of the network. We show how to compute these contributions systematically and derive a set of symmetry relations among subgraphs of differing topologies. We conclude by comparing our results for small sample networks to Monte-Carlo simulations and mean-field approximations.

arXiv link: https://arxiv.org/abs/2109.03530

(This is Part II of a two-part talk.)

Forcing axioms spell out the dictum that if a statement can be forced, then it is already true. The P_max axiom (*) goes beyond that by claiming that if a statement is consistent, then it is already true. Here, the statement in question needs to come from a resticted class of statements, and "consistent" needs to mean "consistent in a strong sense". It turns out that (*) is actually equivalent to a forcing axiom, and the proof is by showing that the (strong) consistency of certain theories gives rise to a corresponding notion of forcing producing a model of that theory. Our result builds upon earlier work of R. Jensen and (ultimately) Keisler's "consistency properties".

Forcing axioms spell out the dictum that if a statement can be forced, then it is already true. The P_max axiom (*) goes beyond that by claiming that if a statement is consistent, then it is already true. Here, the statement in question needs to come from a resticted class of statements, and "consistent" needs to mean "consistent in a strong sense". It turns out that (*) is actually equivalent to a forcing axiom, and the proof is by showing that the (strong) consistency of certain theories gives rise to a corresponding notion of forcing producing a model of that theory. Our result builds upon earlier work of R. Jensen and (ultimately) Keisler's "consistency properties".

(This is Part I of a two-part talk.)

In his very first note on noncommutative differential geometry, Connes
showed that the position and momentum operators on the line could be used to
construct constant curvature connections over an irrational noncommutative

2-torus $\mathcal{A}_\theta$. When $\theta$ is a quadratic irrationality,
this yields, in particular, constant curvature connections on non-trivial
noncommutative line bundles---is there an underlying monopole on some
non-trivial noncommutative principal $U(1)$-bundle? We use this case study
to illustrate how approaches to quantum principal bundles introduced by
Brzeziński–Majid and Đurđević, respectively, can be fruitfully synthesized
to reframe classical gauge theory on quantum principal bundles in terms of
synthesis of total spaces (as noncommutative manifolds) from vertical and
horizontal geometric data.

I shall present joint work with Maxim Kontsevich describing an interesting
domain of complex metrics on a smooth manifold. It is a complexification of
the space of ordinary Riemannian metrics, and has the Lorentzian metrics
(but not metrics of other signatures) on its boundary. Use of the domain
leads to a modified axiom system for QFT which illuminates not only the
special role of Lorentz signature, but also of features such as local
commutativity, unitarity, and global hyperbolicity.

The non-compact exceptional simple group G_2* turns out to be the symmetry group of quantized twistor theory. Certain implications of this remarkable fact will be explored in this talk.

Kernel-based statistical algorithms have found wide success in statistical machine learning in the past ten years as a non-parametric, easily computable engine for reasoning with probability measures. The main idea is to use a kernel to facilitate a mapping of probability measures, the objects of interest, into well-behaved spaces where calculations can be carried out. This methodology has found wide application, for example two-sample testing, independence testing, goodness-of-fit testing, parameter inference and MCMC thinning. Most theoretical investigations and practical applications have focused on Euclidean data. This talk will outline work that adapts the kernel-based methodology to data in an arbitrary Hilbert space which then opens the door to applications for functional data, where a single data sample is a discretely observed function, for example time series or random surfaces. Such data is becoming increasingly more prominent within the statistical community and in machine learning. Emphasis shall be given to the two-sample and goodness-of-fit testing problems.

The roots of a function represented by its Chebyshev expansion are known to be the eigenvalues of the so-called colleague matrix, which is a Hessenberg matrix that is the sum of a symmetric tridiagonal matrix and a rank 1 perturbation. The rootfinding problem is thus reformulated as an eigenproblem, making the computation of the eigenvalues of such matrices a subject of significant practical interest. To obtain the roots with the maximum possible accuracy, the eigensolver used must posess a somewhat subtle form of stability.

In this talk, I will discuss a recently constructed algorithm for the diagonalization of colleague matrices, satisfying the relevant stability requirements.  The scheme has CPU time requirements proportional to n^2, with n the dimensionality of the problem; the storage requirements are proportional to n. Furthermore, the actual CPU times (and storage requirements) of the procedure are quite acceptable, making it an approach of choice even for small-scale problems. I will illustrate the performance of the algorithm with several numerical examples.

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Variational phase-field models of fracture have been at the center of a multidisciplinary effort involving a large community of mathematicians, mechanicians, engineers, and computational scientists over the last 25 years or so.

I will start with a modern interpretation of Griffith's classical criterion as a variational principle for a free discontinuity energy and will recall some of the milestones in its analysis. Then, I will introduce the phase-field approximation per se and describe its numerical implementation. I illustrate how phase-field models have led to major breakthroughs in the predictive simulation of fracture in complex situations.

I then will turn my attention to current issues, with a specific emphasis on crack nucleation in nominally brittle materials. I will recall the fundamental incompatibility between Griffith’s theory and nucleation criteria based on a stress yield surface: the strength vs. toughness paradox. I will then present several attempts at addressing this issue within the realm of phase-fracture and discuss their respective strengths and weaknesses. 

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A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.