Proves correspondence between harmonic and Higgs bundles.
problem Connecting harmonic and Higgs bundles for study.
method Kobayashi-Hitchin correspondence for polystable bundles.
result Establishes correspondence between good wild harmonic bundles and polystable good filtered λ-flat bundles. The purpose of this paper is to produce restrictions on fundamental groups of manifolds admitting good complexifications by proving the following Cheeger-Gromoll type splitting theorem: Any closed manifold M admitting a good complexification has a finite-sheeted regular covering M1 such that M1 admits a fiber b…
New orbifolds found that cover manifolds but not in a finite way.
problem Finding orbifolds that cover manifolds but not finitely.
method Constructing specific orbifolds.
result Examples of orbifolds that cover manifolds but not finitely.
3-manifolds with torsion homology expand in all dimensions.
problem Constructing 3-manifolds with good expansion properties.
method Constructing 3-manifolds with specific homological properties and demonstrating their expansion.
result 3-manifolds with torsion homology expand in all dimensions.
Study shows shortest geodesic length on certain manifolds is limited by volume, diameter, and cover elements.
problem Bounding the length of shortest closed geodesics on Riemannian manifolds with good covers.
method Generalization of previous results using diameter, volume, and cover elements to bound geodesic length.
result Length of shortest closed geodesic is bounded by a function of volume, diameter, and cover elements.
Study of (R,ε)-panted cobordism groups in cusped hyperbolic 3-manifolds.
problem Understanding the structure of (R,ε)-panted cobordism groups in cusped hyperbolic 3-manifolds. method Investigating the abelian group generated by (R,ε)-good curves in M modulo the oriented boundaries of (R,ε)-good pants. result For sufficiently small ε>0 and sufficiently large R>0, the (R,ε)-panted cobordism group of M is isomorphic to H1(extSO(M);Z). Generalizes Kuperberg's invariant to broader algebraic structures.
problem Extending Kuperberg's invariant to a wider class of algebraic objects.
method Constructing a scalar invariant for 3-manifolds using involutory Hopf algebras in arbitrary symmetric monoidal categories with good pairs of morphisms.
result Illustrated examples of the generalized construction for specific algebraic structures.
In this short note, we study the injectivity radius bound for three dimensional complete and non-compact Riemannian manifold with good leaf foliations and with bounded curvature up to first order. We obtain the injectivity bound by using the minimal surface theory and the Gauss-Bonnet theorem.
Proves surface embedding theorem for 4-manifolds with good fundamental group.
problem Surface embedding in 4-manifolds with good fundamental group.
method Introduces dual spheres and Kervaire-Milnor invariant for computation.
result Combinatorial formula for Kervaire-Milnor invariant.
In this note we establish estimates for the harmonic map heat flow from S1 into a closed manifold, and use it to construct sweepouts with the following good property: each curve in the tightened sweepout, whose energy is close to the maximal energy of curves in the sweepout, is itself close to a closed geodesic.
We give a complete proof of Thurston's Orbifold Theorem for very good 3-orbifolds of cyclic type. An orbifold is said to be very good when it has a finite cover which is a manifold. A 3-orbifold is of cyclic type if the singular set is a non-empty 1-manifold transverse to the boundary.
We introduce the cutting construction of possibly non-compact symplectic toric manifolds, in particular, toric symplectic cones that correspond to a weakly convex good cone. Since the symplectization of a toric contact manifold is a toric symplectic cone, we can also construct toric contact manifolds that correspond to…
Optimal persuasion involves projecting state vectors onto lower-dimensional 'optimal information manifolds'.
problem Optimal persuasion of another agent observing multi-dimensional data.
method Performing non-linear dimension reduction by projecting state vectors onto the 'optimal information manifold'.
result Optimal information design splits information into 'good' and 'bad' components, revealing only the direction of good information.
Study on instantons on cylinder manifolds with specific conditions.
problem Analyzing instantons on cylinder manifolds with non-zero smooth forms.
method Examined the instanton equation on cylinder manifolds with specific conditions.
result Instantons on good manifolds decay exponentially at the ends.
We characterize the quasiprojective groups that appear as fundamental groups of compact 3-manifolds (with or without boundary). We also characterize all closed 3-manifolds that admit good complexifications. These answer questions of Friedl--Suciu, \cite{fs}, and Totaro \cite{tot}
New autoencoder uses goodness-of-fit tests for better model performance.
problem Improving the goodness-of-fit in generative models.
method Develops Goodness-of-Fit Autoencoder (GoFAE) incorporating GoF tests at minibatch and global levels.
result GoFAE achieves comparable performance to deep generative models while retaining statistical indistinguishability.
