Leiden University

Many networks exhibit some community structure. There exists a wide variety of approaches to detect communities in networks, each offering different interpretations and associated algorithms. For large networks, there is the additional requirement of speed. In this context, the so-called label propagation algorithm (LPA) was proposed, which runs in near linear time. In partitions uncovered by LPA, each node is ensured to have most links to its assigned community. We here propose a fast variant of LPA (FLPA) that is based on processing a queue of nodes whose neighbourhood recently changed. We test FLPA exhaustively on benchmark networks and empirical networks, finding that it runs up to 700 times faster than LPA. In partitions found by FLPA, we prove that each node is again guaranteed to have most links to its assigned community. Our results show that FLPA is generally preferable to LPA.

It is a well-known fact that Boolean rings, those rings in which $x^2 = x$ for all $x$, are necessarily commutative. There is a short and completely elementary proof of this. One may wonder what the situation is for rings in which $x^n = x$ for all $x$, where $n > 2$ is some positive integer. Jacobson and Herstein proved a very general theorem regarding these rings, and the proof follows a widely applicable strategy that can often be used to reduce questions about general rings to more manageable ones. We discuss this strategy, but will also focus on a different approach: can we also find ''elementary'' proofs of some special cases of the theorem? We treat a number of these explicit computations, among which a few new results.

The observation that the Dirac operator on a spin manifold encodes both the Riemannian metric as well as the fundamental class in K-homology leads to the paradigm of noncommutative geometry: the viewpoint that spectral triples generalise Riemannian manifolds. To encode maps between Riemannian manifolds, one is naturally led to consider the unbounded picture of Kasparov's KK-theory. In this talk I will explain how smooth cycles in KK-theory give a natural notion of noncommutative fibration, encoding morphisms noncommutative geometry in manner compatible with index theory.

We consider the impact of spatial heterogeneities on the dynamics of 
localized patterns in systems of partial differential equations (in one 
spatial dimension). We will mostly focus on the most simple possible 
heterogeneity: a small jump-like defect that appears in models in which 
some parameters change in value as the spatial variable x crosses 
through a critical value -- which can be due to natural inhomogeneities, 
as is typically the case in ecological models, or can be imposed on the 
model for engineering purposes, as in Josephson junctions. Even such a 
small, simplified heterogeneity may have a crucial impact on the 
dynamics of the PDE. We will especially consider the effect of the 
heterogeneity on the existence of defect solutions, which boils down to 
finding heteroclinic (or homoclinic) orbits in an n-dimensional 
dynamical system in `time' x, for which the vector field for x > 0 
differs slightly from that for x < 0 (under the assumption that there is 
such an orbit in the homogeneous problem). Both the dimension of the 
problem and the nature of the linearized system near the limit points 
have a remarkably rich impact on the defect solutions. We complement the 
general approach by considering two explicit examples: a heterogeneous 
extended Fisher–Kolmogorov equation (n = 4) and a heterogeneous 
generalized FitzHugh–Nagumo system (n = 6).

Pancakes.

Let K be a number field and E/K be an elliptic curve. Multiplication by n induces a map from the n^2-Selmer group of E/K to the n-Selmer group. The image of this map contains the image of E(K) in the n-Selmer group and is often smaller. Thus, computing the image of the n^2-Selmer group under multiplication by n can give a tighter bound on the rank of E/K. The Cassels-Tate pairing is a pairing on the n-Selmer group whose kernel is equal to the image of the n^2-Selmer group under multiplication by n. For n=2, Cassels gave an explicit description of the Cassels-Tate pairing as a sum of local pairings and computed the local pairing in terms of the Hilbert symbol. In this talk, I will give a formula for the local Cassels-Tate pairing for n=3 and describe an algorithm to compute it for n an odd prime. This is joint work with Tom Fisher.