Millennium Problem Research Articles

Solutions for three millennium problems.

Solutions for three millennium problems.

Resonance Electroproduction and the Origin of Mass

One of the greatest challenges within the Standard Model is to discover the source of visible mass. Indeed, this is the focus of a “Millennium Problem”, posed by the Clay Mathematics Institute. The answer is hidden within quantum chromodynamics (QCD); and it is probable that revealing the origin of mass will also explain the nature of confinement. In connection with these issues, this perspective will describe insights that have recently been drawn using contemporary methods for solving the continuum bound-state problem in relativistic quantum field theory and how they have been informed and enabled by modern experiments on nucleon-resonance electroproduction.

The new exact solution of the compressible 3D Navier–Stokes equations

The new exact solution of the compressible 3D Navier-Stokes (NS) equations in the case of zero shear viscosity η=0and finite volume viscosityςis obtained. It gives the possibility to avoid the closure problem and has finite time singularity att=t0 which is depended on initial velocity. The regularization of this singularity is possible due to the introduction of a super threshold homogeneous friction (unavoidable arising in all numerical simulations with NS equations) μ=νlmin2>μth=1/t0or by taking into account of any small shear viscosityν=η/ρ0>0;ν=const. This also gives the unexpectedly positive answer on the generalization of the Millennium Prize Problem for the case of compressible Navier-Stokes equations www.claymath.org.

Iwasawa theory and BSD conjecture

In this survey article, we introduce the backgrounds for BSD (Birch and Swinnerton-Dyer) conjecture—one of the seven millennium problems, and Iwasawa theory,which is a main tool of the study of BSD conjecture. Then we discuss the methods and results to study them, focusing on recent progress.

Exact solution of the compressible Euler–Helmholtz equation and the millennium prize problem generalization

SummaryFor the first time the exact strong solution of the three-dimensional (3D) compressible Euler–Helmholtz (EH) vortex equation in unbounded space is obtained. It corresponds to the general solution obtained in Euler variables for the 3D Riemann–Hopf (RH) equation and has a singularity in finite time. The exact closed description for the evolution of all one and multi point moments of the hydrodynamic fields and their spectra is now possible and thus, the mysterious turbulence problem has the exact solution for this case. As an example, for the first time the two-dimensional (2D) energy spectrum in the inertial scale interval is obtained directly from the exact solution of compressible 2D Navier–Stokes (NS) equation. For this purpose, a smooth continuation at all times for these exact solutions to the EH and RH equations is obtained by introducing a super-threshold homogeneous friction or any even extremely low effective viscosity. A new, smooth for all time, analytic solutions to the 1D, 2D and 3D NS equations is obtained. This solution coincides with the above-mentioned modified by viscosity solutions to the EH and RH equations, when also taking into account a linear relation between the pressure and the velocity field divergence, which is well known for the systems far from the equilibrium with relatively large second viscosity. This gives the positive answer to the generalization of the millennium prize problem (www.claymath.org) for the case of the compressible NS equation. Indeed, earlier only a negative answer to the problem of the existence of smooth solutions for any finite time has been a priori considered for the compressible case with the finite initial energy of the smooth velocity field in the unbounded space.

On the Navier-Stokes problem

One of the millennium problems is discussed. The results of the author’s solution to this problem are explained. The problem discussed is the Navier-Stokes problem in the whole space.

SIGEST

The SIGEST article in this issue is “Formation of Finite-Time Singularities in the 3D Axisymmetric Euler Equations: A Numerics Guided Study,” by Guo Luo and Thomas Y. Hou. This work considers the nonexistence of global finite-energy solutions to the 3D Euler equations. The authors propose a class of potentially singular solutions for the axisymmetric case, and present compelling numerical evidence that finite-time blowup occurs. Their contribution is closely related to the Clay Millennium Prize Problem on existence and smoothness of solutions to the Navier--Stokes equation, and hence sheds light on a preeminent open problem in fluid mechanics. At the heart of this work is a meticulous set of computational experiments that use customized Galerkin approximations and adaptive meshes in order to track a nearly singular numerical solution with high accuracy. The numerical results are validated with a range of tests, and the behavior close to the putative point of singularity is used to propose asymptotic scalings for various important quantities. The experiments also lead to a relevant 1D reduced model for which a finite-time singularity can be established rigorously. The original 2014 article, which appeared in SIAM's Multiscale Modeling and Simulation journal, has attracted considerable attention. In preparing this SIGEST version, the authors have made numerous modifications and revisions. These include: expanding the introductory section, streamlining and condensing the presentation of the numerical results, and updating the reference list. They have also highlighted various directions in which these equations, first published in 1757 by Leonhard Euler, may continue to tantalize both applied and numerical analysts.

