Defines relative $\dbar$-complex and studies its curvature properties.
problem Curvature properties of relative $\dbar$-complex and associated vector bundles.
method Definition and study of curvature properties of the relative $\dbar$-complex and associated vector bundles.
result Yamaguchi's theory on subharmonicity of the Green operator can be seen as a curvature property of the quotient bundle.
Fukaya-Yamaguchi conjecture holds in 4D manifolds with nonnegative curvature.
problem Fundamental group of nonnegative curvature manifolds.
method Observation in dimension 4.
result Fukaya-Yamaguchi conjecture holds in 4D.
Study shows nonexistence of certain geometric structures in complex geometries.
problem Failure of Lichnerowicz-type conjectures in specific parabolic geometries.
method Used techniques from Erickson to establish existence of specific geometries.
result Nonexistence of certain geometric structures in Yamaguchi nonrigid parabolic models.
These are lecture notes on the rigidity of submanifolds of projective space "resembling" compact Hermitian symmetric spaces in their homogeneous embeddings. Recent results are surveyed, along with their classical predecessors. The notes include an introduction to moving frames in projective geometry, an exposition of t…
Proves torus sequences can't collapse to intervals under curvature bounds.
problem Proving torus sequences can't collapse to intervals under curvature bounds.
method Contradiction proof using Yamaguchi fibration theorem and covers.
result Proves tori can't collapse to intervals under curvature constraints.
Paper extends Enami-Ozeki-Yamaguchi's work on planar quadrangulations.
problem Finding the maximum number of colors for proper anti-rainbow colorings on planar quadrangulations.
method Introducing half-monochromatic colorings for plane graphs with even polygonal faces and providing an upper bound in terms of the independence number.
result An upper bound on the maximum number of colors for half-monochromatic colorings is given in terms of the independence number.
In this paper, we study the topology of topologically regular 4-dimensional open non-negatively curved Alexandrov spaces. These spaces occur naturally as the blow-up limits of compact Riemannian manifolds with lower curvature bound. These manifolds have also been studied by Yamaguchi in his preprint [Yam2002]. Our main…
We will simplify earlier proofs of Perelman's collapsing theorem for 3-manifolds given by Shioya-Yamaguchi and Morgan-Tian. Among other things, we use Perelman's critical point theory (e.g., multiple conic singularity theory and his fibration theory) for Alexandrov spaces to construct the desired local Seifert fibratio…
This paper extends the Good Covering Theorem and Jordan Curve Theorem for proximal Alexandrov spaces.
problem Extending the Good Covering Theorem and Jordan Curve Theorem to proximal Alexandrov spaces.
method Introducing path cycles and using them to extend the Good Covering Theorem and Jordan Curve Theorem.
result Extensions of the Mitsuishi-Yamaguchi Good Covering Theorem and Jordan Curve Theorem for proximal Alexandrov spaces.
This paper standardizes Cheeger deformations on fiber bundles with compact groups.
problem Developing a unified approach to Cheeger deformations on fiber bundles.
method Systematic introduction and re-proving existing results using Cheeger deformations.
result Unified approach to Cheeger deformations on fiber bundles with compact structure groups.
3D spaces collapse to 8 Thurston geometries.
problem Characterizing collapsed 3D Alexandrov spaces.
method Proving Alexandrov spaces modeled on Thurston geometries.
result Sufficiently collapsed irreducible Alexandrov 3-spaces are geometric.
In this note we discuss the fundamental groups and diameters of positively Ricci curved n-manifolds. We use a method combining the results about equivarient Hausdorff convergence developed by Fukaya and Yamaguchi with the Ricci version of splitting theorem by Cheeger and Colding to give new information on the topolog…
Paper extends theorem on covering spaces and Jordan curves.
problem Covering and extending theorems for Alexandrov spaces.
method Introduces proximal homotopic cycles to extend the Mitsuishi-Yamaguchi theorem.
result Extensions of the Mitsuishi-Yamaguchi Good Covering Theorem and Jordan curve theorem.
Classifies generalized Seifert fiber spaces and their branched covers.
problem Classifying generalized Seifert fiber spaces and their topological properties.
method Symbolic classification and computation of invariants.
result Canonical double branched cover of non-manifold spaces is a Seifert manifold.
Cartan calculus applied to string topology homology.
problem Understanding the structure of free loop spaces.
method Introduced Cartan calculus on loop homology, linked to string topology operations.
result Loop product and bracket behavior under Hodge decomposition.
