Point counting on definable sets in non-archimedean settings has many faces. For sets living in Q_p^n, one can count actual rational points of bounded height, but for sets in C((t))^n, one rather "counts" the polynomials in t of bounded degree. What if the latter is of infinite cardinality? We treat three settings, each with completely different behaviour for point counting : 1) the setting of subanalytic sets, where we show finiteness of point counting but growth can be aribitrarily fast with the degree in t ; 2) the setting of Pfaffian sets, which is new in the non-archimedean world and for which we show an analogue of Wilkie's conjecture in all dimensions; 3) the Hensel minimal setting, which is most general and where finiteness starts to fail, even for definable transcendental curves! In this infinite case, one bounds the dimension rather than the (infinite) cardinality. This represents joint work with Binyamini, Novikov, with Halupczok, Rideau, Vermeulen, and separate work by Cantoral-Farfan, Nguyen, Vermeulen.

 I will be reviewing our recent article arXiv:2101.02218 where we propose a simple method for detecting global (a.k.a. non-perturbative) anomalies for generic quantum field theories. The basic idea is to study how the symmetries are realized on the Hilbert space of the theory. I will present several elementary examples where everything can be solved explicitly. After that, we will use these results to make non-trivial predictions about strongly interacting theories.

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Unsupervised learning, in particular learning general nonlinear representations, is one of the deepest problems in machine learning. Estimating latent quantities in a generative model provides a principled framework, and has been successfully used in the linear case, especially in the form of independent component analysis (ICA). However, extending ICA to the nonlinear case has proven to be extremely difficult: A straight-forward extension is unidentifiable, i.e. it is not possible to recover those latent components that actually generated the data. Recently, we have shown that this problem can be solved by using additional information, in particular in the form of temporal structure or some additional observed variable. Our methods were originally based on "self-supervised" learning increasingly used in deep learning, but in more recent work, we have provided likelihood-based approaches. In particular, we have developed computational methods for efficient maximization of the likelihood for two variants of the model, based on variational inference or Riemannian relative gradients, respectively.

Mixed-precision algorithms combine low- and high-precision computations in order to benefit from the performance gains of reduced-precision while retaining good accuracy. In this talk we focus on explicit stabilised Runge-Kutta (ESRK) methods for parabolic PDEs as they are especially amenable to a mixed-precision treatment. However, some of the concepts we present can be extended more generally to Runge-Kutta (RK) methods in general.

Consider the problem $y' = f(t,y)$ and let $u$ be the roundoff unit of the low-precision used. Standard mixed-precision schemes perform all evaluations of $f$ in reduced-precision to improve efficiency. We show that while this approach has many benefits, it harms the convergence order of the method leading to a limiting accuracy of $O(u)$.

In this talk we present a more accurate alternative: a scheme, which we call $q$-order-preserving, that is unaffected by this limiting behaviour. The idea is simple: by using $q$ high-precision evaluations of $f$ we can hope to retain a limiting convergence order of $O(\Delta t^{q})$. However, the practical design of these order-preserving schemes is less straight-forward.

We specifically focus on ESRK schemes as these are low-order schemes that employ a much larger number of stages than dictated by their convergence order so as to maximise stability. As such, these methods require most of the computational effort to be spent for stability rather than for accuracy purposes. We present new $s$-stage order $1$ and $2$ RK-Chebyshev and RK-Legendre methods that are provably full-order preserving. These methods are essentially as cheap as their fully low-precision equivalent and they are as accurate and (almost) as stable as their high-precision counterpart.

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Standard worst-case rounding error bounds of most numerical linear algebra algorithms grow linearly with the problem size and the machine precision. These bounds suggest that numerical algorithms could be inaccurate at large scale and/or at low precisions, but fortunately they are pessimistic. We will review recent advances in probabilistic rounding error analyses, which have attracted renewed interest due to the emergence of low precisions on modern hardware as well as the rise of stochastic rounding.

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There has been a recent uptick in interest in fermionic CFTs. These mildly generalise the usual notion of CFT to allow dependence on a background spin structure. I will discuss how this generalisation manifests itself in the symmetries, anomalies, and boundary conditions of the theory, using the series of unitary Virasoro minimal models as an example.