Mathematicians frequently present their own work in a diachronic fashion, e.g. by comparing their "modern" methods to those supposedly of the "Ancients," or by situating their latest theories as an "abstract" counterpart to more "classical" ones. The construction of such contrasts entangle mathematical labour and cultural life writ large. Indeed, it involves on the part of mathematicians the shaping up of correspondences between their technical achievements and intellectual discussions taking place on a much broader stage, such as those surrounding the concept of modernity, its relation to an imagined ancient past, or the characterisation of scientific progress as an increase in abstraction. This talk will explore the creation and use of such mathematical diachronies, the focus being on the works of Felix Klein, Hieronymus Zeuthen, and Hermann Schubert.

We conduct a systematic large-scale analysis of order book-driven predictability in high-frequency returns by leveraging deep learning techniques. First, we introduce a new and robust representation of the order book, the volume representation. Next, we carry out an extensive empirical experiment to address various questions regarding predictability. We investigate if and how far ahead there is predictability, the importance of a robust data representation, the advantages of multi-horizon modeling, and the presence of universal trading patterns. We use model confidence sets, which provide a formalized statistical inference framework particularly well suited to answer these questions. Our findings show that at high frequencies predictability in mid-price returns is not just present, but ubiquitous. The performance of the deep learning models is strongly dependent on the choice of order book representation, and in this respect, the volume representation appears to have multiple practical advantages.