Geometric Realization Research Articles

Geometrical realization of low-dimensional complete intersections

This paper aims to give some examples of diffeomorphic (or homeomorphic) low-dimensional complete intersections, which can be considered as a geometrical realization of classification theorems about complete intersections. A conjecture of Libgober and Wood (1982) will be confirmed by one of the examples.

Directed Homology Theories and Eilenberg-Steenrod Axioms

In this paper, we define and study a homology theory, that we call “natural homology”, which associates a natural system of abelian groups to every space in a large class of directed spaces and precubical sets. We show that this homology theory enjoys many important properties, as an invariant for directed homotopy. Among its properties, we show that subdivided precubical sets have the same homology type as the original ones ; similarly, the natural homology of a precubical set is of the same type as the natural homology of its geometric realization. By same type we mean equivalent up to some form of bisimulation, that we define using the notion of open map. Last but not least, natural homology, for the class of spaces we consider, exhibits very important properties such as Hurewicz theorems, and most of Eilenberg-Steenrod axioms, in particular the dimension, homotopy, additivity and exactness axioms. This last axiom is studied in a general framework of (generalized) exact sequences.

Topological substitution for the aperiodic Rauzy fractal tiling

We consider two families of planar self-similar tilings of different nature: the tilings consisting of translated copies of the fractal sets defined by an iterated function system, and the tilings obtained as a geometrical realization of a topological substitution (an object of purely combinatorial nature). We establish a link between the two families in a specific case, by defining an explicit topological substitution and by proving that it generates the same tilings as those associated with the Tribonacci Rauzy fractal.

The cotriple resolution of differential graded algebras

We consider the cotriple resolution of algebras over operads in differential graded modules. We focus, to be more precise, on the example of algebras over the differential graded Barratt-Eccles operad and on the example of commutative algebras. We prove that the geometric realization of the cotriple resolution (in the sense of model categories) gives a cofibrant resolution functor on these categories of differential graded algebras.

Toric matrix Schubert varieties and their polytopes

Given a matrix Schubert variety $\overline{X_\pi}$, it can be written as $\overline{X_\pi}=Y_\pi\times \mathbb{C}^q$ (where $q$ is maximal possible). We characterize when $Y_{\pi}$ is toric (with respect to a $(\mathbb{C}^*)^{2n-1}$-action) and study the associated polytope $\Phi(\mathbb{P}(Y_\pi))$ of its projectivization. We construct regular triangulations of $\Phi(\mathbb{P}(Y_\pi))$ which we show are geometric realizations of a family of subword complexes. Subword complexes were introduced by Knutson and Miller in 2004, who also showed that they are homeomorphic to balls or spheres and raised the question of their polytopal realizations.

On the relation between continuous and combinatorial

Axiomatic Cohesion proposes that the contrast between cohesion and non-cohesion may be expressed by means of a geometric morphism $p :\mathcal{E} \to \mathcal {S}$ (between toposes) with certain special properties that allow to effectively use the intuition that the objects of $\mathcal{E}$ are `spaces' and those of $\mathcal{S}$ are `sets'. Such geometric morphisms are called (pre-)cohesive. We may also say that $\mathcal{E}$ is pre-cohesive (over $\mathcal{S}$). In this case, the topos $\mathcal{E}$ determines an $\mathcal{S}$-enriched `homotopy' category. The purpose of the present paper is to study certain aspects of this homotopy theory. We introduce weakly Kan objects in a pre-cohesive topos, which are analogous to Kan complexes in the topos of simplicial sets. Also, given a geometric morphism $g:\mathcal{F} \to\mathcal{E}$ between pre-cohesive toposes $\mathcal{F}$ and $\mathcal{E}$ (over the same base), we define what it means for $g$ to preserve pieces. We prove that if $g$ preserves pieces then it induces an adjunction between the homotopy categories determined by $\mathcal{F}$ and $\mathcal{E}$, and that the direct image $g_*:\mathcal{F}\to \mathcal{E}$ preserves weakly Kan objects. These and other results support the intuition that the inverse image of $g$ is `geometric realization'. In particular, since Kan complexes are weakly Kan in the pre-cohesive topos of simplicial sets, the result relating $g$ and weakly Kan objects is analogous to the fact that the singular complex of a space is a Kan complex.

