With the introduction of supermarket loyalty cards in recent decades, there has been an ever-growing body of customer-level shopping data. A natural way to represent this data is with a bipartite network, in which customers are connected to products that they purchased. By predicting likely edges in these networks, one can provide personalised product recommendations to customers.
In this talk, I will first discuss a basic approach for recommendations, based on network community detection, that we have validated on a promotional campaign run by our industrial collaborators. I will then describe a multilayer network model that accounts for the fact that customers tend to buy the same grocery items repeatedly over time. By modelling such correlations explicitly, link-prediction accuracy improves considerably. This approach is also useful in other networks that exhibit significant edge correlations, such as social networks (in which people often have repeated interactions with other people), airline networks (in which popular routes are often served by more than one airline), and biological networks (in which, for example, proteins can interact in multiple ways). 
 

Multilayer networks are a way to represent dependent connectivity patterns — e.g., time-dependence, multiple types of interactions, or both — that arise in many applications and which are difficult to incorporate into standard network representations. In the study of multilayer networks, it is important to investigate mesoscale (i.e., intermediate-scale) structures, such as communities, to discover features that lie between the microscale and the macroscale. We introduce a framework for the construction of generative models for mesoscale structure in multilayer networks.  We model dependency at the level of partitions rather than with respect to edges, and treat the process of generating a multilayer partition separately from the process of generating edges for a given multilayer partition. Our framework can admit many features of empirical multilayer networks and explicitly incorporates a user-specified interlayer dependency structure. We discuss the parameters and some properties of our framework, and illustrate an example of its use with benchmark models for multilayer community-detection tools. 

Standard representation theory transforms groups=algebra into vector spaces = (linear) algebra. The modern approach, geometric representation theory constructs geometric objects from algebra and captures various algebraic representations through geometric gadgets/invariants on these objects. This field started with celebrated Borel-Weil-Bott and Beilinson-Bernstein theorems but equally is in rapid expansion nowadays. I will start from the very beginnings of this field and try to get to the recent developments (time permitting).

Formally, the Fourier coefficients of Eisenstein series on Kac-Moody groups contain as yet mysterious automorphic L-functions relevant to open conjectures such as that of Ramanujan and Langlands functoriality. In this talk, we will consider the constant Fourier coefficient, if it even makes sense rigorously, and its relationship to the geometry and combinatorics of a Kac-Moody group. Joint work with Kyu-Hwan Lee.

Unitary highest weight modules were classified in the 1980s by Enright-Howe-Wallach and independently by Jakobsen. The classification is based on a version of the Dirac inequality, but the proofs also require a number of other techniques and are quite involved. We present a much simpler proof based on a different version of the Dirac inequality. This is joint work with Vladimir Soucek and Vit Tucek.
 

Many real-world networks contain small recurring connectivity patterns also known as network motifs. Although network motifs are widely considered to be important structural features of networks that are closely connected to their function methods for characterizing and modelling the local connectivity structure of complex networks remain underdeveloped. In this talk, we will present a non-parametric approach that is based on generative models in which networks are generated by adding not only single edges but also but also copies of larger subgraphs such as triangles to the graph. We show that such models can be formulated in terms of latent states that correspond to subgraph decompositions of the network and derive analytic expressions for the likelihood of such models. Following a Bayesian approach, we present a nonparametric prior for model parameters. Solving the resulting inference problem results in a principled approach for identifying atomic connectivity patterns of networks that do not only identify statistically significant connectivity patterns but also produces a decomposition of the network into such atomic substructures. We tested the presented approach on simulated data for which the algorithm recovers the latent state to a high degree of accuracy. In the case of empirical networks, the method identifies concise sets atomic subgraphs from within thousands of candidates that are plausible and include known atomic substructures.

Following recent papers by Nima Arkani-Hamed.