Abstract We describe the lower algebraic K -theory of the integral group ring of both the pure and full braid groups of the real projective plane $$\mathbb {R}P^2$$ R P 2 with 3 strings, as well as that of the integral group ring of the mapping class group of $$\mathbb {R}P^2$$ R P 2 with 3 marked points. In addition, we give a general formula for the algebraic K -theory groups of the group ring of the mapping class group of non-orientable surfaces with k marked points, where $$k\ge 3$$ k ≥ 3 .

In this study, we explore the geometric construction of the Klein bottle and the real projective plane (RP^{2}) within the framework of tensor networks, focusing on the implementation of crosscap and rainbow boundaries. Previous investigations have applied boundary matrix product state techniques to study these boundaries. We introduce an approach that incorporates such boundaries into the tensor renormalization group methodology, facilitated by an efficient representation of a spatial reflection operator. This advancement enables us to compute the crosscap and rainbow free energy terms and the one-point function on RP^{2} with enhanced efficiency and for larger system sizes. Additionally, our method is capable of calculating the partition function under isotropic conditions of space and imaginary time. The versatility of this approach is further underscored by its applicability to constructing other (non)orientable surfaces of higher genus.

Free gases of spinless fermions moving on a lattice-symmetric geometric background are considered. Their topological properties at zero temperature can be used to classify their Fermi seas and associated boundaries. The flat orbifolds Rd/Γ, where Γ is the crystallographic group of symmetry in d-dimensional momentum space, are used to accomplish this task. Two topological classes exist for d=1: an interval, which is identified as a conductor, and a circumference, which corresponds to an insulator. The number of topological classes increases to 17 for d=2: 8 have the topology of a disk, that are generally recognized as conductors, and 4 correspond to a two-sphere, matching insulators. Both sets eventually contain a finite number of conical singularities and reflection corners at the boundaries. The remaining cases in the listing relate to conductors (annulus, Möbius strip) and insulators (two-torus, real projective plane, Klein bottle). Examples that fall under this list are given, along with physical interpretations of the singularities. It is anticipated that the findings of this classification will be robust under perturbative interactions due to its topological character.

The study of the fractal theory in Euclidean spaces has recently emerged as an intriguing research area. The concept of fractal interpolation yields a method to approximate functions that are both self-affine or non-self-affine and consequently allows substantial flexibility and diversity of the fractal modeling problem. In this article, we introduce non-affine fractal functions on the non-Euclidean real projective plane. To do so, a real projective plane with a linear structure is considered. Then we study some classical approximation results for it. After considering a suitable iterated functions system (IFS) on the real projective plane, we construct non-affine fractal functions on it. Some fractal versions of classical approximation results are proved for the projective plane. Moreover, we prove that the attractor of an IFS on the dual space of the real projective plane is also the graph of a fractal function.

Abstract We give a sharp lower bound for energy in homotopy classes of maps from real projective space to Riemannian manifolds, together with an upper bound for the infimum of the energy in such a homotopy class. We characterize the maps attaining this lower bound for energy, and we explain how the infimum of the energy in a homotopy class of maps of real projective ‐space is determined by an associated class of maps of the real projective plane.

We compute the p p -widths, { ω p } \{\omega _p\} , for the real projective plane with the standard metric.

Abstract Two-dimensional conformal field theories (CFTs) defined on non-orientable Riemann surfaces obey consistency Cardy conditions analogous to those in the orientable case. We revisit those conditions for irrational theories with central charge c > 1 in the context of two-point functions of primaries on the Real Projective plane $$ {\mathbbm{RP}}^2 $$ RP 2 and the partition function on the Klein bottle $$ {\mathbbm{K}}^2 $$ K 2 . Using the irrational versions of the Virasoro fusion and modular kernels we derive universal expressions for the non-orientable CFT data at large conformal dimension, assuming a gap in the spectrum of scalar primaries. In particular, we derive asymptotic formulas at finite central charge for the averaged Light-Light-Heavy product C LLH × Γ H of OPE coefficients with the $$ {\mathbbm{RP}}^2 $$ RP 2 one-point function normalizations, as well as for the parity-weighted density of heavy scalar primaries (or equivalently the density of heavy $$ {\Gamma}_H^2 $$ Γ H 2 ). We discuss the gravitational interpretation of the results.

