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Pawel Nurowski

Publications and source records attributed to Pawel Nurowski.

At least 19 recordsLinked to original sources
arXiv2026

Tight-Binding Spectra of Finite Incidence Geometries: From Spatial Localization to $SU(6)$ Flavor Symmetry

We investigate the spectral properties of tight-binding quantum Hamiltonians defined on the bipartite Levi graphs of finite geometric configurations. In Part I, we analyze the $10_3$ Desargues and Kantor configurations, demonstrating how deterministic spatial deformations induced by real planar embeddings ($\mathbb{R}\mathbb{P}^2$) destroy translational symmetry, leading to structural wave localization. We show how embedding in the complex projective plane ($\mathbb{C}\mathbb{P}^2$) restores Bloch wave propagation via synthetic gauge phases. In Part II, we evaluate the Schl\"afli double six ($12_5, 30_2$) and the Cremona-Richmond ($15_3$) configurations in $\mathbb{R}^3$. We analytically derive their exact spectral decompositions, confirming the existence of macroscopic zero-energy flat bands ($E=0$) driven entirely by geometric frustration. In Part III, we establish a formal structural isomorphism between these discrete tight-binding networks and the flavor symmetry sector of the Standard Model. We map the Schl\"afli graph to the $SU(6)$ flavor multiplets, where the flat band corresponds to the kinematic freezing of ultra-heavy baryons. Finally, we discuss the complementary Cremona-Richmond $15_3$ topology, demonstrating how its distinct geometric nature (based on tritangent planes rather than point intersections) provides a purely algebraic, topological completion to the $W(E_6)$ symmetry of the 27 lines.

math-ph↗
arXiv2026

Nineteen to the Dozen: Embedding the Neo-Riemannian Tonnetz into a Cyclic 19_3 Symmetric Configuration

This paper bridges combinatorial geometry and music theory to solve the fundamental challenge of embedding classical Western harmony into the microtonal 19-tone equal temperament (19-TET). Inspired by Roger Penrose's observations on the mathematical elegance of 19-TET, we provide the theoretical foundation for a physical 19-TET acoustic piano currently under construction. However, playing classical 12-TET music on such an instrument poses a topological problem: emvedding the classical Euler-Riemann Tonnetz into the 19-TET universe inevitably distorts structural chords, creating dissonant ``wolves.'' By formalizing these harmonic spaces as incidence configurations (the 12_3 and 19_3 graphs) and utilizing integer cuts in our optimization model, we exhaustively prove that exactly 32 out of 36 Neo-Riemannian harmonic connections can be preserved. We demonstrate a strict 5-fold degeneracy of this optimum: there exist exactly 5 mathematically equivalent local packings for the wolf chords. Among these, we identify a unique canonical realization in which the 14 excised vertices form a perfectly contiguous geometric void along the primary Hamiltonian cycle. We reveal that the 4 inevitably broken edges represent the exact topological scars of the historical enharmonic diesis, and we formulate the Vicentino Hypothesis regarding 16th-century microtonal composition. Finally, to make this theoretical geometry physically playable, we design a novel 19-TET split-key keyboard, formalized through a biomechanical cost function that optimizes the performer's hand span. This work provides the complete theoretical, historical, and ergonomic blueprint for the next generation of microtonal acoustic instruments.

math.GM↗
arXiv2026

In Search of the Canonical Harmony for 12-TET

Is the specific structure of Western tonal harmony a physical inevitability derived from acoustics, or is it merely one solution among many in a purely algebraic landscape? In this paper, we strip away the physics of vibrating strings and treat harmony as the solution to a simple linear system within the cyclic group $\mathbb{Z}_{12}$. By defining a harmonic system as a partitioning of a generator interval (the "Fifth") into two complementary thirds, we derive a complete classification of all possible harmonic universes in 12-Tone Equal Temperament. We show that every such system corresponds to a specific topological structure, visualized via its Levi graph. Our analysis reveals a counter-intuitive fact: the topological structure of standard Western harmony is not unique. It is one of exactly twelve mathematically isomorphic systems. However, we demonstrate that these shadows are not created equal. We identify a "privileged quartet" of systems that preserve the global connectivity of the Circle of Fifths. Furthermore, by invoking the Chinese Remainder Theorem, we present a compelling argument that the most number-theoretically natural system in $\mathbb{Z}_{12}$ is not the Western one, but a specific "alien" system defined by the partition $(9,4)$. This classification provides a rigorous mathematical foundation for neo-Riemannian theory and offers composers a precise "translation map" to explore syntactically familiar, yet sonically distinct, harmonic worlds.

