Feynman's words “What I cannot create, I do not understand” inspire us to use the power of topology and chirality to experimentally re-produce phenomena and "bring to life" theories from diverse fields like particle physics and cosmology. Even physically-sound models that turned out not describing the real World around us can materialize in the artificial "meta-World" table-top experiments that we meticulously design. I will first discuss how vortex knots in chiral liquid crystals can exhibit atom-like behavior, including fusion, fission and self-assembly into various crystals with giant electrostriction properties. These findings will let us admire the beautiful history of the early model of atoms by Lord Kelvin, as well as the very last poem by Maxwell related to it. I will then show that these vortices interact with light similar to what was predicted for the elusive cosmic strings, with knots and crystalline arrays of vortices allowing to spatially localize beams of light into closed loops and knots. 

We discuss several aspects of geometry and topology of knotted surfaces where the unifying theme is the discrete holonomy groups of corresponding geometric structures, which also involves algebra of varieties of discrete group representations and dynamics of their action in different senses - from dynamics of group orbits in considered spaces to ergodicity of group action, dynamical systems and dynamics of equivariant mappings with bounded distortion (quasiconformal, quasisymmetric and quasiregular). An interesting and unusual aspect is given by the wild properties of obtained knotted surfaces (in particular almost everywhere wildly knotted spheres - cf. A. T. Fomenko's art).

Our approach has a combinatorial flavor based on our method of "hyperbolic block-building", Siamese twins construction resulting in dis-crete representations with arbitrary large kernels (applications of our recently introduced conformal interbreeding generalizing the Gromov-Piatetskii hyperbolic interbreeding). These methods let us construct everywhere wild nontrivial 2-knots and surfaces in 4-sphere and solve well known problems in geometric analysis. Created wild surfaces have ergodic dynamics of uniform hyperbolic lattices and are obtained by constructed wild quasisymmetric maps equivariant with respect to the action of uniform hyperbolic lattices. This is connected to theory of conformal deformations of hyperbolic structures, their Teichmuller spaces (varieties of discrete reprs of hyperbolic lattices) and nontrivial homology 4-cobordisms. For related material (negatively curved locally symmetric rank one spaces, their Teichmuller spaces, reprs-n of uniform hyperbolic lattices, hyperbolic 4-cobordisms and several their appls to algebra, geometry, topology and geom analysis we refer to our new book "Dynamics of Discrete Group Action" published in the series De Gruyter Advances in Analysis and Geometry, 10.

The dynamics of a FK model with a modified law of spring dispersion Φ(x) is considered. It has two local minima. A dynamic structural phase transition between them was observed.

In [1], the FK model (β=1) was written in the continuum approximation, after which an exact analytical solution was found for a soliton moving at speed w (Frenkel-Kontorova dislocation).

In [1] there is no answer to the question about the presence in the system of other non-dislocation solutions that are localized in space and do not decay in time.

Such solutions were found in exact numerical calculations.

Let us write the Hamiltonian as the sum of kinetic K and potential U energies, with the law of elastic dispersion of the general form:

                                                               (1)

Let us consider not only Hooke’s law, but also a function that does not allow the intersection of atoms in space (x≠o), for example, with two local minima x≈α and x=1+β:

                                                                                 (2)

Assuming discrete time - t = j h, j = 1.2 ..., where h is the time step - we have solutions to the system of Newton’s equations for the k-th CH atom of the form:

                   (3)

System of equations (3) is an algorithm for constructing dynamic solutions of the FK model.

In [2], the “Chain” program with algorithm (2)-(3) constructs dynamic solutions of the FK model.

An example of a dynamic solution with Hooke's law of elastic dispersion is considered and in it an energy excitation that does not decay in time, moving with a non-uniform speed and with an energy lower than the rest energy of the dislocation, is found.

When considering the dynamics of the FK model [2]: - α=0.5, β=0, γ=0.044, V₀=0.03, Δ=1, a phase transition from β- to α-phase was found, with a decrease in the size of the CH by almost two times.

