Geometrically realizes Khovanov homology for semiadequate links.
problem Computing Khovanov homology for semiadequate links.
method Introducing partial presimplicial sets and their geometric realization.
result Concrete formula for homotopy type of geometric realization.
We study geometric realization questions of curvature in the affine, Riemannian, almost Hermitian, almost para Hermitian, almost hyper Hermitian, almost hyper para Hermitian, Hermitian, and para Hermitian settings. We also express questions in Ivanov-Petrova geometry, Osserman geometry, and curvature homogeneity in ter…
We show that every Kaehler algebraic curvature tensor is geometrically realizable by a Kaehler manifold of constant scalar curvature. We also show that every para-Kaehler algebraic curvature tensor is geometrically realizable by a para-Kaehler manifold of constant scalar curvature
We study the 8 natural GL equivariant geometric realization questions for the space of generalized algebraic curvature tensors. All but one of them is solvable; a non-zero projectively flat Ricci antisymmetric generalized algebraic curvature is not geometrically realizable by a projectively flat Ricci antisymmetric tor…
We show that every Kaehler affine curvature model can be realized geometrically.
We show any Weyl curvature model can be geometrically realized by a Weyl manifold
We realize Stasheff's multiplihedron geometrically as the moduli space of stable quilted disks. This generalizes the geometric realization of the associahedron as the moduli space of stable disks. We show that this moduli space is the non-negative real part of a complex moduli space of stable scaled marked curves.
Two geometric realizations of F(4) Lie superalgebra found.
problem Geometric realization of F(4) Lie superalgebra. method Explicit geometric realizations as supersymmetries of super-PDE systems.
result Two orders of super-PDE systems found for F(4). We show that a para-Hermitian algebraic curvature model satisfies the para-Gray identity if and only if it is geometrically realizable by a para-Hermitian manifold. This requires extending the Tricerri-Vanhecke curvature decomposition to the para-Hermitian setting. Additionally, the geometric realization can be chosen …
New combinatorial framework for geometric realizations of subword complexes.
problem Proving or disproving geometric realizations of subword complexes of Coxeter groups.
method Algebraic combinatorics and discrete geometry framework, parameter matrices.
result Existence of parameter matrices equivalent to realizability of subword complexes as chirotopes.
We define generalized currents associated with immersions of abstract oriented solenoids with a transversal measure. We realize geometrically the full real homology of a compact manifold with these generalized currents, and more precisely with immersions of minimal uniquely ergodic solenoids. This makes precise and geo…
Geometrically realized polyhedra from directed trees, including associahedra.
problem Understanding the structure of associative algebras with co-inner products.
method Geometric realization of polyhedra using directed planar trees.
result These polyhedra, including associahedra, are homeomorphic to balls.
Paper shows braid groups can't be realized geometrically.
problem Cannot geometrically realize braid groups as homeomorphisms.
method Showed non-existence of a section for the projection map.
result Braid groups cannot be realized geometrically.
The paper studies highest weight representations of Lie superalgebras and their geometric realizations.
problem Understanding highest weight representations of Lie superalgebras and their geometric realizations.
method Analyzes representations of Lie superalgebras and their geometric realizations on Hermitian superspaces.
result Discovers geometric realizations of highest weight representations of Lie superalgebras.
We show that a Hermitian algebraic curvature model satisfies the Gray identity if and only if it is geometrically realizable by a Hermitian manifold. Furthermore, such a curvature model can in fact be realized by a Hermitian manifold of constant scalar curvature and constant *-scalar curvature which satisfies the Kaehl…
We show any Riemannian curvature model can be geometrically realized by a manifold with constant scalar curvature. We also show that any pseudo-Hermitian curvature model, para-Hermitian curvature model, hyper-pseudo-Hermitian curvature model, or hyper-para-Hermitian curvature model can be realized by a manifold with co…
Study supports conjecture about pretzel links' homology.
problem Investigate extreme Khovanov homology of pretzel links.
method Geometric realizations of the independence simplicial complex.
result Supports conjecture that extreme Khovanov homology is torsion-free.
