### Article

## Positive forms on hyperkahler manifolds

The Kobayashi pseudometric on a complex manifold is the maximal pseudometric such that any holomorphic map from the Poincaré disk to the manifold is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi–Yau manifolds. Using ergodicity of complex structures, we prove this for all hyperkähler manifold with b_2\geqslant 7 that admits a deformation with a Lagrangian fibration and whose Picard rank is not maximal. The Strominger-Yau-Zaslow (SYZ) conjecture claims that parabolic nef line bundles on hyperkähler manifolds are semi-ample. We prove that the Kobayashi pseudometric vanishes for any hyperkähler manifold with b_2\geqslant 7 if the SYZ conjecture holds for all its deformations. This proves the Kobayashi conjecture for all K3 surfaces and their Hilbert schemes.

Let M be a closed symplectic manifold of volume V. We say that M admits an unobstructed symplectic packing by balls if any collection of symplectic balls (of possibly different radii) of total volume less than V admits a symplectic embedding to M. In 1994 McDuff and Polterovich proved that symplectic packings of Kahler manifolds can be characterized in terms of the Kahler cones of their blow-ups. When M is a Kahler manifold which is not a union of its proper subvarieties (such a manifold is called Campana simple) these Kahler cones can be described explicitly using the Demailly and Paun structure theorem. We prove that any Campana simple Kahler manifold, as well as any manifold which is a limit of Campana simple manifolds in a smooth deformation, admits an unobstructed symplectic packing by balls. This is used to show that all even-dimensional tori equipped with Kahler symplectic forms and all hyperkahler manifolds of maximal holonomy admit unobstructed symplectic packings by balls. This generalizes a previous result by Latschev-McDuff-Schlenk. We also consider symplectic packings by other shapes and show, using Ratner's orbit closure theorem, that any even-dimensional torus equipped with a Kahler form whose cohomology class is not proportional to a rational one admits a full symplectic packing by any number of equal polydisks (and, in particular, by any number of equal cubes).

Let (M,I,J,K) be a hyperkahler manifold, and Z⊂(M,I) a complex subvariety in (M,I). We say that Z is trianalytic if it is complex analytic with respect to J and K, and absolutely trianalytic if it is trianalytic with respect to any hyperk\"ahler triple of complex structures (M,I,J′,K′) containing I. For a generic complex structure I on M, all complex subvarieties of (M,I) are absolutely trianalytic. It is known that a normalization Z′ of a trianalytic subvariety is smooth; we prove that b2(Z′) is no smaller than b2(M) when M has maximal holonomy (that is, M is IHS). To study absolutely trianalytic subvarieties further, we define a new geometric structure, called k-symplectic structure; this structure is a generalization of the hypersymplectic structure. A k-symplectic structure on a 2d-dimensional manifold X is a k-dimensional space R of closed 2-forms on X which all have rank 2d or d. It is called non-degenerate if the set of all degenerate forms in R is a smooth, non-degenerate quadric hypersurface in R. We consider absolutely trianalytic tori in a hyperkahler manifold M of maximal holonomy. We prove that any such torus is equipped with a non-degenerate k-symplectic structure, where k=b2(M). We show that the tangent bundle TX of a k-symplectic manifold is a Clifford module over a Clifford algebra Cl(k−1). Then an absolutely trianalytic torus in a hyperkahler manifold M with b2(M)≥2r+1 is at least 2r−1-dimensional.

The article considers the Views of L. N. Tolstoy not only as a representative, but also as a accomplisher of the Enlightenment. A comparison of his philosophy with the ideas of Spinoza and Diderot made it possible to clarify some aspects of the transition to the unique Tolstoy’s religious and philosophical doctrine. The comparison of General and specific features of the three philosophers was subjected to a special analysis. Special attention is paid to the way of thinking, the relation to science and the specifics of the worldview by Tolstoy and Diderot. An important aspect is researched the contradiction between the way of thinking and the way of life of the three philosophers.

Tolstoy's transition from rational perception of life to its religious and existential bases is shown. Tolstoy gradually moves away from the idea of a natural man to the idea of a man, who living the commandments of Christ. Starting from the educational worldview, Tolstoy ended by creation of religious and philosophical doctrine, which were relevant for the 20th century.

This important new book offers the first full-length interpretation of the thought of Martin Heidegger with respect to irony. In a radical reading of Heidegger's major works (from *Being and **Time* through the ‘Rector's Address' and the ‘Letter on Humanism' to ‘The Origin of the Work of Art' and the *Spiegel* interview), Andrew Haas does not claim that Heidegger is simply being ironic. Rather he argues that Heidegger's writings make such an interpretation possible - perhaps even necessary.

Heidegger begins *Being and Time* with a quote from Plato, a thinker famous for his insistence upon Socratic irony. *The Irony of Heidegger *takes seriously the apparently curious decision to introduce the threat of irony even as philosophy begins in earnest to raise the question of the meaning of being. Through a detailed and thorough reading of Heidegger's major texts and the fundamental questions they raise, Haas reveals that one of the most important philosophers of the 20th century can be read with as much irony as earnestness.* The Irony of Heidegger* attempts to show that the essence of this irony lies in uncertainty, and that the entire project of onto-heno-chrono-phenomenology, therefore needs to be called into question.

The article is concerned with the notions of technology in essays of Ernst and Friedrich Georg Jünger. The special problem of the connection between technology and freedom is discussed in the broader context of the criticism of culture and technocracy discussion in the German intellectual history of the first half of the 20th century.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.