New class of singular complex manifolds studied with degenerate theory.
problem Understanding singular complex manifolds.
method Developed degenerate Kodaira-Hodge theory for new class.
result New degenerate theory for singular complex manifolds.
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48 results for “complexity class”
Proves Massey's theorems on complex structure obstructions.
problem Finding complex structures on real vector bundles.
method Fractional Stiefel-Whitney classes and combinations of Pontryagin, Chern, and Euler classes.
result Determines the second obstruction for rank six bundles.
Study on biharmonic almost complex structures on compact manifolds.
problem Existence and regularity of biharmonic almost complex structures.
method Analyzes biharmonic almost complex structures on compact almost Hermitian manifolds, focusing on dimension four.
result Existence of energy-minimizing biharmonic almost complex structures for various topologies and homotopy classes.
Study mapping class groups on CAT(0) cube complexes.
problem Asymptotically rigid mapping class groups of infinitely-punctured surfaces.
method Introduced CAT(0) cube complexes and determined their properties.
result Determined CAT(0) properties of cube complexes.
Study of contact-complex Riemannian submersions in various manifold classes.
problem Characterizing submersions in specific manifold classes.
method Analyzing properties of O'Neill invariants and discussing integrability and minimality.
result Main properties of O'Neill invariants and integrability/minimality of fibres discussed.
Mapping class group subgroups yield quasi-isometric curve complex.
problem Understanding the curve complex through coset intersections.
method Proving quasi-isometry and combinatorial equivalence of curve complex and coset intersection complex.
result Automorphism group of coset intersection complex is the extended mapping class group.
Study on complex line fields on almost-complex manifolds, proving existence conditions.
problem Existence of linearly independent complex line fields on almost-complex manifolds.
method Prove necessary and sufficient conditions for the existence of one, two, or three fields over certain manifolds.
result Necessary and sufficient condition for the existence of complex line fields over certain manifolds.
Defines a cell complex for even spin mapping class group.
problem No specific problem stated; focuses on definition.
method Defines a cell complex with an action of the even spin mapping class group.
result Obtains a finite presentation of the even spin mapping class group.
Study of pseudoconvex 3-manifolds in complex surfaces.
problem Understanding pseudoconvex 3-manifolds in complex surfaces.
method Develop tools for constructing topologically pseudoconvex embeddings and classify almost-complex structures.
result Every closed, oriented 3-manifold can be embedded in a compact complex surface realizing any homotopy class of almost-complex structures.
By the work of Harer, the reduced homology of the complex of curves is a fundamental cohomological object associated to all torsion free finite index subgroups of the mapping class group. We call this homology group the Steinberg module of the mapping class group. It was previously known that the curve complex has the …
Formula derived for Bott-Chern classes in complex blow-ups.
problem Calculating Bott-Chern classes in blow-ups of complex manifolds.
method Proved blow-up formula for Bott-Chern classes, established Riemann-Roch without denominators.
result Formula for Bott-Chern classes in blow-ups.
Study Morse complexity of manifolds and homology classes, proving bounds and implications.
problem Understanding Morse complexity of manifolds and homology classes.
method Used surgery theory and index theory to prove upper and lower bounds.
result Locally symmetric spaces of Lie groups with discrete series representations do not admit open book decompositions.
Study on Lee classes of complex surfaces, proving connectedness and bounds.
problem Understanding Lee classes of complex surfaces with LCS structures.
method Analyzing deRham classes of Lee 1-forms and using properties of PSH functions.
result Connectedness of Lee deRham classes and explicit negative upper bound on hyperbolic Kato surfaces.
For a genus g handlebody H a simplicial complex, with vertices being isotopy classes of certain incompressible surfaces in H, is constructed and several properties are established. In particular, this complex naturally contains, as a subcomplex, the complex of curves of the boundary surface of H. As in the classical th…
Complex Balanced Spacesmath.CV
In this paper, the concept of balanced manifolds is generalized to reduced complex spaces: the class B and balanced spaces. Compared with the case of Kahlerian, the class B is similar to the Fujiki class C and the balanced space is similar to the Kahler space. Some properties about these complex spaces are obtained, an…
New class of complex manifolds defined, properties studied.
problem Understanding properties of complex manifolds.
method Introduced and studied wHHR manifolds, proved metric equivalence.
result Bergman and Kobayashi metrics are biLipschitz equivalent for wHHR Stein manifolds.
