Whereas the spectral properties of random matrices has been the subject of numerous studies and is well understood, the statistical properties of the corresponding eigenvectors has only been investigated in the last few years. We will review several recent results and emphasize their importance for cleaning empirical covariance matrices, a subject of great importance for financial applications.

In the twentieth century we leant that the theory of communication is a mathematical theory. Mathematics is able to add to the value of data, for example by removing redundancy (compression) or increasing robustness (error correction). More famously mathematics can add value to data in the presence of an adversary which is my personal definition of cryptography. Cryptographers talk about properties of confidentiality, integrity, and authentication, though modern cryptography also facilitates transparency (distributed ledgers), plausible deniability (TrueCrypt), and anonymity (Tor).
Modern cryptography faces new design challenges as people demand more functionality from data. Some recent requirements include homomorphic encryption, efficient zero knowledge proofs for smart contracting, quantum resistant cryptography, and lightweight cryptography. I'll try and cover some of the motivations and methods for these.
 

As early as the 1920's Marshall suggested that firms co-locate in cities to reduce the costs of moving goods, people, and ideas. These 'forces of agglomeration' have given rise, for example, to the high tech clusters of San Francisco and Boston, and the automobile cluster in Detroit. Yet, despite its importance for city planners and industrial policy-makers, until recently there has been little success in estimating the relative importance of each Marshallian channel to the location decisions of firms.
Here we explore a burgeoning literature that aims to exploit the co-location patterns of industries in cities in order to disentangle the relationship between industry co-agglomeration and customer/supplier, labour and idea sharing. Building on previous approaches that focus on across- and between-industry estimates, we propose a network-based method to estimate the relative importance of each Marshallian channel at a meso scale. Specifically, we use a community detection technique to construct a hierarchical decomposition of the full set of industries into clusters based on co-agglomeration patterns, and show that these industry clusters exhibit distinct patterns in terms of their relative reliance on individual Marshallian channels.

The second part is to use industry relatedness, which we measure via a similar metric to co-location, to better understand the association of industrial emissions to city-industry agglomeration. Specifically, we see that industrial emissions (which are the largest source of greenhouse emissions in the US) are highly tied to certain industries, and furthermore that communities in the industry relatedness network tend to explain the tendency of particular industry clusters to produce emissions. This is important, because it limits cities' abilities to move to a greener industry basket as some cities may be more or less constrained to highly polluting industry clusters, while others have more potential for diversification away from polluting industries.