math.CO↗
arXiv2025

The Beltrami -- de Sitter model

We combine the well-known Beltrami-Klein model of non-Euclidean geometry on a $2-$dimensional disk, where the geodesics are the chords of the disk, with the $2-$dimensional de Sitter space. The geometry of the de Sitter space is defined on the complement of the Beltrami-Klein disk in the plane, with the de Sitter metric being the unique Lorentzian Einstein metric whose light cones form cones tangent to the disk in this complement. This leads to a Beltrami-de Sitter model on the plane ${\bf R}^2$, which is endowed with the Riemannian Beltrami metric on the disk and the Lorentzian de Sitter metric outside the disk in ${\bf R}^2$. We explore the relevance of this model for Penrose's Conformal Cyclic Cosmology, first in the $2-$dimensional setting and subsequently in higher dimensions, including the physically significant case of four dimensions. In this context, we define a Radon-like transform between the de Sitter and Beltrami spaces, facilitating the purely geometric transformation of physical fields from the Lorentzian de Sitter space to the Riemannian Beltrami space. In the $2-$, and $3-$dimensional cases, we also uncover a hidden ${\bf G}_2$ symmetry associated with the de Sitter spaces in these dimensions, which is related to a certain vector distribution naturally defined by the geometry of the model. We suggest the potential for discovering similar hidden symmetries in the $n-$dimensional Beltrami-de Sitter model.

gr-qc↗
arXiv2024

Conformal structures with $G_2$-symmetric twistor distribution

For any 4D split-signature conformal structure, there is an induced twistor distribution on the 5D space of all self-dual totally null 2-planes, which is $(2,3,5)$ when the conformal structure is not anti-self-dual. Several examples where the twistor distribution achieves maximal symmetry (the split-real form of the exceptional simple Lie algebra of type $\mathrm{G}_2$) were previously known, and these include fascinating examples arising from the rolling of surfaces without twisting or slipping. Relaxing the rolling assumption, we establish a complete local classification result among those homogeneous 4D split-conformal structures for which the symmetry algebra induces a multiply-transitive action on the 5D space. Furthermore, we discuss geometric properties of these conformal structures such as their curvature, holonomy, and existence of Einstein representatives.

math.DG↗
arXiv2023

Exceptional real Lie algebras $\mathfrak{f}_4$ and $\mathfrak{e}_6$ via contactifications

In Cartan's PhD thesis, there is a formula defining a certain rank 8 vector distribution in dimension 15, whose algebra of authomorphism is the split real form of the simple exceptional complex Lie algebra $\mathfrak{f}_4$. Cartan's formula is written in the standard Cartesian coordinates in $\mathbb{R}^{15}$. In the present paper we explain how to find analogous formula for the flat models of any bracket generating distribution $\mathcal D$ whose symbol algebra $\mathfrak{n}({\mathcal D})$ is constant and 2-step graded, $\mathfrak{n}({\mathcal D})=\mathfrak{n}_{-2}\oplus\mathfrak{n}_{-1}$. The formula is given in terms of a solution to a certain system of linear algebraic equations determined by two representations $(ρ,\mathfrak{n}_{-1})$ and $(τ,\mathfrak{n}_{-2})$ of a Lie algebra $\mathfrak{n}_{00}$ contained in the $0$th order Tanaka prolongation $\mathfrak{n}_0$ of $\mathfrak{n}({\mathcal D})$. Numerous examples are provided, with particular emphasis on the distributions with symmetries being real forms of simple exceptional Lie algebras $\mathfrak{f}_4$ and $\mathfrak{e}_6$.

math.DG↗
arXiv2023

Ants and bracket generating distributions in dimension 5 and 6

We consider a mechanical system of three ants on the floor, which move according to two independt rules: Rule A - forces the velocity of any given ant to always point at a neighboring ant, and Rule B - forces the velocity of every ant to be parallel to the line defined by the two other ants. We observe that Rule A equips the 6-dimensional configuration space of the ants with a structure of a homogeneous (3,6) distribution, and that Rule B foliates this 6-dimensional configuration space onto 5-dimensional leaves, each of which is equiped with a homogeneous (2,3,5) distribution. The symmetry properties and Bryant-Cartan local invariants of these distributions are determined. In the case of Rule B we study and determine the singular trajectories (abnormal extremals) of the corresponding distributions. We show that these satisfy an interesting system of two ODEs of Fuchsian type.