The main goal of our paper is the study of several classes of submanifolds of generalized complex manifolds. Along with the generalized complex submanifolds defined by Gualtieri and Hitchin (we call these ``generalized Lagrangian submanifolds'' in our paper), we introduce and study three other classes of submanifolds. …
Optimizes energy of mappings from complex projective spaces.
problem Finding energy-minimizing mappings between complex projective spaces and Riemannian manifolds.
method Establishes optimal lower bounds for energy functionals and characterizes optimal mappings.
result Optimal lower bounds for energy functionals are characterized for mappings from real and complex projective spaces.
The study extends cobordism theory to complex sections, defining and calculating cobordism groups.
problem Understanding when almost complex manifolds can have complex sections.
method Defined complex section cobordism, determined groups, and introduced an obstruction.
result The obstruction vanishes for certain multiplicative generators in the complex cobordism ring.
Paper proves polynomial equivalence of quantum complexity metrics.
problem Quantum complexity metrics equivalence.
method Study of right-invariant metrics on unitary group.
result All metrics in the equivalence class have polynomial slowdown in approximation.
Automorphisms of pants complex are shown to be inner.
problem Understanding automorphisms of pants complexes.
method Proving automorphisms are inner for specific groups.
result Automorphism groups are naturally isomorphic to pants complex.
We construct a graph complex calculating the integral ho- mology of the bordered mapping class groups. We compute the ho- mology of the bordered mapping class groups of various surfaces. Using the circle action on this graph complex, we build a double complex and a spectral sequence converging to the homology of the un…
We establish various stability results for solutions of complex Monge-Ampère equations in big cohomology classes, generalizing results that were known to hold in the context of Kähler classes.
Stability results for complex Monge-Ampère equations in various classes.
problem Stability of solutions to complex Monge-Ampère equations.
method Weak stability results followed by stability proofs.
result Proves stability of solutions in relative full mass classes and on quasi-projective varieties.
We use mathematical induction to prove that the horizontal composition in the class of coherently diagonal complexes is indeed a binary operation. That is to say, the embedding of two coherently diagonal complexes in an alternating planar diagram produces a coherently diagonal complex.
This paper deals with the question of J.Morava on existence of canonical complex cobordism class of singular submanifold. We present several solutions of this question for -- the set of points where generic sections of a complex vector bundle are linearly dependent. The corresponding complex co…
We prove the existence of plurisubharmonic functions with prescribed logarithmic singularities on complex 3-folds equipped with a nef class of positive volume. We prove the same result for rational classes on Moishezon n-folds.
S. Gervais gave a finite presentation for the mapping class group of a surface (math.GT/9811162). We show this presentation without using Wajnryb's simple presentation. We use a complex of curves defined by Harvey, in place of Hatcher and Thurston's complex.
Study algebraic obstructions to knot-like complex realizability.
problem Algebraic obstructions to knot-like complex realizability.
method Classification of local equivalence classes over .
result Classification answers a question about knot-like complexes.
New calculations of topological complexity for symplectic CW-complexes.
problem Calculating topological complexity for symplectic CW-complexes.
method Using atoroidal cohomology classes and CW-complexes, proving topological complexity for symplectic spaces.
result Every atoroidally symplectic CW-complex of dimension 2n has topological complexity 4n.
In this paper, we study the complex Landsberg spaces and some of their important subclasses. We introduce and characterize the class of generalized Berwald and complex Landsberg spaces. The intersection of these spaces gives the so called Landsberg class.
Improved analysis for extreme multi-class CRL with better sample complexity.
problem Theoretical sample complexity of CRL in extreme multi-class settings is poorly understood.
method Improved U-Statistics estimator to capture class concentration, proving sample complexity.
result Sample complexity is for extreme multi-class learning, independent of class distribution.
We propose the definition of a Manifold with a complex Lie structure at infinity. The important class of ACH manifolds enters into this class.
Study quotients of curve complex actions by mapping class group.
problem Understanding actions of mapping class group on curve complex quotients.
method Cone off uniformly quasi-convex subspaces to form symmetric curve sets, non-maximal train track sets, and compression body disc sets. Analyze actions of mapping class group on these quotients.
result Actions of mapping class group on quotients are strongly WPD, non-elementary, and have infinite diameter.
