### Working paper

## Topology of nerves and formal concepts

Cluster assessment remains one of the most actual problems in data mining. In this paper, a new approach to the selection of clusters based on a combination of measures of cluster quality is proposed. The new approach incorporates easily expert understanding of “interestingness” of clusters and does not require pre-defined parameters and thresholds. The subset of selected clusters is small and can be analyzed easily by experts.

According to the currently prevalent theory, hippocampal formation constructs and maintains cognitive spatial maps. Most of the experimental evidence for this theory comes from the studies on navigation in laboratory rats and mice, typically male animals. While these animals exhibit a rich repertoire of behaviors associated with navigation, including locomotion, head movements, whisking, sniffing, raring and scent marking, the contribution of these behavioral patterns to hippocampal activity has not been sufficiently studied. Instead, many publications have considered animal position in space as the single variable that affects the firing of hippocampal place cells and entorhinal grid cells. Here we argue that future work should focus on a more detailed examination of different behaviors exhibited during navigation in order to interpret the cause of spatial tuning in hippocampal neurons. As a step in this direction, we have analyzed data from two datasets, shared online, containing recordings from rats navigating in square and round arenas. Our analyses revealed structured, grid-like navigation patterns, evident from the spatial maps of animal position, velocity and acceleration. Moreover, grid cells available in the datasets exhibited the same spatial periodicity as the navigation parameters. These findings cast doubt on the cognitive-map interpretation of grid cells, since they suggest that neuronal spatial patterns could be caused by behaviors associated with navigation instead of representing a hierarchically high spatial map. Additionally, we speculate that scent marks left by navigating animals could contribute to neuronal responses while rats and mice sniff their environment.

The discovery of place-representing neurons in the hippocampal formation has been recognized by the Nobel Committee as a paradigm shift in Neuroscience (Burgess, 2014). Here we call attention to an innovative paper of particular note (Zhang and Manahan-Vaughan, 2015) that added important findings to this field of study.

Zhang and Manahan-Vaughan investigated the contribution of olfactory cues to the formation of place fields in hippocampal neurons. For this purpose, they put male Wistar rats in the darkness into a 80 × 80 cm square box. Four odors (orange, vanilla, almond, and lemon) were placed into the quadrants of the arena. Chocolate crumbs were scattered across the arena to encourage exploratory behavior. The researchers observed the formation of stable place fields in the hippocampal neurons, even though visual cues were unavailable to the rats. The place fields rotated when the odor placements were rotated, and remapped when the odors were shuffled. The authors concluded that “despite the less precise nature of olfactory stimuli compared with visual stimuli, these can substitute for visual inputs to enable the acquisition of metric information about space.”

We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.

We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.

We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.