Searcharxiv⌕ Search

arXiv subjects

Yuri I. Manin

Publications and source records attributed to Yuri I. Manin.

At least 37 records · Page 2Linked to original sources
arXiv2014

Local zeta factors and geometries under Spec Z

The first part of this note shows that the odd period polynomial of each Hecke cusp eigenform for full modular group produces via Rodriguez--Villegas transform ([Ro--V]) a polynomial satisfying the functional equation of zeta type and having nontrivial zeros only on the middle line of its critical strip. The second part discusses Chebyshev lambda--structure of the polynomial ring as Borger's descent data to $\bold{F}_1$ and suggests its role in possible relation of $Γ_{\bold{R}}$--factor to "real geometry over $\bold{F}_1$" (cf. also [CoCons2]).

math.AG↗
arXiv2014

Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology

We introduce some algebraic geometric models in cosmology related to the "boundaries" of space-time: Big Bang, Mixmaster Universe, Penrose's crossovers between aeons. We suggest to model the kinematics of Big Bang using the algebraic geometric (or analytic) blow up of a point $x$. This creates a boundary which consists of the projective space of tangent directions to $x$ and possibly of the light cone of $x$. We argue that time on the boundary undergoes the Wick rotation and becomes purely imaginary. The Mixmaster (Bianchi IX) model of the early history of the universe is neatly explained in this picture by postulating that the reverse Wick rotation follows a hyperbolic geodesic connecting imaginary time axis to the real one. Penrose's idea to see the Big Bang as a sign of crossover from "the end of previous aeon" of the expanding and cooling Universe to the "beginning of the next aeon" is interpreted as an identification of a natural boundary of Minkowski space at infinity with the Big Bang boundary.

gr-qc↗
arXiv2013

Arithmetic Differential Equations of Painleve' VI Type

Using the description of Paileve' VI family of differential equations in terms of a universal elliptic curve, going back to R. Fuchs (cf. [Ma96]), we translate it into the realm of Arithmetic Differential Equations (cf. [Bu05]), where the role of derivative "in the $p$--adic direction" is played by a version of Fermat quotient

math.NT↗
arXiv2013

Numbers as functions

In this survey I discuss A. Buium's theory of ``differential equations in the p-adic direction'' ([Bu05]) and its interrelations with ``geometry over fields with one element'', on the background of various approaches to p-adic models in theoretical physics.

math.NT↗
arXiv2013

Moduli Operad over F1

In this paper we answer a question raised in [25], Sec. 4, by showing that the genus zero moduli operad $\{\bar{M}_{0,n+1}\}$ can be endowed with natural descent data that allow it to be considered as the lift to ${\rm Spec}\,\Z$ of an operad over $\F_1$. The relevant descent data are based on a notion of constructible sets and constructible functions over $\F_1$, which describes suitable differences of torifications with a positivity condition on the class in the Grothendieck ring. More generally, we do the same for the operads $\{T_{d,n+1}\}$ (whose components were) introduced in [5]. Finally, we describe a blueprint structure on $\{\bar{M}_{0,n}\}$ and we discuss from this perspective the genus zero boundary modular operad $\{\bar{M}_{g,n+1}^0\}$.

math.AG↗
arXiv2013

Zipf's law and L. Levin's probability distributions

Zipf's law in its basic incarnation is an empirical probability distribution governing the frequency of usage of words in a language. As Terence Tao recently remarked, it still lacks a convincing and satisfactory mathematical explanation. In this paper I suggest that at least in certain situations, Zipf's law can be explained as a special case of the a priori distribution introduced and studied by L. Levin. The Zipf ranking corresponding to diminishing probability appears then as the ordering determined by the growing Kolmogorov complexity. One argument justifying this assertion is the appeal to a recent interpretation by Yu. Manin and M. Marcolli of asymptotic bounds for error--correcting codes in terms of phase transition. In the respective partition function, Kolmogorov complexity of a code plays the role of its energy. This version contains minor corrections and additions.

cs.IT↗
arXiv2013

Complexity vs Energy: Theory of Computation and Theoretical Physics

This paper is a survey dedicated to the analogy between the notions of {\it complexity} in theoretical computer science and {\it energy} in physics. This analogy is not metaphorical: I describe three precise mathematical contexts, suggested recently, in which mathematics related to (un)computability is inspired by and to a degree reproduces formalisms of statistical physics and quantum field theory.

cs.CC↗
arXiv2013

Non-commutative generalized Dedekind symbols

We define and study generalized Dedekind symbols with values in non--necessarily commutative groups, generalizing constructions of Sh. Fukuhara in [Fu1], [Fu2]. Basic examples of such symbols are obtained by replacing period integrals of modular forms (cf. [Ma1], [Ma2], [Kn1], [Kn2], [ChZ]) by iterated period integrals introduced and studied in [Ma3], [Ma4].

math.NT↗
arXiv2013

Kolmogorov complexity as a hidden factor of scientific discourse: from Newton's law to data mining