In 1992, Osamu Kakimizu defined a complex that has become known as the Kakimizu complex of a knot. Vertices correspond to isotopy classes of minimal genus Seifert surfaces of the knot. Higher dimensional simplices correspond to collections of such classes of Seifert surfaces that admit disjoint representatives. We show…
Treating a conjecture, P^#P != NP, on the separation of complexity classes as an axiom, an implication is found in three manifold topology with little obvious connection to complexity theory. This is reminiscent of Harvey Friedman's work on finitistic interpretations of large cardinal axioms.
The paper examines Kodaira dimensions of specific complex 4-manifolds with torsion first Chern class.
problem Investigating the Kodaira dimension of almost complex 4-manifolds with torsion first Chern class.
method Developed theory of pseudoholomorphic structures on vector bundles, computed tangent spaces of infinitesimal deformations, and proved unobstructedness theorems.
result Proved that Kodaira dimension can only be 0 or -∞ for tamed almost complex structures.
Active learning method balances bias and variance under class imbalance.
problem Active learning under label shift when class proportions differ.
method Mediated Active Learning under Label Shift (MALLS) using a 'medial distribution'.
result MALLS reduces asymptotic sample complexity under arbitrary label shift.
Almost complex structures found on many homotopy complex projective spaces.
problem Finding almost complex structures on homotopy complex projective spaces.
method New proof using Chern classes and homotopy properties.
result Classification of almost complex structures on homotopy for .
The relationship of generalized manifolds to Poincaré duality complexes and topological manifoldsmath.GT
The primary purpose of this paper concerns the relation of (compact) generalized manifolds to finite Poincaré duality complexes (PD complexes). The problem is that an arbitrary generalized manifold is always an ENR space, but it is not necessarily a complex. Moreover, finite PD complexes require the Poincaré dualit…
We show that the Rademacher complexity of any -valued function class composed with an -Lipschitz function is bounded by the maximum Rademacher complexity of the restriction of the function class along each coordinate, times a factor of .
The paper analyzes risk bounds and Rademacher complexity in batch RL.
problem Estimating/minimizing Bellman error with general value function approximation.
method Characterizes generalization performance using Rademacher complexities of function classes.
result Risk bounds and Rademacher complexities provide insights into batch RL.
For compact complex manifolds with vanishing first Chern class that are compact torus principal bundles over Kähler manifolds, we prove that all holomorphic geometric structures on them, of affine type, are locally homogeneous. For a compact simply connected complex manifold in Fujiki class , whose dimensio…
A complex analytic approach to orbifold Chern classes on singular varieties and its applicationsmath.AG
The article proves a complex analytic inequality for stable Q-sheaves on Kähler varieties.
problem Proving a Bogomolov-Gieseker inequality for stable Q-sheaves on Kähler varieties.
method Complex analytic approach, including a new purely analytical proof and novel interpretation of orbifold Chern classes.
result Characterization of the equality case in the Bogomolov-Gieseker inequality and novel interpretation of the second orbifold Chern class.
Classifies Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.
problem Identifying Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.
method Comprehensive classification of commensurability classes of cusped, arithmetic, and non-arithmetic complex hyperbolic 2-manifolds.
result Some Nil 3-manifolds are cross-sections in every commensurability class, while others are cross-sections in only one.
The paper studies complexes of hypersurfaces in homology classes and proves their connectedness and simple connectedness.
problem Investigating complexes of hypersurfaces in homology classes and proving their topological properties.
method Defining and analyzing simplicial complexes and for properly embedded hypersurfaces in -manifolds, proving connectedness and simple connectedness.
result Proves connectedness and simple connectedness of the complexes and .
Strongly Gauduchon spacesmath.CV
We define strongly Gauduchon spaces and the class SG which are generalization of strongly Gauduchon manifolds in complex spaces. Comparing with the case of Kahlerian, the strongly Gauduchon space and the class SG are similar to the Kahler space and the Fujiki class C respectively. Some properties about these complex sp…
Comparative learning combines realizable and agnostic settings for two hypothesis classes, reducing sample complexity.
problem Learning with two hypothesis classes in a more general setting than single hypothesis classes.
method Introduces comparative learning, defines mutual VC dimension and Littlestone dimension, and applies insights to multiaccuracy and multicalibration.
result Sample complexity of comparative learning is characterized by mutual VC dimension and Littlestone dimension.