University of Helsinki

Conformal Field Theories (CFT) are believed to be exactly solvable once their primary scaling fields and their 3-point functions are known. This input is called the spectrum and structure constants of the CFT respectively. I will review recent work where this conformal bootstrap program can be rigorously carried out for the case of Liouville CFT, a theory that plays a fundamental role in 2d random surface theory and many other fields in physics and mathematics. Liouville CFT has a probabilistic formulation on an arbitrary Riemann surface and the bootstrap formula can be seen as a "quantization" of the plumbing construction of surfaces with marked points axiomatically discussed earlier by Graeme Segal. Joint work with Colin Guillarmou, Remi Rhodes and Vincent Vargas

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.