Coordinated Robot Navigation via Hierarchical Clustering

We introduce the use of hierarchical clustering for relaxed deterministic coordination and control of multiple robots. Traditionally, an unsupervised learning method, hierarchical clustering offers a formalism for identifying and representing spatially cohesive and segregated robot groups at different resolutions by relating the continuous space of configurations to the combinatorial space of trees. We formalize and exploit this relation, developing computationally effective reactive algorithms for navigating through the combinatorial space in concert with geometric realizations for a particular choice of the hierarchical clustering method. These constructions yield computationally effective vector field planners for both hierarchically invariant as well as transitional navigation in the configuration space. We apply these methods to the centralized coordination and control of $n$ perfectly sensed and actuated Euclidean spheres in a $d$ -dimensional ambient space (for arbitrary $n$ and $d$ ). Given a desired configuration supporting a desired hierarchy, we construct a hybrid controller that is quadratic in $n$ and algebraic in $d$ and prove that its execution brings all but a measure zero set of initial configurations to the desired goal, with the guarantee of no collisions along the way.

M2-brane surface operators and gauge theory dualities in Toda

We give a microscopic two dimensional ${\cal N}=(2,2)$ gauge theory description of arbitrary M2-branes ending on $N_f$ M5-branes wrapping a punctured Riemann surface. These realize surface operators in four dimensional ${\cal N}=2$ field theories. We show that the expectation value of these surface operators on the sphere is captured by a Toda CFT correlation function in the presence of an additional degenerate vertex operator labelled by a representation ${\cal R}$ of $SU(N_f)$, which also labels M2-branes ending on M5-branes. We prove that symmetries of Toda CFT correlators provide a geometric realization of dualities between two dimensional gauge theories, including ${\cal N}=(2,2)$ analogues of Seiberg and Kutasov--Schwimmer dualities. As a bonus, we find new explicit conformal blocks, braiding matrices, and fusion rules in Toda CFT.

Symmetries on the Lattice of k-Bounded Partitions

In 2002, Suter [25] identified a dihedral symmetry on certain order ideals in Young’s lattice and gave a combinatorial action on the partitions in these order ideals. Viewing this result geometrically, the order ideals can be seen to be in bijection with the alcoves in a 2- fold dilation in the geometric realization of the affine symmetric group. By considering the m-fold dilation we observe a larger set of order ideals in the k-bounded partition lattice that was considered by Lapointe, Lascoux, and Morse [14] in the study of k-Schur functions. We identify the order ideal and the cyclic action on it explicitly in a geometric and combinatorial form.

Band spectrum is D-brane

We show that band spectrum of topological insulators can be identified as the shape of D-branes in string theory. The identification is based on a relation between the Berry connection associated with the band structure and the ADHM/Nahm construction of solitons whose geometric realization is available with D-branes. We also show that chiral and helical edge states are identified as D-branes representing a noncommutative monopole.

3D Printing of Buildings and Building Components as the Future of Sustainable Construction?

The paper presents the state-of-the-art concerning the current achievements in the field of 3D printing of buildings and building components. The 3D printing technologies, comparing to traditional techniques of constructing the buildings, could be considered as environmental friendly derivative giving almost unlimited possibilities for geometric complexity realizations. Two kinds of technologies were described in this paper with pointing to Contour Crafting as a promising technique that may be able to revolutionize construction industry in near future. Numerous advantages of this technology, such as reduction of the costs and time, minimizing the pollution of environment and decrease of injuries and fatalities on construction sites could be cited. Despite many advantages and hopes, some concerns are summarized in the conclusions, as the technology still has many limitations. A brief description of few examples of pioneering usage of 3D printing in construction industry are presented (Canal House in Amsterdam, WinSun company and printing application for building carried out by Skanska company). Creating a model that will be appropriate for 3D printers is possible in many different modelling programs. One of the most popular formats for sharing such models is STL format. In the paper sample models crated in Autodesk Inventor are shown, but also other tools suitable for preparing models for 3D printing are briefly discussed.