Conclusions:

1 The dynamics of the FK model and the dynamics of its continuum approximation do not always and everywhere coincide.

2 For the original discrete model, the result of an exact solution of the string limit may turn out to be erroneous. For example, in [3] an exact expression was obtained for a statistical sum of the FK model in the continuum approximation. Based on the above, it can be argued that this solution is not applicable to the original FK model.

3 If we accept that in local field theories ∇φ - this is a gradient analogue of Hooke’s chain, then we assume that in the center of the black hole matter with a changed metric and with fields collapsing to the size ɑN is grouped.

From the fact that the stretching of the elastically periodic Hooke chain (CH) is symmetrical relative to its center (odd symmetry), it follows that the coordinate of the center of gravity CH remains unchanged. This makes it possible to accurately solve the system of nonlinear equations for the stationary states of the FK model and select the ground state from them.

In 1938, the authors: Yakov Ilyich Frenkel and Tatyana Abramovna Kontorova (LFTI) put forward the Frenkel-Kontorova model (FK model) [1], which served as the basis for the creation of many theories of highly nonlinear processes [2].

бThe FK model is an elastically periodic chain of atoms (CH) in a periodic potential.

CH - is a one-dimensional sequence of N point atoms of masses m with coordinates {x_i} and period β, interconnected by elastic springs with the law of elastic dispersion Φ(x). Φ(x) - most often this is Hooke's law, .

The periodic potential V(x) has even symmetry and period a=1.

In most previous works [1,2], it was assumed N = ∞ that solutions would be simplified by eliminating boundary effects from consideration. As it turned out [3,4], this assumption is wrong: - in the discrete FK model there is no small parameter 1/N, so it is necessary to find exact solutions taking into account the position of its boundary atoms.

The potential energy of the FK system has the form:

                                         (1)

where N is the number of atoms in the chain, N=2K+1; x_i is the distance of the i-th atom to the center of the chain.

In [1]

If β=0, then in the ground state of the FK model the point CH is at the minimum of the potential V(x=0). When β≠0 the position of the central atom does not change x₀=0. The positions of the remaining atoms relative to the center are not even and are determined from a system of nonlinear equilibrium equations.

  - system of N equilibrium equations. Taking into account x_-i = - x_i, we have:

                                                                                  (2)

At x₀=0, all coordinates x_i and β are functions of x₁, x₁ ∈ [0, 1], β ∈ [0, 1], therefore, solutions for the ground state of the FK model can be obtained by minimizing U(x₁) with respect to x₁ [3,4]. Comparing the exact solutions [3,4] with their continuum approximations [5], we found that the properties of the ground state of the discrete FK model coincide with the properties of its continuum approximation only in the homogeneity region.

The work is devoted to the theory of nucleation and growth of crystals in a supersaturated (supercooled) liquid. An integro-differential model of the kinetic and balance equations for the crystal-size distribution function and supersaturation (supercooling) of the liquid has been formulated and analytically solved, taking into account the following factors: (i) fluctuations in crystal growth rates leading to diffusion of the distribution function in the particle size space, (ii) nonstationary growth rates of individual crystals, and (iii) arbitrary initial crystal size distribution. The problem is solved for arbitrary crystal nucleation kinetics, and the Weber-Volmer-Frenkel-Zel'dovich and Meirs kinetic mechanisms are considered as special cases for calculations. Analytical solutions of the nonstationary problem are derived: the crystal-size distribution function and the supersaturation (supercooling) of the liquid. Some biomedical applications of the developed theory for crystal growth from supersaturated solutions are discussed. The theory is compared with experimental data on protein crystallization of lysozyme and canavalin, as well as bovine and porcine insulin. The time-dependent dynamics of solution supersaturation and a bell-shaped particle-size distribution function are studied for these substances.