The isoperimetric spectrum of finitely presented groups

The isoperimeric spectrum consists of all real positive numbers \alpha such that n^{\alpha} is equivalent to the Dehn function of a finitely presented group. In this note we show how a recent result of Olshanskii completes the description of the isoperimetric spectrum modulo the celebrated Computer Science conjecture (and one of the seven Millennium Problems) \mathbf {P = NP} and even a formally weaker conjecture.

Old mathematical challenges: Precedents to the millennium problems

The Millennium Problems stated by the Clay Mathematics Institute became a stimulus for mathematical research. The aim of this article is to show some previous challenges that were also a stimulus to prove interesting results. With this pretext, we present three moments in the history of mathematics that were important for the development of new lines of research. We briefly analyse the Tartaglia challenge, which allowed experts to discover a formula for third degree equations; Johan Bernoulli’s problem of the curve of fastest descent, which originated the calculus of variations; and the incidence of the problems posed by David Hilbert in 1900, focusing on the first problem in the list: the continuum hypothesis.

The Hodge conjecture: The complications of understanding the shape of geometric spaces

The Hodge conjecture is one of the seven millennium problems, and is framed within differential geometry and algebraic geometry. It was proposed by William Hodge in 1950 and is currently a stimulus for the development of several theories based on geometry, analysis, and mathematical physics. It proposes a natural condition for the existence of complex submanifolds within a complex manifold. Manifolds are the spaces in which geometric objects can be considered. In complex manifolds, the structure of the space is based on complex numbers, instead of the most intuitive structure of geometry, based on real numbers.

Navier-Stokes equations: Unpredictability even without butterflies?

Mathematically, the motion of a fluid is described by the so-called Navier-Stokes equations. In the spirit of Newtonian mechanics, these equations should determine the future motion of the fluid out of its initial state. However, despite the significant effort made for more than a century, this determinism has not yet been mathematically proved nor disproved. This paper offers a general perspective on the Navier-Stokes equations, the fourth millennium problem.

Old mathematical challenges: Precedents to the millennium problems

Old mathematical challenges: Precedents to the millennium problems

The new analytical solution of the 3D Navier-Stokes equation for compressible medium clarifies the sixth Millennium Prize problem

The new analytical solution of the 3D Navier-Stokes equation for compressible medium clarifies the sixth Millennium Prize problem

Hodges Conjecture Clay Institute Millennium Problem Solution

Hodges Conjecture Clay Institute Millennium Problem Solution

MAN vs. COMPUTER: DIFFERENCE OF THE ESSENCES. THE PROBLEM OF THE SCIENTIFIC CREATION

The critical analysis of the problem of creation of Artificial Intelligence (AI) and of Artificial General Intelligence (AGI) is proposed. The unity of formal logic and of rational dialectics is the methodological basis of the analysis. The main results of the analysis are as follows: (1) the model of man represents the unity of the two material aspects: “physiological body” (controllable aspect) and “psychical body” (controlling aspect); (2) the “psychical body” is the subsystem “subconsciousness + consciousness”; (3) in the comprehensive sense of the word, the thinking is an attribute of the complete system “physiological body + psychical body + environment”. (3) in the broad sense of the word, thinking and creativity are an essential feature of the subsystem “subconsciousness + consciousness”; (4) in the narrow (concise) sense of the word, thinking and creativity are the attribute of the instinct of the conservation (preservation, retention, maintenance) of life (i.e., the self-preservation instinct, the survival instinct); the instinct of the conservation of life exists in subconsciousness; (5) the instinct of the conservation of life is a system of elementary (basic) instincts; thinking is the attribute of the each elementary instinct; (6) the mechanism of thinking and the essence of creation cannot be cognized by man; (7) a computer as a device cannot think and create (in particular, it cannot prove theorems) because a computer does not have the subconsciousness; (8) the modeling of human thinking, Human Intellect, and the creation of AI and AGI are the impossible because the essential properties of the complete system “man + environment” cannot be cognized and modeled; (9) the existence of AI and AGI conflicts with the essence of the thinking; (10) the existence of AI and AGI contradict to formal-logical and rational-dialectical laws. Keywords: human intellect, artificial intelligence, artificial general intelligence, artificial psychology, computer science, psychology, linguistics, mathematics, P versus NP problem, Millennium Problems (Clay Mathematics Institute), formal logic, dialectics, philosophy, metaphysics.

