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Ioannis P. Zois

Publications and source records attributed to Ioannis P. Zois.

17 recordsLinked to original sources
arXiv2026

The Noncommutative Foliation Invariant (NCFI): extension to the odd codimension case and computed examples

This article extends the definition of the Noncommutative Foliation Invariant (NCFI) for foliations of odd codimension and computes certain key examples for both even and odd codimension cases: fibrational foliations, irrational Kronecker, rational Kronecker (both the vertical and the horzontal foliation using a flat connection) and weighted Hopf/orbifold cases. We also prove some more general results along the way. More concretely, we compute the Noncommutative Foliation Invariant (NCFI) for several basic families of foliated manifolds. In even codimension the invariant is the Chern--Connes pairing of Connes' transverse fundamental cyclic cocycle with the \(K_0\)-class associated with the transverse geometric module. In odd codimension the transverse cocycle has odd degree, so a numerical pairing also requires a specified odd \(K_1\)-class; the odd-codimensional value is therefore an invariant of the foliation together with this chosen odd-favourable structure.

math.OA↗
arXiv2026

Cyclic source pairings for Penrose--Sparling non-Hausdorff twistor spaces

We introduce noncommutative geometry techniques in order to reinterpret the Penrose--Sparling non-Hausdorff twistor space of the anti-self-dual Coulomb field by means of an explicit etale gluing groupoid and its convolution algebra. This algebraic model keeps track of both the identified open part and the two non-separated copies of the source quadric. We compute two kinds of Chern--Connes pairings. The strict tangent-module analogue, obtained from \([T_{\mathbb R}\CP^3\otimes\C]\), vanishes because \[ \operatorname{ch}_3(T^{1,0}\CP^3)+ \operatorname{ch}_3(T^{0,1}\CP^3)=0. \] By contrast, the Penrose--Sparling Coulomb line bundle \(\calC_n\) defines a \(K_0(A_Q)\)-class, and the relative cyclic cocycle supported on the two non-separated copies of a ruling line \(L\subset Q\) gives \[ \mathcal Q(\calC_n)= \frac12\left\langle φ_L^+-φ_L^-,[\calC_n]\right\rangle=n. \] Thus the source-adapted cyclic pairing recovers the Coulomb charge. We also formulate the non-abelian version in principal-bundle language. For a connected complex reductive group \(G\), a maximal torus \(T\subset G\), and a cocharacter \(λ:\C^*\to T\), the source is a principal \(G\)-bundle modification of type \(λ\) along \(Q\).

math.DG↗
arXiv2026

A Boundary--Residue Incidence Coalgebra for Associahedral Scattering Forms

We introduce a boundary--residue incidence coalgebra associated with the face poset of a positive geometry and apply it to associahedral scattering forms. The construction is motivated by the analogy between the Connes--Kreimer coproduct on Feynman graphs and the recursive residue structure of canonical forms. For the Stasheff associahedron \(K_n\), whose faces are indexed by non-crossing dissections of an \((n+1)\)-gon, we prove that the incidence coproduct records all intermediate nested planar factorisation channels of the corresponding tree-level scalar amplitude. The residue of the canonical form on a face labelled by a dissection factorises as the exterior product of canonical forms on the lower associahedra associated with the resulting subpolygons. We illustrate the construction explicitly for the pentagon associahedron \(K_4\), corresponding to the five-point planar scalar amplitude. We then formulate a loop-level extension: whenever a planar loop integrand is represented by a positive geometry, the associahedral face poset is replaced by the boundary poset of the corresponding loop geometry. The one-loop halohedron gives a concrete scalar example, while in the non-planar case we define the associated incidence coalgebra at the level of logarithmic singularity strata. Finally, we compare the boundary--residue coalgebra with the cellular incidence coalgebra of a triangulated or regular CW spacetime. The face poset of a finite regular CW complex reconstructs its barycentric subdivision, and hence its underlying polyhedron, while in positive geometry the same incidence mechanism organises canonical-form residues. This yields an incidence-first bridge between cellular spacetime topology and positive-geometric amplitude factorisation, without assuming that metric or causal data are determined by topology.

math-ph↗
arXiv2016

Quantum Metrology: Towards an alternative definition for the meter

The motivation for this article came from an attempt to give an alternative definition for the meter, the SI unit for measuring length. As a starting point towards this goal, in this piece of work we present the underlying theory behind our approach which uses ideas from quantum field theory and noncommutative geometry, in particular the notion of an odd K-cycle which is based on the Dirac operator (and its inverse, the Dirac propagator). Using (the perhaps more familiar) physics terminology, the key point in our strategy is this: instead of measuring length directly in space-time we measure the "algebraic (spectral) length" in the space of the corresponding quantum states of some particle (fermion) acted upon by the Dirac propagator. This approach shares the spirit of the unanimus vote of the 24th General Conference of Standards and Measures (21st October 2011) in Serves, France for the redefinition of the fundamental units using Planck's constant.

math-ph↗
arXiv2014

Noncommutative Topological Quantum Field Theory, Noncommutative Floer Homology, Noncommutative Hodge Theory

We present some ideas for a possible Noncommutative Topological Quantum Field Theory (NCTQFT) and Noncommutative Floer Homology (NCFH). Our motivation is two-fold and it comes both from physics and mathematics: On the one hand we argue that NCTQFT is the correct mathematical framework for a quantum field theory of all known interactions in nature (including gravity). On the other hand we hope that a possible NCFH will apply to practically every 3-manifold (and not only to homology 3-spheres as ordinary Floer Homology currently does). The two motivations are closely related since, at least in the commutative case, Floer Homology Groups constitute the space of quantum observables of (3+1)-dim Topological Quantum Field Theory. Towards this goal we present some "Noncommutative" Versions of Hodge Theory for noncommutative differentail forms and tangential cohomology for foliations.