The theory we have developed is important for describing the bulk crystallization of insulin, proteins and other vital chemicals. For example, the study of the protein lysozyme, most commonly released from chicken egg whites, is important because this enzyme hydrolyses polysaccharides on bacterial cell walls. It is used as an antiseptic and also as a food additive. It should be noted that the rate of decomposition of protein supersaturation in crystallizing solutions of lysozyme was investigated in Ref. [1] when the crystallized protein is more stable than the dissolved one. The growth dynamics of another important protein, canavalin, was studied in Ref. [2]. In this tudy, we have compared our theory and experimental data on crystallization of lysozyme and canavalin proteins. For a more precise description of bulk crystallzation it is necessary to take the crystal anisotropy into account. The simplest way to do this is to use an ellipsoidal coordinate system. To generalise the present theory to the case of anisotropic particle growth, an approach recently developed in Refs. [3,4] can be used.

The answer to Wheeler's question ``How come the quantum?'' given by Kauffman is presented and explored. The answer, going back to an approach by Dirac, proposes a topological origin of Planck's quantum of action h-bar. The proposal assumes that space, particles and wave functions consist of unobservable strands of Planck radius, and that their crossing switches define h-bar. The proposal is checked against all quantum effects, including non-commutativity, spinor wave functions, entanglement, Heisenberg's indeterminacy relation, and the Schrödinger and Dirac equations. The principle of least action is deduced. The spectra of elementary particles, the gauge interactions, and general relativity are derived. Estimates for elementary particle masses and for coupling constants, as well as numerous experimental predictions are deduced. Complete agreement with observations is found. The derivations also appear to eliminate alternatives and thus provide arguments for the uniqueness of the proposal.

These talks are a short introduction to knot theory from a combinatorial point of view and with an eye towards applications to Natural Science.  The talks are self-contained and form a short introductory course in knot theory.

1. Introduction to knots and unknots, Reidemeister moves, linking numbers, Fox coloring to detect knotting and linking. Rational tangles and fractions via coloring. Link,Twist, Writhe and DNA.

These talks are a short introduction to knot theory from a combinatorial point of view and with an eye towards applications to Natural Science.  The talks are self-contained and form a short introductory course in knot theory.

2. Introduction to the Kauffman bracket polynomial, Many examples. Discussion of other knot polynomials. Relationships with graph theory and with statistical mechanics, state summation models, the Potts model, tensor networks and categories.

These talks are a short introduction to knot theory from a combinatorial point of view and with an eye towards applications to Natural Science.  The talks are self-contained and form a short introductory course in knot theory.

3. Introduction to the Khovanov Homology via working with the bracket polynomial and  cube categories and applications of Rasmussen invariant to reconnection numbers for knotted vortices. Discussion of other applications of knot theory and knot homology.

These talks are a short introduction to knot theory from a combinatorial point of view and with an eye towards applications to Natural Science.  The talks are self-contained and form a short introductory course in knot theory.

4. Knot theory and quantum computing. This talk will discuss how quantum algorithms that compute the Jones polynomial can be constructed, how the Fibonacci model - based in knot theoretic recoupling theory - can be used to create universal quantum computation, and how this model is related to the Quantum Hall effect. We will also discuss the role of the Dirac equation and Majorana fermions in topological quantum computing.

In this talk we discuss how, starting with John Horton Conway’s skein theory of the Alexander polynomial, that new invariants of knots arose (the Jones polynomial among them) that are related to statistical mechanics. I will tell the story of how I discovered statistical mechanics summation models for the Alexander and Jones polynomials. The will continue, via the work of Ed Witten, to gauge theory and quantum field theory. 

This relationship of knot theory and physical theory is intimately tied with a mystery discovered by Herman Weyl in the early part of the 20th century. Weyl discovered that if one takes a line element A in spacetime as a differential 1-form, and writes down dA in the sense of the differential forms of Grassmann, then dA expresses the mathematical form of the Electromagnetic Field.