math.DG↗
arXiv2022

Apparent evidence for Hawking points in the CMB Sky

This paper presents strong observational evidence of numerous previously unobserved anomalous circular spots, of significantly raised temperature, in the CMB sky. The spots have angular radii between 0.03 and 0.04 radians (i.e. angular diameters between about 3 and 4 degrees). There is a clear cut-off at that size, indicating that each anomalous spot would have originated from a highly energetic point-like source, located at the end of inflation -- or else point-like at the conformally expanded Big Bang, if it is considered that there was no inflationary phase. The significant presence of these anomalous spots, was initially noticed in the Planck 70 GHz satellite data by comparison with 1000 standard simulations, and then confirmed by extending the comparison to 10000 simulations. Such anomalous points were then found at precisely the same locations in the WMAP data, their significance confirmed by comparison with 1000 WMAP simulations. Planck and WMAP have very different noise properties and it seems exceedingly unlikely that the observed presence of anomalous points in the same directions on both maps may come entirely from the noise. Subsequently, further confirmation was found in the Planck data by comparison with 1000 FFP8.1 MC simulations (with $l \leq 1500$). The existence of such anomalous regions, resulting from point-like sources at the conformally stretched-out big bang, is a predicted consequence of conformal cyclic cosmology (CCC), these sources being the Hawking points of the theory, resulting from the Hawking radiation from supermassive black holes in a cosmic aeon prior to our own.

astro-ph.CO↗
arXiv2021

Homogeneous C21 Models

Fels-Kaup (Acta Mathematica 2008) classified homogeneous $\mathfrak{C}_{2,1}$ hypersurfaces $M^5 \subset \mathbb{C}^3$ and discovered that they are all biholomorphic to tubes $S^2 \times i \mathbb{R}^3$ over some affinely homogeneous surface $S^2 \subset \mathbb{R}^3$. The second and third authors in 2003.08166, by performing highly non-straightforward calculations, conducted the Cartan method of equivalence to classify homogeneous models of PDE systems related to such $\mathfrak{C}^{2,1}$ hypersurfaces $M^5 \subset \mathbb{C}^3$. Kolar-Kossovskiy 1905.05629 and the authors 2003.01952 constructed a formal and a convergent Poincaré-Moser normal form for $\mathfrak{C}_{2,1}$ hypersurfaces $M^5 \subset \mathbb{C}^3$. But this was only a first, preliminary step. Indeed, the invariant branching tree underlying Fels-Kaup's classification was still missing in the literature, due to computational obstacles. The present work applies the power series method of equivalence, confirms Fels-Kaup 2008, and finds a differential-invariant tree. To terminate the middle (thickest) branch, it is necessary to compute up to order $10$ with $5$ variables. Again, calculations, done by hand, are non-straightforward.

math.CV↗
arXiv2021

Radiative Poincare type eon and its follower

We consider two consecutive eons $\hat{M}$ and $\check{M}$ from Penrose's Conformal Cyclic Cosmology and study how the matter content of the past eon ($\hat{M}$) determines the matter content of the present eon ($\check{M}$) by means of the reciprocity hypothesis. We assume that the only matter content in the final stages of the past eon is a spherical wave described by Einstein's equations with the pure radiation energy momentum tensor $$\hat{T}^{ij} = \hatΦK^iK^j, \quad \hat{g}_{ij} K^iK^j = 0,$$ and with cosmological constant $\hatΛ$ . We solve these Einstein's equations associating to $\hat{M}$ the metric $\hat{g}=t^{-2}\big(-d t^2+h_t\big)$, which is a Lorentzian analog of the Poincaré-Einstein metric known from the theory of conformal invariants. The solution is obtained under the assumption that the 3-dimensional conformal structure $[h]$ on the $\mathscr{I}^+$ of $\hat{M}$ is flat, that the metric $\hat{g}$ admits a power series expansion in the time variable $t$, and that $h_0\in [h]$. Such solution depends on one real arbitrary function of the radial variable $r$. Applying the reciprocal hypothesis, $\hat{g}\to \check{g}=t^4\hat{g}$, we show that the new eon $(\check{M},\check{g})$ created from the one containing a single spherical wave, is filled at its initial state with three types of radiation: (i) the damped spherical wave which continues its life from the previous eon, (ii) the in-going spherical wave obtained as a result of a collision of the wave from the past eon with the Bang hypersurface and (3) randomly scattered waves that could be interpreted as perfect fluid with the energy density $\checkρ$ and the isotropic pressure $\check{p}$ such that $\check{p}=\tfrac13\checkρ$.