Study quantifies convergence of Alexandrov spaces without collapsing.
problem Quantifying convergence of Alexandrov spaces without collapsing.
method Lipschitz homotopy convergence for Alexandrov spaces.
result Lipschitz homotopies can be chosen to preserve singular strata.
We show that a complete Riemannian manifold of dimension n with $\Ric\geq n{-}1$ and its n-st eigenvalue close to n is both Gromov-Hausdorff close and diffeomorphic to the standard sphere. This extends, in an optimal way, a result of P. Petersen. We also show that a manifold with $\Ric\geq n{-}1$ and volume close…
The paper proves fibration theorems for manifolds with almost nonnegative Ricci curvature.
problem Proving fibration theorems for manifolds with specific curvature conditions.
method Using equivariant regularity theorems and Gromov-Hausdorff convergence.
result Closed manifolds with certain curvature conditions fiber over a b1-torus. Uniform bounds found for extremal subsets in specific Alexandrov spaces.
problem Bounding properties of extremal subsets in Alexandrov spaces with constraints on dimension, curvature, and diameter.
method Application of essential coverings introduced by Yamaguchi.
result Uniform upper bounds on the number, Betti numbers, and volume of extremal subsets.
We will simplify the earlier proofs of Perelman's collapsing theorem of 3-manifolds given by Shioya-Yamaguchi and Morgan-Tian. Among other things, we use Perelman's semi-convex analysis of distance functions to construct the desired local Seifert fibration structure on collapsed 3-manifolds. The verification of Perelma…
The paper improves collapsing Alexandrov spaces results using good coverings.
problem Collapsing Alexandrov spaces with no proper extremal subsets.
method Application of good coverings of Alexandrov spaces.
result Construction of an infinitely long exact sequence of homotopy groups and a spectral sequence of cohomology groups.
The paper proves a fibration theorem for collapsing sequences of Alexandrov spaces.
problem Understanding the structure of collapsing sequences of Alexandrov spaces.
method Constructing almost Lipschitz submersions and proving locally trivial fibrations.
result A locally trivial fibration is established under certain conditions on volumes and singularity weakness.
Unified proof of smooth fibration theorems for collapsed manifolds.
problem Smooth fibration theorems for collapsed manifolds with Ricci curvature bounded below.
method Generalized Reifenberg condition and transformation technique for almost splitting maps.
result Unified proof of smooth fibration theorems in many previous works.
The article proves estimates on homotopy and cohomology dimensions in fibrations.
problem Estimates on dimensions of homotopy and cohomology groups in fibrations.
method Proves estimates in formal elliptic spaces and specific cases.
result Proves estimates on dimensions of homotopy and cohomology groups in fibrations.
We prove a general extrinsic rigidity theorem for homogeneous varieties in CPN. The theorem is used to show that the adjoint variety of a complex simple Lie algebra g (the unique minimal G orbit in Pg) is extrinsically rigid to third order. In contrast, we show that the ad…
Paper studies rigid properties of closed CSL submanifolds in unit spheres.
problem Understanding geometric properties of closed CSL submanifolds in unit spheres.
method Analyzes the rigidity of closed CSL submanifolds in the unit sphere using geometric methods.
result Closed CSL submanifolds in S5 with specific properties are either totally geodesic or flat minimal Legendrian tori. New proof of Suita's conjecture without L2 extension theorem.
problem Equality in Suita's conjecture for open Riemann surfaces.
method Exploration of Maitani and Yamaguchi's variation formula for the Bergman kernel and harmonicity.
result Characterization of surfaces by constant Gaussian curvature properties of the Bergman kernel or metric.
Unified framework for Alexandrov 3-spaces, extending manifold results.
problem Extend manifold topology results to Alexandrov 3-spaces.
method Generalize connected sum, prime decomposition, and Dehn surgery.
result Unified framework for Alexandrov 3-spaces, including non-manifold spaces.
Proves functional equation for twisted Ruelle zeta function on hyperbolic surfaces.
problem Determines the functional equation for twisted Ruelle zeta functions.
method Analyzes scattering matrix and uses topological data of hyperbolic surfaces.
result Determines the order of the divisor of R(s,χ) at s=0 and computes its Laurent expansion.
Proves uniform index bound for loops in Alexandrov spaces.
problem Uniform index bound for loops in Alexandrov spaces.
method Based on ideas from Kapovitch, Petrunin, and Tuschmann; uses Hurewicz fibration and gradient push.
result Uniform index bound of w(n) for loops in Alexandrov spaces. Discrete Morse-Bott theory on CW complexes generalizes Forman's theory.
problem No specific problem stated; focuses on theory development.
method Derived a discrete Morse-Bott theory on CW complexes.
result Discrete Morse-Bott theory is a generalization of Forman's theory.