The field is expressed by the holonomy of the form A  around loops in spacetime. Weyl was so impressed with his observation that he suggested building a Geometry that would unify his line element A and the metric of General Relativity to make a unified field theory. But Einstein asked why should spacetime lengths change under transport? And the Weyl theory did not quite succeed. Yet it did succeed by the quantum reformulation of Fritz London, where the key was to see that the holonomy could represent a phase change in the quantum wave-function. Experimental confirmation of the influence of a gauge potential A on quantum interference came much later with the Aharonov-Bohm effect. Theoretical influence of this idea came with the generalization of A to a Lie algebra valued 1-form and the corresponding generalized gauge theories such as Yang-Mills theory. Then the physical field is not dA but dA + A^A and the holonomy remains important. Witten suggested the use of a spatial gauge A so that measuring its holonomy along a knot K would produce invariants such as the Jones polynomial. Witten understood that a formal answer required integration over all  the connections A. This integral of Witten is a functional integral in the quantum field theory associated with A. It has deep formal properties that inform indeed not only the Jones polynomial, but a host of other invariants as well, and the seeds of relationships with the three manifold invariants of Reshetikhin and Turaev, and the Vassiliev invariants of knots and links. Topological Quantum Field Theory was born. This is the story of a revolution in knot theory that started with Conway in 1969, focused by Jones and Kauffman, and began again with Witten in 1988. This talk will discuss these matters and will mention more recent developments such as Khovanov homology whose physical interpretations are not yet fully articulated. The talk will be self-contained and suitable for a general scientific audience. We will illustrate these key ideas in the relationship of knots and natural science with geometry, diagrams and dynamics.

Aspects of gauge theory, Hamiltonian mechanics and quantum mechanics arise naturally in the mathematics of a non-commutative framework for calculus and differential geometry. This talk discusses our general results in this domain. We give a derivation of a generalization of the Feynman-Dyson derivation of electromagnetism using the non-commutative context and diagrammatic techniques. We then discuss, in more depth, relationships with gauge theory and differential geometry.  The key aspect of this approach is the representation of derivatives as commutators. This creates the context of a non-commutative world and allows a synoptic view of patterns in mathematical physics.

Finally, we explore the relationship of general relativity and non-commutative algebra via the expression of covariant derivatives in terms of commutators and articulate the Bianchi Identity in terms of the Jacobi Identity. In this way we can formulate curvature in terms of commutators and formulate algebraic constraints on curvature so that our curvature tensors have the requisite symmetries to produce a divergence free Einstein tensor. The talk will discuss these structures and their relationship with classical general relativity.

Aspects of gauge theory, Hamiltonian mechanics and quantum mechanics arise naturally in the mathematics of a non-commutative framework for calculus and differential geometry. This talk discusses our general results in this domain. We give a derivation of a generalization of the Feynman-Dyson derivation of electromagnetism using the non-commutative context and diagrammatic techniques. We then discuss, in more depth, relationships with gauge theory and differential geometry.  The key aspect of this approach is the representation of derivatives as commutators. This creates the context of a non-commutative world and allows a synoptic view of patterns in mathematical physics.

Finally, we explore the relationship of general relativity and non-commutative algebra via the expression of covariant derivatives in terms of commutators and articulate the Bianchi Identity in terms of the Jacobi Identity. In this way we can formulate curvature in terms of commutators and formulate algebraic constraints on curvature so that our curvature tensors have the requisite symmetries to produce a divergence free Einstein tensor. The talk will discuss these structures and their relationship with classical general relativity.