Order complex of ideals in a commutative ring with identity

Order complex is an important object associated to a partially ordered set. Following a suggestion from V. A. Vassiliev (1994), we investigate an order complex associated to the partially ordered set of nontrivial ideals in a commutative ring with identity. We determine the homotopy type of the geometric realization for the order complex associated to a general commutative ring with identity. We show that this complex is contractible except for semilocal rings with trivial Jacobson radical when it is homotopy equivalent to a sphere.

Geometric point-circle pentagonal geometries from Moore graphs

We construct isometric point-circle configurations on surfaces from uniform maps. This gives one geometric realisation in terms of points and circles of the Desargues configuration in the real projective plane, and three distinct geometric realisations of the pentagonal geometry with seven points on each line and seven lines through each point on three distinct dianalytic surfaces of genus 57. We also give a geometric realisation of the latter pentagonal geometry in terms of points and hyperspheres in 24 dimensional Euclidean space. From these, we also obtain geometric realisations in terms of points and circles (or hyperspheres) of pentagonal geometries with k circles (hyperspheres) through each point and k-1 points on each circle (hypersphere).

Froggatt-Nielsen meets Mordell-Weil: a phenomenological survey of global F-theory GUTs with U(1)s

In F-theory, U(1) gauge symmetries are encoded in rational sections, which generate the Mordell-Weil group of the elliptic fibration of the compactification space. Recently the possible U(1) charges for global SU(5) F-theory GUTs with smooth rational sections were classified [1]. In this paper we utilize this classification to probe global F-theory models for their phenomenological viability. After imposing an exotic-free MSSM spectrum, anomaly cancellation (related to hypercharge flux GUT breaking in the presence of U(1) gauge symmetries), absence of dimension four and five proton decay operators and other R-parity violating couplings, and the presence of at least the third generation top Yukawa coupling, we generate the remaining quark and lepton Yukawa textures by a Froggatt-Nielsen mechanism. In this process we require that the dangerous couplings are forbidden at leading order, and when re-generated by singlet vevs, lie within the experimental bounds. We scan over all possible configurations, and show that only a small class of U(1) charge assignments and matter distributions satisfy all the requirements. The solutions give rise to the exact MSSM spectrum with realistic quark and lepton Yukawa textures, which are consistent with the CKM and PMNS mixing matrices. We also discuss the geometric realization of these models, and provide pointers to the class of elliptic fibrations with good phenomenological properties.

Classification of discrete series by minimal 𝐾-type

Following the proof by Hecht and Schmid of Blattner’s conjecture forKKmultiplicities of representations belonging to the discrete series it turned out that some results which were earlier known with some hypothesis on the Harish-Chandra parameter of the discrete series representation could be extended removing those hypotheses. For example this was so for the geometric realization problem. Occasionally a few other results followed by first proving them for Harish-Chandra parameters which are sufficiently regular and then using Zuckerman translation functors, wall crossing methods, etc. Recently, Hongyu He raised the question (private communication) of whether the characterization of a discrete series representation by its lowestKK-type, which was proved by this author and R. Hotta with some hypothesis on the Harish-Chandra parameter of the discrete series representations, can be extended toalldiscrete series representations excluding none, using a combination of these powerful techniques. In this article we will answer this question using Dirac operator methods and a result of Susana Salamanca-Riba.