Introduces sheaf quantization, a topological approach to geometric quantization.
problem Topological realization of WKB-states in geometric quantization.
method Enhancement of constructible sheaves, Betti counterpart of Fukaya--Floer theory.
result Introduction to sheaf quantization as a topological realization of WKB-states.
This paper develops a general method for constructing Poisson integrators.
problem Lack of a general theory for Poisson integrators due to geometric challenges.
method Adapting structural results about symplectic realizations to create geometric approximations.
result Developed a general approach for constructing geometric integrators on Poisson manifolds.
This paper develops a new homology theory for biquandles and discusses geometric realizations.
problem Constructing knot invariants using set-theoretic Yang-Baxter equation.
method Developed a normalized (co)homology theory for biquandles and geometrically realized them.
result Geometric realization of biquandles has finitely generated second homotopy group for finite biquandles.
The covariant derivative of the Kähler form of an almost pseudo-Hermitian or of an almost para-Hermitian manifold satisfies certain algebraic relations. We show, conversely, that any 3-tensor which satisfies these algebraic relations can be realized geometrically.
The paper explores symplectic foliations and their leaves on manifolds.
problem Which manifolds can be realized as leaves of codimension-1 symplectic foliations?
method Observations and deformations of symplectic structures; examples of manifolds.
result Examples of manifolds that can be realized as leaves but not as symplectic leaves.
The study constructs a hyperbolic orbifold to show how certain Salem numbers can be realized geometrically.
problem Constructing a geometric model for Salem numbers of degree 4.
method Constructing an arithmetic hyperbolic 6-orbifold and proving its properties.
result Any square-rootable Salem number of degree at most 4 can be realized as the exponential of a closed geodesic length in the constructed orbifold.
We define generalized currents associated with immersions of abstract solenoids with a transversal measure. We realize geometrically the full real homology of a compact manifold with these generalized currents, and more precisely with immersions of minimal uniquely ergodic solenoids. This makes precise and geometric De…
Maps configuration spaces to groups for large k.
problem Understanding connections between configuration spaces and groups Gnk. method Constructs a map from fundamental groups of complements to Gnk. result Discovers connections between different Gnk groups. In this paper, we study face vectors of simplicial posets that are the face posets of cell decompositions of topological manifolds without boundary. We characterize all possible face vectors of simplicial posets whose geometric realizations are homeomorphic to the product of spheres. As a corollary, we obtain the chara…
New integrators preserve geometric structure in Hamiltonian systems.
problem Preserving geometric structure in Hamiltonian systems on Jacobi manifolds.
method Combining Poissonization and symplectic bi-realizations to construct structure-preserving integrators.
result Explicit construction and application of Jacobi Hamiltonian integrators.
Rational maps structure theorem with geometric decomposition and realizability proof.
problem Realizability of rational maps branch data.
method Geometric decomposition of pullback metric into footballs and application to realizability.
result Realizability of branch data for rational maps when k>l+1. Geometric framework for signed multivariate tail-dependence compatibility at various thresholds.
problem Modeling and analyzing signed multivariate tail-dependence across different thresholds.
method Developed a geometric witness framework to represent and invert signed tail families, identifying nonnegative weights and normalized masses.
result Characterization and synthesis of signed multivariate tail-dependence at finite thresholds, preserving the complete signed tail family throughout.
New framework tackles geometric structure existence and classification.
problem Existence and classification of geometric structures.
method Developed a new framework of relative algebroids.
result New framework addresses geometric structure problems.
We examine geometric representability results for various classes of equiaffine curvature operators. We show every Ricci flat algebraic curvature operator is geometrically realizable by a Ricci flat torsion free connection on the tangent bundle of some smooth manifold.
New geometric approach realizes 5D bulk theories with 4D edge modes.
problem Realizing novel higher-dimensional junctions of theories coupled to localized edge modes.
method M-theory on singular, asymptotically conical G2-holonomy orbifolds.
result Geometric approach shows how bulk generalized symmetries are inherited in the boundary system.
Three perspectives on quantizing magnetic Poisson structures are explored.
problem Quantization of magnetic Poisson structures with non-associativity.
method Deformation quantization, symplectic realization, geometric quantization using bundle gerbes.
result Comparison and contrast of different quantization approaches.
