The Hyperbolic Lattice Point Count in Infinite Volume with Applications to Sieves

We develop novel techniques using abstract operator theory to obtain asymptotic formulae for lattice counting problems on infinite-volume hyperbolic manifolds, with error terms which are uniform as the lattice moves through "congruence" subgroups. We give the following application to the theory of affine linear sieves. In the spirit of Fermat, consider the problem of primes in the sum of two squares, f(c,d)=c^2+d^2, but restrict (c,d) to the orbit O = (0,1).Gamma, where Gamma is an infinite-index non-elementary finitely-generated subgroup of SL(2,Z). Assume that the Reimann surface Gamma\H^2 has a cusp at infinity. We show that the set of values f(O) contains infinitely many integers having at most R prime factors for any R>4/(delta-theta), where theta>1/2 is the spectral gap and delta<1 is the Hausdorff dimension of the limit set of Gamma. If delta>149/150, then we can take theta=5/6, giving R=25. The limit of this method is R=9 for delta-theta>4/9. This is the same number of prime factors as attained in Brun's original attack on the twin prime conjecture.Comment: 33 pages, 1 figure, minor corrections. To appear, Duke Math

Iovem Imperium, or Sacred Aspects of Roman “Globalization”

The article deals with the question of the “globalization” project of the Roman civilization. Author asserts that the Romans had a specific “globalization” project. The construct “Iovem imperium” can explain the phenomenon of the Roman self-government and “sacred claim” of Roman community to domination in other lands. Pax Romana was conceived as an expression of Roman power (imperium), the boundaries of the Roman Republic were perceived as the border of the civilized world. Augustus was a brilliant manager, who could implement the Roman idea (an essential element of which was “Iovem imperium”) in the best way and create an almost perfect model of “globalization” in the ancient world. Forms of government were subordinated to the general concept of the Roman idea, and when to implement it in the new historical conditions required concentration of the supreme power in the same hands, the Romans willingly agreed to this, seeing in a World Empire highest embodiment of the republic as a “common cause” of its citizens

On some dimensional properties of 4-manifolds

It is shown, under the assumption of Jensen's principle $\lozenge$, that if for a complex L with $[L] \geq [S^{4}]$ there exists a metrizable compactum whose extension dimension is L, then there exists a differentiable, countably compact, perfectly normal and hereditarily separable 4-manifold whose extension dimension is also [L].Comment: 13 page

On the Nature of the Gods, or “Epistemological Polytheism” as History Comprehension Method

The article is devoted to the issue of history comprehension of the ancient societies in the context of their religious identity. Religion is one of the fundamental elements of civilization idea (“ontological project”); it constructs “universe” that is distinguished by the “laws of nature”, specific only for it. To make “communication” with ancient people maximally authentic, the researcher should not only recognize their right to look at the “world” in its own way, but also accept its “laws”, that means – religion as well. Since the latter is almost impossible, the scientist is deprived of the possibility to comprehend another cultural and historical reality as vivid establishment of human spirit; he/she will see only scheme or fable in it. The proposed in this paper method of “epistemological polytheism”, based on thought experiment, gives the possibility to bypass difficulties of perception and to approximate the understanding of meanings that define the ancient people worldview

Urbis et Orbis: Non-Euclidean Space of History

Social space is superimposed on the civilization map of the world whereas the social time is correlated with the duration of civilization existence. Within own civilization the concept space is non-homogeneous, there are “singled out points” — “concept factories”. As social structures, cities may exist rather long, sometimes during several millennia, but as concept centres they are limited by the duration of civilization existence. If civilization is a “concept universe”, nobody and nothing may cross the boundaries, which include cities as well. Death of civilization leads to reboot cultural and historical space-time. On the other hand, reformatted olds concepts are not preserved, but there may be reception of old concepts and their new interpretation. However, even in case of genetic links presence, they are the other concepts and not modified old ones. Under certain circumstances may take place “rebranding” when attractive name is connected to the concept of absolutely different order to attach to it authority of the past

Gods of Transhumanism

Purpose of the article is to identify the religious factor in the teaching of transhumanism, to determine its role in the ideology of this flow of thought and to identify the possible limits of technology interference in human nature. Theoretical basis. The methodological basis of the article is the idea of transhumanism. Originality. In the foreseeable future, robots will be able to pass the Turing test, become “electronic personalities” and gain political rights, although the question of the possibility of machine consciousness and self-awareness remains open. In the face of robots, people create their assistants, evolutionary competition with which they will almost certainly lose with the initial data. For successful competition with robots, people will have to change, ceasing to be people in the classical sense. Changing the nature of man will require the emergence of a new – posthuman – anthropology. Conclusions. Against the background of scientific discoveries, technical breakthroughs and everyday improvements of the last decades, an anthropological revolution has taken shape, which made it possible to set the task of creating inhumanly intelligent creatures, as well as changing human nature, up to discussing options for artificial immortality. The history of man ends and the history of the posthuman begins. We can no longer turn off this path, however, in our power to preserve our human qualities in the posthuman future. The theme of the soul again reminded of itself, but from a different perspective – as the theme of consciousness and self-awareness. It became again relevant in connection with the development of computer and cloud technologies, artificial intelligence technologies, etc. If a machine ever becomes a "man", then can a man become a "machine"? However, even if such a hypothetical probability would turn into reality, we cannot talk about any form of individual immortality or about the continuation of existence in a different physical form. A digital copy of the soul will still remain a copy, and I see no fundamental possibility of isolating a substrate-independent mind from the human body. Immortality itself is necessary not so much for stopping someone’s fears or encouraging someone’s hopes, but for the final solution of a religious issue. However, the gods hold the keys to heaven hard and are unlikely to admit our modified descendants there

The Measure of All Gods: Religious Paradigms of the Antiquity as Anthropological Invariants

Purpose of the article is the reconstruction of ancient Greek and ancient Roman models of religiosity as anthropological invariants that determine the patterns of thinking and being of subsequent eras. Theoretical basis. The author applied the statement of Protagoras that "Man is the measure of all things" to the reconstruction of the religious sphere of culture. I proceed from the fact that each historical community has a set of inherent ideas about the principles of reality, which found unique "universes of meanings". The historical space acquires anthropological properties that determine the specific mythology of the respective societies, as well as their spiritual successors. In particular, the religious models of ancient Greece and ancient Rome had a huge influence on formation of the worldview of the Christian civilization of the West. Originality. Multiplicity of the Olympic mythology contributed to the diversity of the expression forms of the Greek genius, which manifested itself in different fields of cultural activity, not reducible to political, philosophical or religious unity. The poverty of Roman mythology was compensated by a clear awareness of the unity of the community, which for all historical vicissitudes had always remained an unchanging ideal, and which was conceived as a reflection of the unity of the heavens. These two approaches to the divine predetermined the formation of two interacting, but conceptually different anthropological paradigms of Antiquity. Conclusions. Western concepts of divinity are invariants of two basic theological concepts – "Greek" (naturalism and paganism) and "Roman" (transcendentalism and henotheism). These are ideal types, so these two tendencies can co-exist in one society. The Roman trend continued to be realized by the anti-Roman religion, which took Roman forms and Roman name. Iconoclasm was a Byzantine version of the Reformation, promoted by the Isaurian emperors and failed due to the strong Hellenistic naturalistic lobby. Modern "Romans" are trying to get rid of the last elements of religious naturalism, and modern "Greeks" are trying to preserve the Hellenic elements in Christianity. Patterns can be transformed, but the observational view will still be able to identify their lineage. The developed model allows a deeper understanding of the culture of both ancient societies, as well as the outlook of Western man