In this talk we will show how the floating point errors in the simulation of SDEs (stochastic differential equations) can be modelled as stochastic. Furthermore, we will show how these errors can be corrected within a multilevel Monte Carlo approach which performs most calculations with low precision, but a few calculations with higher precision. The same procedure can also be used to correct for errors in converting from uniform random numbers to approximate Normal random numbers. Numerical results will be generated on both CPUs (using single/double precision) and GPUs (using half/single precision).

We propose a novel class of network models for temporal dyadic interaction data. Our objective is to capture important features often observed in social interactions: sparsity, degree heterogeneity, community structure and reciprocity. We use mutually-exciting Hawkes processes to model the interactions between each (directed) pair of individuals. The intensity of each process allows interactions to arise as responses to opposite interactions (reciprocity), or due to shared interests between individuals (community structure). For sparsity and degree heterogeneity, we build the non time dependent part of the intensity function on compound random measures following (Todeschini et al., 2016). We conduct experiments on real- world temporal interaction data and show that the proposed model outperforms competing approaches for link prediction, and leads to interpretable parameters.

Link to paper: https://papers.nips.cc/paper/7502-modelling-sparsity-heterogeneity-reci…

The Grothendieck ring of varieties over a field is a simple idea that formalizes various cut-and-paste arguments in algebraic geometry. We will explain how this intuitive construction leads to nontrivial results, such as computing Euler characteristics, counting points of varieties over finite fields, and determining Hodge numbers. As an example, we will investigate cubic hypersurfaces, especially the varieties parametrizing lines on them. If time permits, we will discuss some of the stranger properties of the Grothendieck ring.

In a joint work with Matt Tointon, we study the fine structure of approximate groups. We deduce various applications on growth, isoperimetry and quantitative estimates for the the simple random walk on finite vertex transitive graphs.