gr-qc↗
arXiv2020

Five-dimensional para-CR manifolds and contact projective geometry in dimension three

We study invariant properties of $5$-dimensional para-CR structures whose Levi form is degenerate in precisely one direction and which are $2$-nondegenerate. We realize that two, out of three, primary (basic) para-CR invariants of such structures are the classical differential invariants known to Monge (1810) and to Wuenschmann (1905) \[ M(G) := 40G_{ppp}^3-45G_{pp}G_{ppp}G_{pppp}+9G_{pp}^2G_{ppppp}, \quad W(H) := 9D^2H_r-27DH_p-18H_rDH_r+18H_pH_r+4H_r^3+54H_z. \] The vanishing $M(G) \equiv 0$ provides a local necessary and sufficient condition for the graph of a function in the $(p,G)$-plane to be contained in a conic, while the vanishing $W(H) \equiv 0$ gives an if-and-only-if condition for a 3rd order ODE to define a natural Lorentzian geometry on the space of its solutions. Mainly, we give a geometric interpretation of the third basic invariant of our class of para-CR structures, the simplest one, of lowest order, and of mixed nature $N(G,H):=2G_{ppp}+G_{pp}H_{rr}$. We establish that the vanishing $N(G,H) \equiv 0$ gives an if-and-only-if condition for the two $3$-dimensional quotients of the para-CR manifold by its two canonical integrable rank-$2$ distributions, to be equipped with contact projective geometries. A curious transformation between the Wuenschmann invariant and the Monge invariant, first noted by us in arXiv:2003.08166, is also discussed, and its mysteries are further revealed.

math.DG↗
arXiv2020

On degenerate para-CR structures: Cartan reduction and homogeneous models

Motivated by recent works in Levi degenerate CR geometry, this article endeavours to study the wider and more flexible para-CR structures for which the constraint of invariancy under complex conjugation is relaxed. We consider $5$-dimensional para-CR structures whose Levi forms are of constant rank $1$ and that are $2$-nondegenerate both with respect to parameters and to variables. Eliminating parameters, such structures may be represented modulo point transformations by pairs of PDEs $z_y=F(x, y, z, z_x)$ $\,\,\&\,\,$ $z_{xxx}=H(x,y,z,z_x,z_{xx})$, with $F$ independent of $z_{xx}$ and $F_{z_xz_x} \neq 0$, that are completely integrable $D_x^3 F=Δ_y H$, Performing at an advanced level Cartan's method of equivalence, we determine all concerned homogeneous models, together with their symmetries: (i) $z_y=\tfrac14 (z_x)^2\quad \&\quad z_{xxx}=0$; (ii) $z_y=\tfrac14 (z_x)^2\quad \& \quad z_{xxx}=(z_{xx})^3$; (iiia) $z_y=\tfrac14 (z_x)^b\,\, \& \,\,z_{xxx} = (2-b)\frac{(z_{xx})^2}{z_x}$ with $z_x>0$ for any real $b\in[1,2)$; (iiib) $z_y = f(z_x)\quad \& \quad z_{xxx}=h(z_x)\big(z_{xx}\big)^2$, where the function $f$ is determined by the implicit equation: \[ (z_x^2+f(z_x)^2)\, \mathrm{exp} \left( 2b\,\mathrm{arctan}\tfrac{bz_x-f(z_x)}{z_x+bf(z_x)} \right) = 1+b^2 \] and where: \[ h(z_x) := \frac{(b^2-3)z_x-4bf(z_x)}{(f(z_x)-bz_x)^2}, \] for any real $b>0$.

math.DG↗
arXiv2018

Aerobatics of flying saucers

Starting from the observation that a flying saucer is a nonholonomic mechanical system whose 5-dimensional configuration space is a contact manifold, we show how to enrich this space with a number of geometric structures by imposing further nonlinear restrictions on the saucer's velocity. These restrictions define certain `manoeuvres' of the saucer, which we call `attacking,' `landing,' or `G2 mode' manoeuvres, and which equip its configuration space with three kinds of flat parabolic geometry in five dimensions. The attacking manoeuvre corresponds to the flat Legendrean contact structure, the landing manoeuvre corresponds to the flat hypersurface type CR structure with Levi form of signature (1,1), and the most complicated G2 manoeuvre corresponds to the contact Engel structure with split real form of the exceptional Lie group G2 as its symmetries. A celebrated double fibration relating the two nonequivalent flat 5-dimensional parabolic G2 geometries is used to construct a `G2 joystick,' consisting of two balls of radii in ratio 1:3 that transforms the difficult G2 manoeuvre into the pilot's action of rolling one of joystick's balls on the other without slipping nor twisting.