math-ph↗
arXiv2013

Solar Activity and Transformer Failures in the Greek National Electric Grid

We study both the short term and long term effects of solar activity on the large transformers (150kV and 400kV) of the Greek national electric grid. We use data analysis and various analytic and statistical methods and models. Contrary to the common belief in PPC Greece, we see that there are considerable both short term (immediate) and long term effects of solar activity onto large transformers in a mid-latitude country (latitude approx. 35 - 41 degrees North) like Greece. Our results can be summarized as follows: For the short term effects: During 1989-2010 there were 43 stormy days (namely days with for example Ap larger or equal to 100) and we had 19 failures occurring during a stormy day plus or minus 3 days and 51 failures occurring during a stormy day plus or minus 7 days. All these failures can be directly related to Geomagnetically Induced Currents (GICs). Explicit cases are presented. For the long term effects we have two main results: The annual transformer failure number for the period of study 1989-2010 follows the solar activity pattern (11 year periodicity, bell-shaped graph). Yet the maximum number of transformer failures occur 3-4 years after the maximum of solar activity. There is statistical correlation between solar activity expressed using various newly defined long term solar activity indices and the annual number of transformer failures. These new long term solar activity indices were defined using both local (from geomagnetic stations in Greece) and global (planetary averages) geomagnetic data. Applying both linear and non-linear statistical regression we compute the regression equations and the corresponding coefficients of determination.

physics.data-an↗
arXiv2010

18 Lectures on K-Theory

We present 18 Introductory Lectures on K-Theory covering its basic three branches, namely topological, analytic (K-Homology) and Higher Algebraic K-Theory, 6 lectures on each branch. The skeleton of these notes was provided by the author's personal notes from a graduate summer school on K-Theory organised by the London Mathematical Society (LMS) back in 1995 in Lancaster, UK.

math.KT↗
arXiv2008

The Mobius Band and the Mobius Foliation

This article presents some computations for a new topological invariant for foliations introduced some years ago by the author using techniques from noncommutative geometry, in particular the pairing between K-Theory and cyclic cohomology. The motivation came from theoretical physics, more specifically from flat vector bundles, in physics terminology from potentials which are pure gauge.

math-ph↗
arXiv2001

Black Hole Entropy, Topological Entropy and the Baum-Connes Conjecture in K-Theory

We shall try to exhibit a relation between black hole entropy and topological entropy using the famous Baum-Connes conjecture for foliated manifolds which are particular examples of noncommutative spaces. Our argument is qualitative and it is based on the microscopic origin of the Beckenstein-Hawking area-entropy formula for black holes, provided by superstring theory, in the more general noncommutative geometric context of M-Theory following the Connes- Douglas-Schwarz article.

math-ph↗
arXiv2001

Black Hole Entropy, Topological Entropy and Noncommutative Geometry

Foliated manifolds are particular examples of noncommutative spaces. In this article we try to give a qualitative description of the Godbillon-Vey class and its relation on the one hand to the holonomy and on the other hand to the topological entropy of a foliation, using a remarkable theorem proved recently by G. Duminy relating these three notions in the case of codim-1 foliations. Moreover we shall investigate its possible relation with the black hole entropy adopting the superstring theory origin of the black hole entropy in the extremal case. This situation we believe has some striking similarities with the explanation due to Bellissard of the integrality of the Hall conductivity in the quantum Hall effect. Our starting point is the Connes-Douglas-Schwarz article on compactifications of matrix models to noncommutative tori.

hep-th↗
arXiv2000

Multiple "parallel" D-branes seen as leaves of foliations and Duminy's theorem

We try to give a qualitative description of the Godbillon-Vey class and its relation on the one hand to the holonomy and on the other hand to the topological entropy of a foliation, using a remarkable theorem proved recently by G. Duminy (which still remains unpublished), relating these three notions in the case of codim-1 foliations. Moreover we shall investigate its possible consequences on string theory. In particular we shall present a conceptual argument according to which the curvature of the B-field (rank two antisymmetric tensor) of open strings might be related to the Godbillon-Vey class using a suitable generalisation of ``Non-Abelian Geometry'' which has just appeared in physics literature. Our starting point again is the Connes-Douglas-Schwarz article on compactifications of matrix models to noncommutative tori.

hep-th↗
arXiv2000

The Godbillon-Vey class, invariants of manifolds and linearised M-Theory

We apply the Godbillon-Vey class to compute the transition amplitudes between some non-commutative solitons in M-Theory; our context is that of Connes-Douglas-Schwarz where they considered compactifications of matrix models on noncommutative tori. Two important consequences follow: we describe a new normalisation for the Abelian Chern-Simons theory using symplectic 4-manifolds as providing cobordisms for tight contact 3-manifolds and we construct a new(?) invariant for 3-manifolds. Moreover we modify the topological Lagrangian density suggested for M-Theory in a previous article to a \textsl{quadratic} one using the fact that the \emph{functor of immersions is a linearisation (or ``the differential'') of the functor of embeddings}

hep-th↗
arXiv1999

A new invariant for sigma models

We introduce a new invariant for foliations using the even pairing between K-homology and cyclic homology and discuss some possible applications in physics.

hep-th↗