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.

Synchronized activity of neurons is important for many aspects of brain function. Synchronization is affected by both network-level parameters, such as connectivity between neurons, and neuron-level parameters, such as firing rate. Many of these parameters are not static but may vary slowly in time. In this talk we focus on neuron-level parameters. Our work centres on the neurotransmitter acetylcholine, which has been shown to modulate the firing properties of several types of neurons through its affect on potassium currents such as the muscarine-sensitive M-current.  In the brain, levels of acetylcholine change with activity.  For example, acetylcholine is higher during waking and REM sleep and lower during slow wave sleep. We will show how the M-current affects the bifurcation structure of a generic conductance-based neural model and how this determines synchronization properties of the model.  We then use phase-model analysis to study the effect of a slowly varying M-current on synchronization.  This is joint work with Victoria Booth, Xueying Wang and Isam Al-Darbasah

An insightful exercise might be to ask what is the most important idea in linear algebra. Our first answer would not be eigenvalues or linearity, it would be “matrix factorizations.” We will discuss a blueprint to generate 53 inter-related matrix factorizations (times 2) most of which appear to be new. The underlying mathematics may be traced back to Cartan (1927), Harish-Chandra (1956), and Flensted-Jensen (1978) . We will discuss the interesting history. One anecdote is that Eugene Wigner (1968) discovered factorizations such as the SVD in passing in a way that was buried and only eight authors have referenced that work. Ironically Wigner referenced Sigurður Helgason (1962) but Wigner did not recognize his results in Helgason's book. This work also extends upon and completes open problems posed by Mackey² & Tisseur (2003/2005).

Classical results of Random Matrix Theory concern exact formulas from the Hermite, Laguerre, Jacobi, and Circular distributions. Following an insight from Freeman Dyson (1970), Zirnbauer (1996) and Duenez (2004/5) linked some of these classical ensembles to Cartan's theory of Symmetric Spaces. One troubling fact is that symmetric spaces alone do not cover all of the Jacobi ensembles. We present a completed theory based on the generalized Cartan distribution. Furthermore, we show how the matrix factorization obtained by the generalized Cartan decomposition G=K₁AK₂ plays a crucial role in sampling algorithms and the derivation of the joint probability density of A.

Joint work with Sungwoo Jeong

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In this talk, we present some effective methods for distributing points for approximating analytic functions with prescribed decay on a strip region including the real axis. Such functions appear when we use numerical methods with variable transformations. Typical examples of such methods are provided by single-exponential (SE) or double-exponential (DE) sinc formulas, in which variable transformations yield single- or double-exponential decay of functions on the real axis. It has been known that the formulas are nearly optimal on a Hardy space with a single- or double-exponential weight on the strip region, which is regarded as a space of transformed functions by the variable transformations.

Recently, we have proposed new approximation formulas that outperform the sinc formulas. For constructing them, we use an expression of the error norm (a.k.a. worst-case error) of an n-point interpolation operator in the weighted Hardy space. The expression is closely related to potential theory, and optimal points for interpolation correspond to an equilibrium measure of an energy functional with an external field. Since a discrete version of the energy becomes convex in the points under a mild condition about the weight, we can obtain good points with a standard optimization technique. Furthermore, with the aid of the formulation with the energy, we can find approximate distributions of the points theoretically.

[References]
- K. Tanaka, T. Okayama, M. Sugihara: Potential theoretic approach to design of accurate formulas for function approximation in symmetric weighted Hardy spaces, IMA Journal of Numerical Analysis Vol. 37 (2017), pp. 861-904.

- K. Tanaka, M. Sugihara: Design of accurate formulas for approximating functions in weighted Hardy spaces by discrete energy minimization, IMA Journal of Numerical Analysis Vol. 39 (2019), pp. 1957-1984.

- S. Hayakawa, K. Tanaka: Convergence analysis of approximation formulas for analytic functions via duality for potential energy minimization, arXiv:1906.03133.

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