The paper connects function theory, dynamics, and ergodic theory via Thurston's theory.
problem Function theory on Teichmüller space and dynamics of mapping class groups.
method Utilizes Thurston's theory and Sullivan's theory on discrete subgroups of hyperbolic space.
result Establishes connections between function theory, dynamics, and ergodic theory.
Survey of Floer theories and their connections.
problem None explicitly stated; focuses on surveying theories.
method None explicitly stated; focuses on surveying theories.
result None explicitly stated; focuses on surveying theories.
New homology theory for semi-groups with specific properties.
problem Developing a homology theory for semi-groups with self-distributivity or idempotency.
method Constructing a new homology theory and comparing it with existing theories.
result Comparison and connections with rack homology and knot theory.
New classes from 4D gauge theories.
problem Constructing characteristic classes for 4-manifold bundles.
method Using SO(3)-Yang-Mills theory and Seiberg-Witten theory for families. result Characteristic classes of 4-manifold bundles constructed.
Lectures on topological field theories and differential cohomology.
problem Exploring topological field theories and their connections to differential cohomology.
method Introduction to topological field theory and generalized Abelian gauge theories.
result Explains the relationship between topological field theories and differential cohomology.
New theory couples Chern-Simons to matter, topological.
problem Developing a new topological theory in 3D.
method Coupling Chern-Simons to matter, using transverse holomorphic foliation.
result The theory is equivalent to an N=2 supersymmetric Chern-Simons matter theory.
The paper defines strong emergence in field theories and proves it exists between certain theories.
problem Defining and proving the existence of strong emergence phenomena between field theories.
method Formal definition and sufficient conditions for emergence, proving existence in Euclidean background.
result Strong emergence exists between certain parameterized Lagrangian field theories.
Topological string theory derived from string geometry for non-perturbative effects.
problem Deriving non-perturbative effects in string theory.
method Formulating topological string geometry theory and deriving the partition function from fluctuations around a classical solution.
result Perturbative partition function of topological string theory derived.
New cohomology theory for Lie 2-algebras extends classical theory.
problem Classical cohomology theory limitations for Lie 2-algebras.
method Introduced a new cohomology theory for Lie 2-algebras.
result Second cohomology group classifies extensions of Lie 2-algebras.
Researchers solve M-theory's gauge enhancement problem using advanced homotopy theory.
problem Lift nonabelian gauge fields from D-branes to M-theory.
method Universal constructions in super homotopy theory, focusing on the cyclification adjunction and fiberwise stabilization.
result Gauge enhancement in M-theory is explained by lifting against the fiberwise stabilization of the unit of the cyclification adjunction.
Unified Higgs bundle vacua from M-theory on Spin(7) spaces.
problem Unifying Higgs bundle vacua from different string compactifications.
method Developed formalism for M-theory on local Spin(7) spaces and constructed explicit solutions.
result Unified 3D effective field theory from 4D M- and F-theory vacua.
This is the first paper in a series introducing a generalized Fredholm theory in a new class of smooth spaces called polyfolds. The theory will be illustrated in upcoming papers by applications to Floer Theory, Gromov-Witten Theory and Symplectic Field Theory.
Develops Hodge theory for foliations using perturbed Laplacians.
problem Creating a Hodge theory for foliations.
method Mimicking Witten's approach to Morse theory with perturbations of the Laplacian.
result Establishes a Hodge theory for tangential cohomology of foliations.
Researchers find new G2-conifolds in M-theory with potential field theory duals.
problem Exploring the field theory interpretation of M-theory G2-conifolds. method Constructing G2-holonomy orbifolds from circle bundles over Calabi-Yau cones. result Many UV perturbative gauge theories have an infrared dual described by smooth G2-holonomy backgrounds in M-theory. New theory captures framing anomaly in gauge theory.
problem Capturing framing anomaly in gauge theory.
method Constructs a relative Crane-Yetter theory from non-semisimple data.
result Establishes invertibility property for the theory.
We survey three different ways in which K-theory in all its forms enters quantum field theory. In Part 1 we give a general argument which relates topological field theory in codimension two with twisted K-theory, and we illustrate with some finite models. Part 2 is a review of pfaffians of Dirac operators, anomalies, a…
Researchers compute K-theory for cohomogeneity-one actions.
problem Computing equivariant K-theory for cohomogeneity-one actions.
method Equivariant homotopy theory, representation theory, Lie theory.
result Derived generators and relations for K-theory ring.