Study of Ricci flow on discrete surfaces of revolution with constant Gaussian curvature.
problem Understanding Ricci flow on discrete surfaces of revolution.
method Explicit parametrizations and Ricci flow analysis for discrete surfaces of revolution.
result Discrete surfaces of revolution approach constant Gaussian curvature under Ricci flow.
Defines an equivariant cobordism category for finite groups.
problem Understanding geometric structures with group actions.
method Introduces a new category for (d−1)-dimensional closed smooth G-manifolds and their cobordisms. result Identifies the homotopy type of the classifying space of the equivariant cobordism category as fixed points of an infinite loop space.
String theory connects lattice models, links, and geometric Langlands.
problem Connecting lattice models, links, and geometric Langlands.
method T-duality and worldvolume theories in string theory.
result Unified understanding of various mathematical concepts.
This paper connects geometric diagrams to spherical T-duality.
problem Establishing equivalence between geometric and string-theory frameworks.
method Introducing ⋆-diagrams and proving their equivalence to spherical T-dual pairs. result Concrete geometric realization of spherical T-duality.
Updates and rewrites a 1974 AMS Memoir on Lie groups.
problem Real reductive Lie groups and their representations.
method Rewriting and updating a 1974 AMS Memoir.
result Ties with recent approaches to geometric realization of unitary representations.
Study intersection polynomials of long virtual knots with supporting genera.
problem Characterize long virtual knots using geometric invariants.
method Define and analyze 1- and 2-supporting genera, and use them to filter long virtual knots. result Provide complete realizability criteria for all twelve intersection polynomials.
Discuss Alan Schoen's I-WP minimal surface with geometric realizations.
problem Characterize and visualize Alan Schoen's I-WP minimal surface.
method Examine various geometric realizations of the I-WP surface and analyze the associate family.
result The associate family of the I-WP surface contains six congruent surfaces at specific angles.
Given an orientable weakly self-dual manifold X of rank two, we build a geometric realization of the Lie algebra sl(6,C) as a naturally defined algebra L of endomorphisms of the space of differential forms of X. We provide an explicit description of Serre generators in terms of natural generators of L. This constructio…
The paper characterizes toroidal sets as attractors for flows and homeomorphisms of R^3.
problem Characterizing toroidal sets as attractors for flows and homeomorphisms in R^3.
method Defined self-geometric index and genus to analyze toroidal sets and their attractor properties.
result Characterized toroidal sets that can be realized as attractors for flows but not for homeomorphisms.
We prove three optimal conformal geometric inequalities of Blatter type on the Klein bottle. These inequalities provide conformal lower bounds of the volume and involve lengths of homotopy classes of curves that are candidates to realize the systole.
In this paper we present an overview of the connection between completely integrable systems and the background geometry of the flow. This relation is better seen when using a group-based concept of moving frame introduced by Fels and Olver in [Acta Appl. Math. 51 (1998), 161-213; 55 (1999), 127-208]. The paper discuss…
We prove the existence of Lagrangian fillings for Dn-type Legendrian links.
problem Exact Lagrangian fillings of Legendrian links of Dn-type. method Legendrian weave calculus and construction of 1-cycles.
result Existence of a Lagrangian filling represented by a weave.
The paper introduces lattice homology for integrally closed submodules and applies it to geometric invariants.
problem Computing numerical invariants of geometric objects.
method Introduces lattice homology for integrally closed submodules and applies it to geometric invariants.
result Well-defined lattice homology associated to quotient modules of type M/N. Paper proposes a deep learning method for better covariance matrix forecasting.
problem Suboptimal predictive performance in traditional matrix volatility forecasting.
method Riemannian-geometry-aware deep learning framework for symmetric positive definite matrices.
result Our method outperforms traditional approaches in predictive accuracy.
New geometric examples of involutions on Hilbert square K3 surfaces.
problem Existence of non-symplectic involutions on Hilbert square K3 surfaces.
method Geometric constructions of involutions on IHS manifolds.
result New examples of involutions proven to exist.