Superstring scattering from orientifold planes requires considering string amplitudes on world-sheets with crosscaps with the lowest order case (in string coupling constant) having the topology of the real projective plane. While amplitudes on the latter have been formulated for the trivial one- and two-point cases in this work we go beyond these cases thereby solving various technicalities. The latter include reducing the complex world-sheet integration of closed string insertions over the real projective plane to pure real open string integrals. As a result we find that scattering of n closed strings on the real projective plane can be expressed in terms of disk amplitudes involving 2n open strings. In this work we explicitly work out in pure spinor formalism the case n = 3 which can be written as a linear combination of two (gauge-invariant) six open string amplitudes. We also present the low-energy expansion of this result necessary to construct closed string couplings on orientifold planes.

Abstract We study curvatures of the groups of measure-preserving diffeomorphisms of non-orientable compact surfaces. For the cases of the Klein bottle and the real projective plane we compute curvatures, their asymptotics and the normalised Ricci curvatures in many directions. Extending the approach of Arnold and Lukatskii we provide estimates of weather unpredictability for natural models of trade wind currents on the Klein bottle and the projective plane.

Abstract Five observations compose the main results of this note. The first records the existence of a smoothly embedded 2‐sphere inside such that performing a Gluck twist on produces a manifold that is homeomorphic but not diffeomorphic to the total space of the nontrivial 2‐sphere bundle over the real projective plane . The second observation is that there is a 5‐dimensional cobordism with a single 2‐handle between the 4‐manifold and a mapping torus that was used by Cappell–Shaneson to construct an exotic . This construction of is similar to the one of the Cappell–Shaneson homotopy 4‐spheres. The third observation is that twisting an embedded real projective plane inside produces a manifold that is homeomorphic but not diffeomorphic to the circle sum of two copies of . The fourth observation records new examples of pairs of homeomorphic but not diffeomorphic closed 4‐manifolds with Euler characteristic one. These include the total space of the nontrivial ‐bundle over . Knotting phenomena of 2‐spheres in nonorientable 4‐manifolds that stands in glaring contrast with known phenomena in the orientable domain is pointed out in the fifth observation.

Diffusion MRI (dMRI) is an imaging technique widely used in neuroimaging research, where the signal carries directional information of underlying neuronal fibres based on the diffusivity of water molecules. One of the shortcomings of dMRI is that numerous images, sampled at gradient directions on a sphere, must be acquired to achieve a reliable angular resolution for model-fitting, which translates to longer scan times, higher costs, and barriers to clinical adoption. In this work we introduce gauge equivariant convolutional neural network (gCNN) layers for dMRI that overcome the challenges associated with the signal being acquired on a sphere with antipodal points identified. This is done by noting that the domain is equivalent to the real projective plane, , which is a non-euclidean and a non-orientable manifold. This is in stark contrast to a rectangular grid which typical convolutional neural networks (CNNs) are designed for. We apply our method to upsample angular resolution for predicting diffusion tensor imaging (DTI) parameters from just six diffusion gradient directions. The symmetries introduced allow gCNNs the ability to train with fewer subjects as compared to a baseline model that involves only 3D convolutions.

We propose a holographic dual for 2D CFT defined on closed non-orientable manifolds, such as the real projective plane ℝℙ2 and the Klein bottle \U0001d5422. Such CFT can be constructed by introducing antipodally identified cuttings, i.e. crosscaps, to a sphere and hence called crosscap CFT (XCFT). The gravity dual is AdS3 spacetime with dS2 end-of-the-world branes. In particular, the Lorentzian spacetime with a global dS2 brane is dual to the unitary time evolution of a crosscap state in CFT, post-selected on the CFT ground state. We compute the holographic ℝℙ2 partition function (or the p-function), one-point function, and \U0001d5422 partition function, and see that they successfully reproduce the XCFT results. We also show a holographic p-theorem as an application.

We prove that the median hypersimplex ?2k,k is Minkowski indecomposable, i.e. it cannot be expressed as a non-trivial Minkowski sum ?2k,k = P + Q, where P, ??2k,k, Q. Since ?2k,k is a deformed permutahedron, we obtain as a corollary that ?2k,k represents a ray in the submodular cone (the deformation cone of the permutahedron). Building on the previously developed geometric methods and extensive computer search, we exhibit a twelve vertex, 4-dimensional polytopal realization of the Bier sphere of the hemi-icosahedron, the vertex minimal triangulation of the real projective plane.

We compute the Bieri-Neumann-Strebel invariants Σ 1 for the full and pure braid groups of the sphere S 2 , the real projective plane R P 2 , the torus T and the Klein bottle K . For M = T or M = K , n ≥ 2 , we show that the action by homeomorphisms of Out ( P n ( M ) ) on S ( P n ( M ) ) contains certain permutations, under which Σ 1 ( P n ( M ) ) c is invariant. Furthermore, Σ 1 ( P n ( T ) ) c , and Σ 1 ( P n ( S 2 ) ) c (with n ≥ 5 ) are finite unions of circles, and Σ 1 ( P n ( K ) ) c is finite. This implies the existence of H⏧ Aut ( P n ( K ) ) with | Aut ( P n ( K ) ) : H | < ∞ such that R ( φ ) = ∞ for every φ ∈ H .