math.DG↗
arXiv2018

Aerodynamics of flying saucers

We identify various structures on the configuration space C of a flying saucer, moving in a three-dimensional smooth manifold M. Always C is a five-dimensional contact manifold. If M has a projective structure, then C is its twistor space and is equipped with an almost contact Legendrean structure. Instead, if M has a conformal structure, then the saucer moves according to a CR structure on C. With yet another structure on M, the contact distribution in C is equipped with a cone over a twisted cubic. This defines a certain type of Cartan geometry on C (more specifically, a type of `parabolic geometry') and we provide examples when this geometry is `flat,' meaning that its symmetries comprise the split form of the exceptional Lie algebra G2.

math.DG↗
arXiv2018

The geometry of marked contact Engel structures

A contact twisted cubic structure (M,C,S) is a 5-dimensional manifold M together with a contact distribution C and a bundle S of twisted cubics that is compatible with the conformal symplectic form on C. In Engel's classical work, the Lie algebra of the exceptional Lie group G_2 was realized as the symmetry algebra of the most symmetric contact twisted cubic structure; we thus refer to this one as the contact Engel structure. In the present paper we equip the contact Engel structure with a smooth section s: M-> S that `marks' a point in each twisted cubic. We study the local geometry of the resulting structures (M,C,S,s), which we call marked contact Engel structures. Equivalently, our study can be viewed as a study of foliations of M by curves whose tangent directions are everywhere contained in S. We provide a complete set of local invariants of marked contact Engel structures, we classify all homogeneous models with symmetry groups of dimension greater than 5, and we prove an analogue of the classical Kerr theorem from relativity.

math.DG↗
arXiv2017

Conformal Walker metrics and linear Fefferman-Graham equations

The conformal Fefferman-Graham ambient metric construction is one of the most fundamental constructions in conformal geometry. It embeds a manifold with a conformal structure into a pseudo-Riemannian manifold whose Ricci tensor vanishes up to a certain order along the original manifold. Despite the general existence result of such ambient metrics by Fefferman and Graham, not many examples of conformal structures with Ricci-flat ambient metrics are known. Motivated by previous examples, for which the Fefferman-Graham equations for the ambient metric to be Ricci-flat reduce to a system of linear PDEs, in the present article we develop a method to find ambient metrics for conformal classes of metrics with two-step nilpotent Schouten tensor. Using this method, for metrics for which the image of the Schouten tensor is invariant under parallel transport, i.e., certain types of Walker metrics, we obtain explicit ambient metrics. This includes certain left-invariant Walker metrics as well as pp-waves.

math.DG↗
arXiv2016

How the green light was given for gravitational wave search

The recent detection of gravitational waves by the LIGO/VIRGO team is an incredibly impressive achievement of experimental physics. It is also a tremendous success of the theory of General Relativity. It confirms the existence of black holes; shows that binary black holes exist; that they may collide and that during the merging process gravitational waves are produced. These are all predictions of General Relativity theory in its fully nonlinear regime. The existence of gravitational waves was predicted by Albert Einstein in 1916 within the framework of linearized Einstein theory. Contrary to common belief, even the very \emph{definition} of a gravitational wave in the fully nonlinear Einstein theory was provided only after Einstein's death. Actually, Einstein had arguments against the existence of nonlinear gravitational waves (they were erroneous but he did not accept this), which virtually stopped development of the subject until the mid 1950s. This is what we refer to as the \emph{Red Light} for gravitational waves research. In the following years, the theme was picked up again and studied vigorously by various experts, mainly Herman Bondi, Felix Pirani, Ivor Robinson and Andrzej Trautman, where the theoretical obstacles concerning gravitational wave existence were successfully overcome, thus giving the `Green Light' for experimentalists to start designing detectors, culminating in the recent LIGO/VIRGO discovery. In this note we tell the story of this theoretical breakthrough. Particular attention is given to the fundamental 1958 papers of Trautman, which seem to be lesser known outside the circle of General Relativity experts. A more detailed technical description of these 2 papers is given in the Appendix.

physics.hist-ph↗