We propose a method for learning topology-preserving data representations (dimensionality reduction). The method aims to provide topological similarity between the data manifold and its latent representation via enforcing the similarity in topological features (clusters, loops, 2D voids, etc.) and their localization. The core of the method is the minimization of the Representation Topology Divergence (RTD) between original high-dimensional data and low-dimensional representation in latent space. RTD minimization provides closeness in topological features with strong theoretical guarantees. We develop a scheme for RTD differentiation and apply it as a loss term for the autoencoder. The proposed method "RTD-AE" better preserves the global structure and topology of the data manifold than state-of-the-art competitors as measured by linear correlation, triplet distance ranking accuracy, and Wasserstein distance between persistence barcodes.

We review Heisenberg homology of configurations in once bounded surfaces and extend the construction to the regular thickening of a finite graph with ribbon structure.

In this paper we formulate the Positivity Problem for nearly linear recurrent sequences. This is a generalisation of the Positivity Problem for linear recurrence sequences and is a special case of the non-reachability problem for linear time-invariant systems. Our main contribution is a decision procedure for the Positivity Problem for nearly linear recurrences of order at most 3 whose characteristic roots have absolute value at most one. The decision procedure relies on a new transcendence result for infinite series that is of independent interest.

Neutrinos from dense environments are unique laboratories for astrophysics, particle physics and many-body physics. They tell us about the last stages of the gravitational core-collapse and the explosion of massive stars. These elusive particles are also tightly linked to heavy elements synthesis in gravitational core-collapse supernovae and binary neutron star mergers, or play a pivotal role at the MeV epoch during the Universe expansion. We highlight theoretical and observational aspects of this interesting domain, in particular for the future measurement of neutrinos from the next core-collapse supernova, and of the diffuse supernova background, whose discovery might lie in the forthcoming future.