The original Lie linear algebra [1-3] is the only true linear algebra since it is the line that is the building element, a real “Ding” [1] that does the job in the constructions it puts together bit by bit, step by step, ”as a partial differential equation itself” in the “form f( x y z dx dy dz) = 0” [2,3]. Both the xyz coordinate axes (at the limit scale behaving like quarks) and the dx dy dz partial derivatives are “straight lines of length equal to zero” [Ib.] and thus the infinitesimal generators of what they singly or in constellations outline in a sequential crystallization of “Figuren” [1] in one “Nullstreifen” [Ib.], two “algebraische Fläche” [Ib.] or three - conforming with Schrödinger wavepackets -  “complex-cone” [2-3] dimensions to, for instance, “in that we restrict ourselves to the linear transformations of r, we find between the corresponding transformations of R: all movements (translation movement, rotation-movement, and the helicoidal movement), semblability transformations, transformation by reciprocal radii, parallel transformation…etc”. [Ib.] Each of these is a Lie group  shaped by the algebra  over a range corresponding e.g. to infinitely small displacements about some angle θ,  and since what is presently called Lie algebra  was obtained by deriving such ‘infinitesimal actions’ from the group the baby initially lost by lack of translation is now lost in translation, too, to an analog tail-of-the-dog resultant “linear vector space” whose however majestic coordinate matrices and renormalization factors  never lead back to the original infinitesimal generator “curve net” [2,3] in which the Standard Model can exactly and exhaustively be tracked down by serial interior volume-preserving lattice transformations [4], and the Periodic System equally exactly and exhaustively by its exterior space-filling modular Aufbau [5-8], all of which at the same time providing accurate images and models as well as online interactive computer program bits and algorithms [6-8]. 

We study the recent series of papers by the Italian-American physicist, Ruggero Maria Santilli based on the Lie-isotopic branch of hadronic mechanics, wherein  a system of extended protons and neutrons in conditions of partial mutual penetration in a  nuclear structure verifies the following properties: 1) Admits, for the first time, explicit and concrete realizations of Bohm’s hidden variables. 2) Violates Bell’s inequalities  3) Verifies the broadening of Heisenberg’s indeterminacy principle for electromagnetic interactions of point-like particles in vacuum into the isouncertainty principle of hadronic mechanics, also called Einstein’s isodeterminism for extended hadrons in conditions of partial mutual penetration. The new principle allows a progressive recovering of Einstein’s determinism in the transition from hadrons to nuclei and stars and its full recovering at the limit of Schwartzschild’s horizon. Some far reaching advances are also indicated.

Physics at its most fundamental is codified into a single operator. In the first instance this appears as a component of the Dirac equation, the relativistic quantum mechanical equation that describes the most fundamental physical state, the fermion or fundamental particle. At first sight, this equation contains mathematical objects which are not otherwise seen in physics, and most people who use the equation use them in a form close to that given by Dirac in his discovery paper of 1928. In this form they give the impression of being the result of a guess or reverse engineering which happen to give the correct results when applied to physical situations. However, Dirac’s 4 × 4 matrices disguise a more fundamental algebraic structure, whose form derives from the nature of the most fundamental quantities in physics and which has been derived from first principles of computation using a universal rewrite structure. When we apply this algebra in its most efficient form, we see that the Dirac equation is not sui generis but can be derived by a standard quantization procedure from classical relativistic energy-momentum conservation, with the operator alone, once specified, determining everything that follows. We also see immediate explanations of many previously unexplained physical facts without further assumptions.

This paper introduces a new theoretical and empirical framework that precisely defines fundamental physical quantities such as mass, length, frequency as quantum intervals. Secondary units of electric current, voltage, resistance, charge squared, and magnetic flux are herein redefined. The reason for the fine-structure constant (α), Newton's Gravitational Constant (G), are defined by equations describing wave-particle duality, field and flux. The core of this work utilizes a series of algebraic equations to elucidate the true underlying physical significance of these terms. Empirical equations depict the reason for electromagnetism measured as amperes of electric current flowing through a copper wire. The analysis describes Newton's Gravitational Constant (G) as intrinsic to the metrology of electromagnetism within the copper conductor. It is demonstrated that G plays a critical, unifying role in the subatomic dynamics governing electric current flow, suggesting a fundamental link between gravity and electromagnetism that extends from the subatomic realm to galactic scales.