We discuss properties of fuzzy de Sitter space defined by means of algebra of the de Sitter group SO(1,4) in unitary irreducible representations. It was shown before that this fuzzy space has local frames with metrics that reduce, in the commutative limit, to the de Sitter metric. Here we determine spectra of the embedding coordinates for (ρ, s = 1 2 ) unitary irreducible representations of the principal continuous series of the SO (1, 4). The result is obtained in the Hilbert space representation, but using representation theory it can be generalized to all representations of the principal continuous series. * majab@ipb.ac.rs, latas@ipb.ac.rs, lnenadovic@ipb.ac.rs 1

We analyse the spinor action on a curved noncommutative space, the so-called truncated Heisenberg algebra, and in particular, the nonminimal coupling of spinors to the torsion. We find that dimensional reduction of the Dirac action gives the noncommutative extension of the Gross-Neveu model, the model which is, as shown by Vignes-Tourneret, fully renormalisable. * majab@ipb.ac.rs, madore@th.u-psud.fr, lnenadovic@ipb.ac.rs * As we do not specify the representation here, we have only one product: the one which defines the algebra, (2.6).

We calculate divergent one-loop corrections to the propagators of the U (1) gauge theory on the truncated Heisenberg space, which is one of the extensions of the Grosse-Wulkenhaar model. The model is purely geometric, based on the Yang-Mills action; the corresponding gaugefixed theory is BRST invariant. We quantize perturbatively and, along with the usual wave-function and mass renormalizations, we find divergent nonlocal terms of the −1 and −2 type. We discuss the meaning of these terms and possible improvements of the model.

Humans are able to conceive physical reality by jointly learning different facets thereof. To every pair of notions related to a perceived reality may correspond a mutual relation, which is a notion on its own, but one-level higher. Thus, we may have a description of perceived reality on at least two levels and the translation map between them is in general, due to their different content corpus, one-to-many. Following success of the unsupervised neural machine translation models, which are essentially one-toone mappings trained separately on monolingual corpora, we examine further capabilities of the unsupervised deep learning methods used there and apply some of these methods to sets of notions of different level and measure. Using the graph and word embedding-like techniques, we build one-to-many map without parallel data in order to establish a unified vector representation of the outer world by combining notions of different kind into a unique conceptual framework. Due to their latent similarity, by aligning the two embedding spaces in purely unsupervised way, one obtains a geometric relation between objects of cognition on the two levels, making it possible to express a natural knowledge using one description in the context of the other. * Author names are given in alphabetical order.