Tensor algebras give rise to one of the most powerful measures of similarity for sequences of arbitrary length called the signature kernel accompanied with attractive theoretical guarantees from stochastic analysis. Previous algorithms to compute the signature kernel scale quadratically in terms of the length and the number of the sequences. To mitigate this severe computational bottleneck, we develop a random Fourier feature-based acceleration of the signature kernel acting on the inherently non-Euclidean domain of sequences. We show uniform approximation guarantees for the proposed unbiased estimator of the signature kernel, while keeping its computation linear in the sequence length and number. In addition, combined with recent advances on tensor projections, we derive two even more scalable time series features with favourable concentration properties and computational complexity both in time and memory. Our empirical results show that the reduction in computational cost comes at a negligible price in terms of accuracy on moderate-sized datasets, and it enables one to scale to large datasets up to a million time series.

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Perverse equivalences, introduced by Chuang and Rouquier, are derived equivalences with a particularly nice combinatorial description. This generalised an earlier construction, with which they proved Broué’s abelian defect group conjecture for blocks of the symmetric groups. Perverse equivalences are of much wider significance in the representation theory of finite dimensional symmetric algebras. Grant has shown that periodic algebras admit perverse autoequivalences. In a similar vein, I will present some perverse equivalences arising from certain periodic modules, with an application to the setting of the symmetric groups.

Positive logic is a generalisation of full first-order logic, where negation is not built in, but can be added as desired. In joint work with Jan Dobrowolski we succesfully generalised the recent development on Kim-independence in NSOP1 theories to the positive setting. One of the important theorems in this development is the independence theorem, whose statement is very similar to the well-known statement for simple theories, and allows us to amalgamate independent types. In this talk we will have a closer look at the proof of this theorem, and what needs to be changed to make the proof work in positive logic compared to full first-order logic.

The Hardy–Littlewood circle method is a well-known analytic technique that has successfully solved several difficult counting problems in number theory. More recently, a version of the method over function fields, combined with spreading out techniques, has led to new results about the geometry of moduli spaces of rational curves on hypersurfaces of low degree. I will explain how one can implement a circle method with an even more geometric flavour, where the computations take place in a suitable Grothendieck ring of varieties, leading thus to a more precise description of the geometry of the above moduli spaces. This is joint work with Tim Browning.