It is often fruitful to study an infinite discrete group via its finite quotients.  For this reason, conditions that guarantee many finite quotients can be useful.  One such notion is residual finiteness.
A group is residually finite if for any non-identity element g there is a homomorphism onto a finite group, which doesn’t map g to e. I will mention how this relates to topology, present an argument why the surface groups are residually finite and I’ll show that in this case it is enough to consider homomorphisms onto alternating groups.

The disruptive drone activity at airports requires an early warning system and Aveillant make a radar system that can do the job. The main problem is telling the difference between birds and drones where there may be one or two drones and 10s or 100s of birds. There is plenty of data including time series for how the targets move and the aim is to improve the discrimination capability of tracker using machine learning.

Specifically, the challenge is to understand whether there can be sufficient separability between birds and drones based on different features, such as flight profiles, length of the track, their states, and their dominance/correlation in the overall discrimination. Along with conventional machine learning techniques, the challenge is to consider how different techniques, such as deep neural networks, may perform in the discrimination task.

The quantum unipotent coordinate ring has a cluster algebra structure. On the other hand, this ring is isomorphic to the Grothendieck ring of the module category of quiver Hecke algebras (QHA). We can prove that cluster monomials of the quantum unipotent coordinate ring correspondi to real simple modules. This is a joint work with Seok-Jin Kang, Myungho Kim and Se-jin Oh.

In recent years, much attention has been given to single-cell RNA sequencing techniques as they allow researchers to examine the functions and relationships of single cells inside a tissue. In this study, we combine single-cell RNA sequencing data with protein–protein interaction networks (PPINs) to detect active modules in cells of different transcriptional states. We achieve this by clustering single-cell RNA sequencing data, constructing node-weighted PPINs, and identifying the maximum-weight connected subgraphs with an exact Steiner-Tree approach. As a case study, we investigate RNA sequencing data from human liver spheroids but the techniques described here are applicable to other organisms and tissues. The benefits of our novel method are two-fold: First, it allows us to identify important proteins (e.g., receptors) which are not detected from a differential gene-expression analysis as they only interact with proteins that are transcribed in higher levels. Second, we find that different transcriptional states have different subnetworks of the PPIN significantly overexpressed. These subnetworks often reflect known biological pathways (e.g., lipid metabolism and stress response) and we obtain a nuanced picture of cellular function as we can associate them with a subset of all analysed cells.

The advent of social media and the resulting ability to instantaneously communicate ideas and messages to connections worldwide is one of the great consequences arising from the telecommunications revolution over the last century. Individuals do not, however, communicate only upon a single platform; instead there exists a plethora of options available to users, many of whom are active on a number of such media. While each platform offers some unique selling point to attract users, e.g., keeping up to date with friends through messaging and statuses (Facebook), photo sharing (Instagram), seeing information from friends, celebrities and numerous other outlets (Twitter) or keeping track of the career paths of friends and past colleagues (Linkedin), the platforms are all based upon the fundamental mechanisms of connecting with other users and transmitting information to them as a result of this link.

In this talk a model for the spreading of online information or “memes" on multiplex networks is introduced and analyzed using branching-process methods. The model generalizes that of [Gleeson et al., Phys. Rev. X., 2016] in two ways. First, even for a monoplex (single-layer) network, the model is defined for any specific network defined by its adjacency matrix, instead of being restricted to an ensemble of random networks. Second, a multiplex version of the model is introduced to capture the behavior of users who post information from one social media platform to another. In both cases the branching process analysis demonstrates that the dynamical system is, in the limit of low innovation, poised near a critical point, which is known to lead to heavy-tailed distributions of meme popularity similar to those observed in empirical data.

[1] J. P. Gleeson et al. “Effects of network structure, competition and memory time on social spreading phenomena”. Physical Review X 6.2 (2016), p. 021019.

[2] J. D. O’Brien et al. "Spreading of memes on multiplex networks." New Journal of Physics 21.2 (2019): 025001.

This is a report on work in progress with Jingyin Huang. The complement of an arrangement of linear hyperplanes in a complex vector space has a natural “Borel-Serre bordification” as a smooth manifold with corners. Its universal cover is analogous to the Borel-Serre bordification of an arithmetic lattice acting on a symmetric space as well as to the Harvey bordification of Teichmuller space. In the first case the boundary of this bordification is homotopy equivalent to a spherical building; in the second case it is homotopy equivalent to curve complex of the surface. In the case of a reflection arrangement the boundary of its universal cover is the “curve complex” of the corresponding spherical Artin group. By definition this is the simplicial complex of all conjugates of proper, irreducible, spherical parabolic subgroups in the Artin group. A cohomological method is used to show that the curve complex of a spherical Artin group has the homotopy type of a wedge of spheres.