A conjecture of Malle predicts an asymptotic formula for the number of number fields with given Galois group and bounded discriminant. Malle conjectured the shape of the formula but not the leading constant. We present a new conjecture on the leading constant motivated by a version for algebraic stacks of Peyre's constant from Manin's conjecture. This is joint work with Tim Santens.

String topology is an umbrella under which lives a family of algebraic structures on the homology of the (compact-open) loop space of a closed smooth manifold, M. Of great interest are the string product and coproduct, in view of the failure of the latter to be a homotopy invariant. We will discuss some existing algebraic and geometric perspectives on these operations, and give some examples that probe the extent to which the string coproduct fails to be a homotopy invariant. We will sketch an alternative point of view on string topology as the study of the derived bornological smooth loop stack and explain why this is a promising model for the observed phenomena of string topology.

We build statistical models to describe how market participants choose the direction, price, and volume of orders. Our dataset, which spans sixteen weeks for four shares traded in Euronext Amsterdam, contains all messages sent to the exchange and includes algorithm identification and member identification. We obtain reliable out-of-sample predictions and report the top features that predict direction, price, and volume of orders sent to the exchange. The coefficients from the fitted models are used to cluster trading behaviour and we find that algorithms registered as Liquidity Providers exhibit the widest range of trading behaviour among dealing capacities. In particular, for the most liquid share in our study, we identify three types of behaviour that we call (i) directional trading, (ii) opportunistic trading, and (iii) market making, and we find that around one third of Liquidity Providers behave as market markers.

This is based on work with Álvaro Cartea, Saad Labyad, Leandro Sánchez-Betancourt and Leon van Veldhuijzen. View the working paper here.
 

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