The word "complexity" is most often used as a meta--linguistic expression referring to certain intuitive characteristics of a natural system and/or its scientific description. These characteristics may include: sheer amount of data that must be taken into account; visible "chaotic" character of these data and/or space distribution/time evolution of a system etc. This talk is centered around the precise mathematical notion of "Kolmogorov complexity", originated in the early theoretical computer science and measuring the degree to which an available information can be compressed. In the first part, I will argue that a characteristic feature of basic scientific theories, from Ptolemy's epicycles to the Standard Model of elementary particles, is their splitting into two very distinct parts: the part of relatively small Kolmogorov complexity ("laws", "basic equations", "periodic table", "natural selection, genotypes, mutations") and another part, of indefinitely large Kolmogorov complexity ("initial and boundary conditions", "phenotypes", "populations"). The data constituting this latter part are obtained by planned observations, focussed experiments, and afterwards collected in growing databases (formerly known as "books", "tables", "encyclopaedias" etc). In this discussion Kolomogorov complexity plays a role of the central metaphor. The second part and Appendix 1 are dedicated to more precise definitions and examples of complexity.

math.HO↗
arXiv2012

Towards motivic quantum cohomology of $\bar{M}_{0,S}$

We explicitly calculate some Gromov--Witten correspondences determined by maps of labeled curves of genus zero to the moduli spaces of labeled curves of genus zero. We consider these calculations as the first step towards studying the self--referential nature of motivic quantum cohomology.

math.AG↗
arXiv2012

Kolmogorov complexity and the asymptotic bound for error-correcting codes

The set of all error--correcting block codes over a fixed alphabet with $q$ letters determines a recursively enumerable set of rational points in the unit square with coordinates $(R,δ)$:= (relative transmission rate, relative minimal distance). Limit points of this set form a closed subset, defined by $R\le α_q(δ)$, where $α_q(δ)$ is a continuous decreasing function called asymptotic bound. Its existence was proved by the first--named author in 1981 ([Man1]), but no approaches to the computation of this function are known, and in [Man5] it was even suggested that this function might be uncomputable in the sense of constructive analysis. In this note we show that the asymptotic bound becomes computable with the assistance of an oracle producing codes in the order of their growing Kolmogorov complexity. Moreover, a natural partition function involving complexity allows us to interpret the asymptotic bound as a curve dividing two different thermodynamic phases of codes.

cs.IT↗
arXiv2012

Foundations as Superstructure. (Reflections of a practicing mathematician)

This talk presents foundations of mathematics as a historically variable set of principles appealing to various modes of human intuition and devoid of any prescriptive/prohibitive power. At each turn of history, foundations crystallize the accepted norms of interpersonal and intergenerational transfer and justification of mathematical knowledge.

math.HO↗
arXiv2011

A computability challenge: asymptotic bounds and isolated error-correcting codes

Consider the set of all error--correcting block codes over a fixed alphabet with $q$ letters. It determines a recursively enumerable set of points in the unit square with coordinates $(R,δ)$:= {\it (relative transmission rate, relative minimal distance).} Limit points of this set form a closed subset, defined by $R\le α_q(δ)$, where $α_q(δ)$ is a continuous decreasing function called {\it asymptotic bound.} Its existence was proved by the author in 1981, but all attempts to find an explicit formula for it so far failed. In this note I consider the question whether this function is computable in the sense of constructive mathematics, and discuss some arguments suggesting that the answer might be negative.

cs.IT↗
arXiv2009

Error-correcting codes and phase transitions

The theory of error-correcting codes is concerned with constructing codes that optimize simultaneously transmission rate and relative minimum distance. These conflicting requirements determine an asymptotic bound, which is a continuous curve in the space of parameters. The main goal of this paper is to relate the asymptotic bound to phase diagrams of quantum statistical mechanical systems. We first identify the code parameters with Hausdorff and von Neumann dimensions, by considering fractals consisting of infinite sequences of code words. We then construct operator algebras associated to individual codes. These are Toeplitz algebras with a time evolution for which the KMS state at critical temperature gives the Hausdorff measure on the corresponding fractal. We extend this construction to algebras associated to limit points of codes, with non-uniform multi-fractal measures, and to tensor products over varying parameters.

cs.IT↗
arXiv2009

Renormalization and computation I: motivation and background

In this paper I argue that infinities in the classical computation theory such as the unsolvability of the Halting Problem can be addressed in the same way as Feynman divergences in Quantum Field Theory, and that meaningful versions of renormalization in this context can be devised. Connections with quantum computation are also touched upon.

math.QA↗
arXiv2009

Renormalization and Computation II: Time Cut-off and the Halting Problem

This is the second installment to the project initiated in [Ma3]. In the first Part, I argued that both philosophy and technique of the perturbative renormalization in quantum field theory could be meaningfully transplanted to the theory of computation, and sketched several contexts supporting this view. In this second part, I address some of the issues raised in [Ma3] and provide their development in three contexts: a categorification of the algorithmic computations; time cut--off and Anytime Algorithms; and finally, a Hopf algebra renormalization of the Halting Problem.

math.QA↗
arXiv2009

Cyclotomy and analytic geometry over F_1

Geometry over non--existent "field with one element" $F_1$ conceived by Jacques Tits [Ti] half a century ago recently found an incarnation, in at least two related but different guises. In this paper I analyze the crucial role of roots of unity in this geometry and propose a version of the notion of "analytic functions" over $F_1$. The paper combines a focused survey with some new constructions. In new version, several local additions and changes are made, references added.

math.AG↗