Abstract In the standard category of directed graphs, graph morphisms map edges to edges. By allowing graph morphisms to map edges to finite paths (path homomorphisms of graphs), we obtain an ambient category in which we determine subcategories enjoying covariant functors to categories of algebras given by constructions of path algebras, Cohn path algebras, and Leavitt path algebras, respectively. Thus, we obtain new tools to unravel homomorphisms between Leavitt path algebras and between graph C*‐algebras. In particular, a graph‐algebraic presentation of the inclusion of the C*‐algebra of a quantum real projective plane into the Toeplitz algebra allows us to determine a quantum CW‐complex structure of the former. It comes as a mixed‐pullback theorem where two ‐homomorphisms are covariantly induced from path homomorphisms of graphs and the remaining two are contravariantly induced by admissible inclusions of graphs. As a main result and an application of new covariant‐induction tools, we prove such a mixed‐pullback theorem for arbitrary graphs whose all vertex‐simple loops have exits, which substantially enlarges the scope of examples coming from noncommutative topology.

Abstract We show that there is a constant such that any 3‐uniform hypergraph with vertices and at least edges contains a triangulation of the real projective plane as a subgraph. This resolves a conjecture of Kupavskii, Polyanskii, Tomon and Zakharov. Furthermore, our work, combined with prior results, asymptotically determines the Turán number of all surfaces.

We classify birational involutions of the real projective plane up to conjugation. In contrast with an analogous classification over the complex numbers (due to E. Bertini, G. Castelnuovo, F. Enriques, L. Bayle and A. Beauville), which includes four different classes of involutions, we discover 12 different classes over the reals, and provide many examples when the fixed curve of an involution does not determine its conjugacy class in the real plane Cremona group.

Abstract For , let be a two‐dimensional algebraically defined graph, that is, a bipartite graph where each partite set is a copy of and two vertices and are adjacent if and only if . It is known that has girth 6 and can be extended to the point‐line incidence graph of the classical real projective plane. However, it was unknown whether there exists such that has girth 6 and is nonisomorphic to . This paper answers this question affirmatively and thus provides a construction of a nonclassical real projective plane. This paper also studies the diameter and girth of for families of bivariate functions .

Abstract In machine learning and computer graphics, a fundamental task is the approximation of a probability density function through a well‐dispersed collection of samples. Providing a formal metric for measuring the distance between probability measures on general spaces, Optimal Transport (OT) emerges as a pivotal theoretical framework within this context. However, the associated computational burden is prohibitive in most real‐world scenarios. Leveraging the simple structure of OT in 1D, Sliced Optimal Transport (SOT) has appeared as an efficient alternative to generate samples in Euclidean spaces. This paper pushes the boundaries of SOT utilization in computational geometry problems by extending its application to sample densities residing on more diverse mathematical domains, including the spherical space 𝕊d, the hyperbolic plane ℍd, and the real projective plane ℙd. Moreover, it ensures the quality of these samples by achieving a blue noise characteristic, regardless of the dimensionality involved. The robustness of our approach is highlighted through its application to various geometry processing tasks, such as the intrinsic blue noise sampling of meshes, as well as the sampling of directions and rotations. These applications collectively underscore the efficacy of our methodology.

Given a cubic K in the real projective plane. Then for each point P there is a conic CP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{{P}}$$\\end{document} associated to P. The conic CP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{{P}}$$\\end{document} is called the polar conic of K with respect to the pole P. We investigate the situation when three conics C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_1$$\\end{document}, C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_2$$\\end{document}, and C3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_3$$\\end{document} are polar conics of K with respect to the poles P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_1$$\\end{document}, P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_2$$\\end{document}, and P3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_3$$\\end{document}, respectively. In particular, we give an elementary proof—without using any results from algebraic geometry—that any three conics C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_1$$\\end{document}, C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_2$$\\end{document}, C3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_3$$\\end{document} in general position, satisfying only a non-degeneracy condition, determine a unique cubic K and three points P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_1$$\\end{document}, P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_2$$\\end{document}, P3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_3$$\\end{document}, such that C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_1$$\\end{document}, C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_2$$\\end{document}, C3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_3$$\\end{document} are polar conics of K with respect to the three poles P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_1$$\\end{document}, P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_2$$\\end{document}, P3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_3$$\\end{document}. This can be seen as a higher degree variant of von Staudt’s Theorem.