A metapopulation model, composed of subpopulations and pairwise connections, is a particle-network framework for epidemic dynamics study. Individuals are well-mixed within each subpopulation and migrate from one subpopulation to another, obeying a given mobility rule. While different mobility rules in metapopulation models have been studied, few efforts have been made to compare the effects of simple (i.e., unbiased) random walks and more complex mobility rules. In this talk, we study susceptible-infectious-susceptible (SIS) dynamics in a metapopulation model, in which individuals obey a second-order parametric random-walk mobility rule called the node2vec. We transform the node2vec mobility rule to a first-order Markov chain whose state space is composed of the directed edges and then derive the epidemic threshold. We find that the epidemic threshold is larger for various networks when individuals avoid frequent backtracking or visiting a neighbor of the previously visited subpopulation than when individuals obey the simple random walk. The amount of change in the epidemic threshold induced by the node2vec mobility is generally not as significant as, but is sometimes comparable with, the one induced by the change in the diffusion rate for individuals.

arXiv links: https://arxiv.org/abs/2006.04904 and https://arxiv.org/abs/2106.08080

The moments of Hardy’s function have been of interest to number theorists since the early 20th century, and to random matrix theorists especially since the seminal work of Keating and Snaith, who were able to conjecture the leading order behaviour of all moments. Studying joint moments offers a unified approach to both moments and derivative moments. In his 2006 thesis, Hughes made a version of the Keating-Snaith conjecture for joint moments of Hardy’s function. Since then, people have been calculating the joint moments on the random matrix side. I will outline some recent progress in these calculations. This is joint work with Theo Assiotis, Benjamin Bedert, and Mustafa Alper Gunes.

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).