In this paper we show that any good toric contact manifold has well defined cylindrical contact homology and describe how it can be combinatorially computed from the associated moment cone. As an application we compute the cylindrical contact homology of a particularly nice family of examples that appear in the work of…
Reductive quotients preserve klt singularities in algebraic geometry.
problem Preserving klt singularities in quotients of klt singularities.
method Proving that the quotient of a klt type singularity by a reductive group is of klt type.
result The quotient of a klt variety by a reductive group results in a klt variety with a suitable boundary.
The study explores extensions of Kähler manifolds and their properties.
problem Analyzing extensions of Kähler manifolds and their properties.
method Formulating conditions for affine bundles and studying their properties.
result Established a relation to adapted complex structures and good complexifications.
Any closed, oriented, hyperbolic three-manifold with nontrivial second homology has many quasigeodesic flows, where quasigeodesic means that flow lines are uniformly efficient in measuring distance in relative homotopy classes. The flows are pseudo-Anosov flows which are almost transverse to finite depth foliations in …
New method to calculate 3-manifold invariants via skew-racks.
problem Calculating invariants of 3-manifolds.
method Introducing skew-racks with good involution and Property FR, defining cocycle invariants.
result Established new approach to obtain 3-manifold invariants via Dehn surgery.
Develops goodness-of-fit tests for noisy submanifold samples.
problem Testing non-linear models on noisy submanifold samples.
method Non-linear least-square problem solution and χ2 distribution application. result Residual follows χ2 distribution with parameters related to model order and dimension. Paper develops techniques for singular metrics on vector bundles.
problem Developing techniques for singular metrics on vector bundles.
method Introducing non-pluripolar products and defining I-good singularities. result Derives a Chern--Weil type formula for Hermitian vector bundles with I-good singularities. The paper establishes conditions for optimal sampling configurations on complex manifolds.
problem Finding optimal sampling configurations on complex manifolds.
method Analyzes point configurations on compact complex manifolds using tensor powers of Hermitian ample line bundles.
result Necessary and sufficient conditions for the existence of asymptotically Fekete sequences.
New metrics found on toric LCS manifolds.
problem Finding compatible complex structures on toric LCS manifolds.
method Proved a bijective correspondence between toric LCS manifolds and pairs (C,a). result Compact toric LCS manifolds have a positive potential.
3-manifolds virtually 1-dominated by hyperbolic ones.
problem Domination of 3-manifolds by hyperbolic ones.
method Enhanced connection principle in good pants constructions.
result Every 3-manifold is virtually 1-dominated by a hyperbolic one.
Researchers define quantiles on Riemannian manifolds using optimal transport.
problem Defining quantiles on nonlinear manifolds.
method Measure-transportation-based approach.
result Theoretical and empirical properties of quantile functions on manifolds.
Unified-GAN improves SSL with both good and bad samples.
problem Improve semi-supervised learning performance with limited labeled data.
method Unified-GAN combines adversarial training with good and bad samples.
result Unified-GAN achieves state-of-the-art performance and robustness to varying labeled data.
Novel proof technique for Gelfand-Fuks cohomology.
problem Comparing sheaf-like data over manifold Cartesian powers.
method Local-to-global analysis through generalized good covers and factorization algebras.
result Unified approach to Gelfand-Fuks cohomology.
Spark complexes defined on good effective orbifold atlases.
problem Constructing a structured representation for effective orbifolds.
method Defining good atlases and constructing spark complexes categorically.
result Spark character 2-functor factors through the constructed 2-functor.
We construct pairs and continuous families of isospectral yet locally non-isometric orbifolds via an equivariant version of Sunada's method. We also observe that if a good orbifold O and a smooth manifold M are isospectral, then they cannot admit non-trivial finite Riemannian covers M1→O …
Cobordism and signatures of manifolds with similar fundamental groups.
problem Understanding cobordism and signatures of manifolds with specific fundamental groups.
method Analyzing smooth closed connected aspherical manifolds with good fundamental groups.
result Cobordism and signatures of manifolds are preserved under certain conditions on their fundamental groups.
Proves sufficient condition for 2D orbifolds to be good.
problem Characterizing 2D orbifolds as good.
method Analyzes orbifold fundamental groups for goodness.
result Connected 2D orbifolds with infinite orbifold fundamental group are good.