The L-Functions and Modular Forms Database Project

The Langlands Programme, formulated by Robert Langlands in the 1960s and since much developed and refined, is a web of interrelated theory and conjectures concerning many objects in number theory, their interconnections, and connections to other fields. At the heart of the Langlands Programme is the concept of an L-function. The most famous L-function is the Riemann zeta-function, and as well as being ubiquitous in number theory itself, L-functions have applications in mathematical physics and cryptography. Two of the seven Clay Mathematics Million Dollar Millennium Problems, the Riemann Hypothesis and the Birch and Swinnerton-Dyer Conjecture, deal with their properties. Many different mathematical objects are connected in various ways to L-functions, but the study of those objects is highly specialized, and most mathematicians have only a vague idea of the objects outside their specialty and how everything is related. Helping mathematicians to understand these connections was the motivation for the L-functions and Modular Forms Database (LMFDB) project. Its mission is to chart the landscape of L-functions and modular forms in a systematic, comprehensive and concrete fashion. This involves developing their theory, creating and improving algorithms for computing and classifying them, and hence discovering new properties of these functions, and testing fundamental conjectures. In the lecture I gave a very brief introduction to L-functions for non-experts, and explained and demonstrated how the large collection of data in the LMFDB is organized and displayed, showing the interrelations between linked objects, through our website www.lmfdb.org. I also showed how this has been created by a world-wide open source collaboration, which we hope may become a model for others.

Hilbert's Fourth Problem as a Possible Candidate on the MILLENNIUM PROBLEM in Geometry

Hilbert’s Fourth Problem is one of the most important mathema tical problems, formulated by Hilbert in 1900. Unfortunately, attempts to solve this problem during 20 th century did not lead to the generally recognized solution, and now modern mathematicians believe that the problem has been formulated by Hilbert very vague and therefore it can not be solved. Th e main purpose of this article is to develop a new view on authors’ original solution to this problem a nd to interpret this problem as MILLENNIUM PROBLEM in Geometry what has an interdisciplinary importance and affects not only on geometry, but also on all theoretical natural sciences. The source of a new approach to solving this problem is a new branch of mathematics, the Mathematics of Harmony, which goes back in its origins to Euclid’s Elements and has interdisciplinary importance for modern science.

The Fine-Structure Constant as the Physical- Mathematical MILLENNIUM PROBLEM

The Fine-Structure Constant as the Physical- Mathematical MILLENNIUM PROBLEM

About the Riemann Hypothesis

The Riemann hypothesis is part of Hilbert’s eighth problem in David Hilbert’s list of 23 unsolved problems. It is also one of the Clay Mathematics Institute’s Millennium Prize Problems. Some mathematicians consider it the most important unresolved problem in pure mathematics. Many mathematicians made a lot of efforts; they don’t have to prove the Riemann hypothesis. In this paper, I use the analytic methods to deny the Riemann Hypothesis; if there’s something wrong, please criticize and correct me.

Navier-Stokes Equations—Millennium Prize Problems

In this work, we present final solving Millennium Prize Problems formulated by Clay Math. Inst., Cambridge. A new uniform time estimation of the Cauchy problem solution for the Navier-Stokes equations is provided. We also describe the loss of smoothness of classical solutions for the Navier-Stokes equations.