Danzer’s configuration revisited

Abstract We revisit the configuration DCD(4) of Danzer, a great inspiration for our work. This configuration of type (354) falls into an in_nite series of geometric point-line configurations DCD(n). Each DCD(n) is characterized combinatorially by having the Kronecker cover over the Odd graph On as its Levi graph. Danzer’s configuration is deeply rooted in Pascal’s Hexagrammum Mysticum. Although the combinatorial configuration is highly symmetric, we conjecture that there are no geometric point-line realizations with 7- or 5-fold rotational symmetry; on the other hand, we found a point-circle realization having the symmetry group D7, the dihedral group of order 14.

Geometric realizations of the basic representation of $$\widehat{\mathfrak {gl}}_r$$ gl ^ r

The realizations of the basic representation of \(\widehat{\mathfrak {gl}}_r\) are known to be parametrized by partitions of r and have an explicit description in terms of vertex operators on the bosonic/fermionic Fock space. In this paper, we give a geometric interpretation of these realizations in terms of geometric operators acting on the equivariant cohomology of certain Nakajima quiver varieties.

On two geometric realizations of an affine Hecke algebra

The article is a contribution to the local theory of geometric Langlands duality. The main result is a categorification of the isomorphism between the (extended) affine Hecke algebra associated to a reductive group $G$ and Grothendieck group of equivariant coherent sheaves on Steinberg variety of Langlands dual group ${G\check { }}$ ; this isomorphism due to Kazhdan–Lusztig and Ginzburg is a key step in the proof of tamely ramified local Langlands conjectures. The paper is a continuation of the author’s joint work with Arkhipov, it relies on the technical material developed in a joint work with Yun.

Reconstruction of super-resolution ocean <i>p</i>CO<sub>2</sub> and air–sea fluxes of CO<sub>2</sub> from satellite imagery in the southeastern Atlantic

Abstract. An accurate quantification of the role of the ocean as source/sink of greenhouse gases (GHGs) requires to access the high-resolution of the GHG air–sea flux at the interface. In this paper we present a novel method to reconstruct maps of surface ocean partial pressure of CO2 ( pCO2) and air–sea CO2 fluxes at super resolution (4 km, i.e., 1/32° at these latitudes) using sea surface temperature (SST) and ocean color (OC) data at this resolution, and CarbonTracker CO2 fluxes data at low resolution (110 km). Inference of super-resolution pCO2 and air–sea CO2 fluxes is performed using novel nonlinear signal processing methodologies that prove efficient in the context of oceanography. The theoretical background comes from the microcanonical multifractal formalism which unlocks the geometrical determination of cascading properties of physical intensive variables. As a consequence, a multi-resolution analysis performed on the signal of the so-called singularity exponents allows for the correct and near optimal cross-scale inference of GHG fluxes, as the inference suits the geometric realization of the cascade. We apply such a methodology to the study offshore of the Benguela area. The inferred representation of oceanic partial pressure of CO2 improves and enhances the description provided by CarbonTracker, capturing the small-scale variability. We examine different combinations of ocean color and sea surface temperature products in order to increase the number of valid points and the quality of the inferred pCO2 field. The methodology is validated using in situ measurements by means of statistical errors. We find that mean absolute and relative errors in the inferred values of pCO2 with respect to in situ measurements are smaller than for CarbonTracker.

Geometry of 6D RG flows

We study renormalization group flows between six-dimensional superconformal field theories (SCFTs) using their geometric realizations as singular limits of F-theory compactified on elliptically fibered Calabi-Yau threefolds. There are two general types of flows: one corresponds to giving expectation values to scalars in the tensor multiplets (tensor branch flow) realized as resolving the base of the geometry. The other corresponds to giving expectation values to hypermultiplets (Higgs branch flow) realized as complex structure deformations of the geometry. To corroborate this physical picture we calculate the change in the anomaly polynomial for these theories, finding strong evidence for a flow from a UV fixed point to an IR fixed point. Moreover, we find evidence against non-trivial dualities for 6D SCFTs. In addition we find non-trivial RG flows for theories realizing small E 8 instantons on ALE spaces.