The paper defines conditions for good involutions in generalized Alexander quandles.
problem Determining conditions for good involutions in generalized Alexander quandles.
method Analyzing the structure of generalized Alexander quandles and their involutions.
result Classification of all good involutions in connected generalized Alexander quandles.
There are certain families of words and word sequences (words in the generators of a two-generator group) that arise frequently in the Teichm{ü}ller theory of hyperbolic three-manifolds and Kleinian and Fuchsian groups and in the discreteness problem for two generator matrix groups. We survey some of the families of su…
The paper extends Liouville theorems to sub-Riemannian manifolds.
problem Generalizing Liouville theorems to sub-Riemannian manifolds.
method Constructing 'good' cut-off functions and applying a nonnegative generalized curvature-dimension inequality.
result The Liouville theorems are extended to sub-Riemannian manifolds.
Regularizes complex functions on manifolds using coordinate balls.
problem Regularizing quasi-plurisubharmonic functions on compact Kaehler manifolds.
method Regularize on local coordinate balls, then glue, focusing on vanishing higher order terms at centers.
result Higher order terms vanish at centers of coordinate balls, forming a delta-net.
We prove an existence theorem for gauge invariant L2-normal neighborhoods of the reduction loci in the space Aa(E) of oriented connections on a fixed Hermitian 2-bundle E. We use this to obtain results on the topology of the moduli space Ba(E) of (non-necessarily irreducible) oriented connectio…
We show, up to h-cobordism, that the existence and uniqueness of connected sum decompositions of oriented 4-dimensional manifolds is an invariant of homotopy equivalence, assuming that the fundamental group of each summand is "good" in the sense of Freedman and Quinn. On a separate note, we observe that the Borel Conje…
We show that the Prüfer surface, which is a separable non-metrizable 2-manifold, has not the homotopy type of a CW-complex. This will follow easily from J. H. C. Whitehead's result: if one has a good approximation of an arbitrary space by a CW-complex, which fails to be a homotopy equivalence, then the given space is n…
A canonical branched covering over each sufficiently good simplicial complex is constructed. Its structure depends on the combinatorial type of the complex. In this way, each closed orientable 3-manifold arises as a branched covering over the 3-sphere from some triangulation of S^3. This result is related to a theorem …
3D good continuation model explains stereo vision using neurogeometry.
problem Understanding how the brain processes 3D visual correspondence.
method Developed a neurogeometric model involving spatial and orientation disparities.
result Provides insight into neural organization and correspondence problem.
We study convex risk measures describing the upper and lower bounds of a good deal bound, which is a subinterval of a no-arbitrage pricing bound. We call such a convex risk measure a good deal valuation and give a set of equivalent conditions for its existence in terms of market. A good deal valuation is characterized …
Directly approximates functions on unknown data manifolds without complex computations.
problem Function approximation on unknown data-defined manifolds with conservative results from traditional methods.
method Direct approach using graph Laplacian and local approximation techniques without eigen-decomposition or atlas.
result Universal estimates for smooth functions without prior knowledge of the target function.
We study the relationship between the geometry and the Laplace spectrum of a Riemannian orbifold O via its heat kernel; as in the manifold case, the time-zero asymptotic expansion of the heat kernel furnishes geometric information about O. In the case of a good Riemannian orbifold (i.e., an orbifold arising as the orbi…
Study on mean Euler characteristic of Gorenstein toric contact manifolds.
problem Calculating the mean Euler characteristic of Gorenstein toric contact manifolds.
method Using the relationship between mean Euler characteristic and the normalized volume of the toric diagram, and applying results from Batyrev and Dais.
result Twice the mean Euler characteristic of a Gorenstein toric contact manifold equals the Euler characteristic of any crepant toric symplectic filling.
The study describes good involutions in quandles and Alexander quandles.
problem Characterizing and enumerating good involutions in quandles and Alexander quandles.
method Completely describing good involutions of free and subquandles of twisted conjugation quandles of groups, including Alexander quandles.
result Explicit mappings for good involutions of linear quandles up to order 23.
Study shows typical scales for manifolds with lower Ricci bounds.
problem Understanding typical scales in manifolds with lower Ricci curvature bounds.
method Analysis of collapsing sequences of Riemannian manifolds with uniform lower Ricci curvature bounds.
result Rescaled manifolds subconverge to a product of a Euclidean and a compact space.