Study virtual fundamental classes of derived manifolds, proving invariant vanishes.
problem Computing virtual fundamental classes for derived manifolds.
method Combining derived differential geometry and cosection localization.
result Stable pair invariants of hyperkähler fourfolds are zero.
Continues work on derived manifolds and symplectic schemes, constructing virtual classes.
problem Constructing virtual fundamental classes for derived manifolds and schemes.
method Cosection localization, reduced virtual fundamental classes, and applications to Donaldson-Thomas theory.
result Virtual fundamental classes for (−2)-shifted symplectic derived schemes are consistent with algebraic and differential geometric constructions. Algorithm finds finite fundamental bikei for virtual knots.
problem Determining which virtual knots have finite fundamental bikei.
method Implemented an algorithm to complete presentation matrices to operation tables.
result Computed fundamental bikei for all prime virtual knots with up to four crossings.
New groups from virtual link stacks distinguish Kishino knots.
problem Detecting and distinguishing Kishino knots from the unknot and from each other.
method Constructing groups from virtual link stacks and using them to distinguish knots.
result The groups constructed using this method distinguish all Kishino knots from the unknot and from each other.
In general, a Kobayashi-Hitchin correspondence establishes an isomorphism between a moduli space of stable algebraic geometric objects and a moduli space of solutions of a certain (generalized) Hermite-Einstein equation. We believe that, for a large class of moduli problems, this correspondence respects the virtual fun…
This paper defines Gromov-Witten invariants for exploded manifolds.
problem Defining Gromov-Witten invariants for a new class of manifolds.
method Construction of a virtual fundamental class for Kuranishi categories.
result Independence and compatibility of the invariants with various operations.
In this paper, we discuss filamentations on oriented chord diagrams. When a filamentation cannot be realized on an oriented chord diagram, then the corresponding flat virtual knot is non-trivial. If a flat knot diagram is non-trivial, then any virtual diagram whose shadow is the flat diagram must also be non-trivial. W…
The paper studies Kähler groups and their virtual algebraic fibrations, finding implications for their fundamental groups.
problem Understanding the structure of Kähler groups and their finite index subgroups.
method Analyzing the virtual algebraic fibrations and Albanese dimension of Kähler groups.
result Groups with virtual Albanese dimension ≤ 1 have strong relations with surface groups and coincide with Green--Lazarsfeld sets.
Let p be a prime. In this paper, we classify the geometric 3-manifolds whose fundamental groups are virtually residually p. Let M=M3 be a virtually fibered 3-manifold. It is well-known that G=π1(M) is residually solvable and even residually finite solvable. We prove that G is always virtually residually p…
Defines virtual immersions to characterize symmetric spaces.
problem Characterizing symmetric spaces using virtual immersions.
method Defines virtual immersions as a generalization of isometric immersions, proves characterization of symmetric spaces.
result A manifold admits a virtual immersion with skew symmetric second fundamental form if and only if it is a symmetric space.
New classification and realization results for 3D and higher PD complexes with highly connected covers.
problem Classifying and realizing PD complexes with (n−2)-connected universal cover. method Using fundamental group, orientation class, and homology class, proving classification, realization, and splitting results.
result Triple (G,ω,μ) can be realized by a PDn-complex with (n−2)-connected universal cover if and only if certain conditions are met. Let (X,ωX∗) be a separated, −2-shifted symplectic derived C-scheme, in the sense of Pantev, Toen, Vezzosi and Vaquie arXiv:1111.3209, of complex virtual dimension vdimCX=n∈Z, and Xan the underlying complex analytic topological space. We prove that …
New invariant for virtual n-links defined and studied.
problem Detecting virtual trefoil and other virtual links.
method Defining a new virtual link invariant VD(K) and studying its geometric properties. result The Dehn space DD(K) is an invariant of K and can detect the virtual trefoil. This paper connects virtual biquandles to biquandles for virtual link colorings.
problem Extending invariants from biquandles to virtual biquandles.
method Establishing equivalence between two representations of virtual braid groups and introducing new labeling rules.
result The number of colorings of a virtual link by virtual biquandles can be recovered from colorings by biquandles.
Let M be a graph manifold. We prove that fundamental groups of embedded incompressible surfaces in M are separable in the fundamental group of M, and that the double cosets for crossing surfaces are also separable. We deduce that if there is a "sufficient" collection of surfaces in M, then the fundamental group of M is…
Computes virtually cyclic dimension for 3-manifold groups.
problem Computing the virtually cyclic geometric dimension of 3-manifold groups.
method Prime and JSJ decompositions of M, push-out type constructions, Bredon cohomology computations.
result Explicit computation of virtually cyclic geometric dimension for 3-manifold groups.
Alternative definition of Casson invariants using virtual counting.
problem Defining Casson invariants in a new way.
method Virtual counting of moduli space of representations into SU(2).
result Alternative method to compute Casson invariants.
We show that a closed, connected, oriented, Riemannian n-manifold, admitting a branched cover of bounded length distortion from Rn, has a virtually Abelian fundamental group.
Proves hyperbolized groups are virtually compact special and linear.
problem Proving hyperbolized groups are virtually compact special and linear.
method Constructing an action of a hyperbolized group on a dual CAT(0) cubical complex.
result Proves hyperbolized groups are virtually compact special and linear.
Given a prime p, a group is called residually p if the intersection of its p-power index normal subgroups is trivial. A group is called virtually residually p if it has a finite index subgroup which is residually p. It is well-known that finitely generated linear groups over fields of characteristic zero are …
The paper proves nonvanishing cohomology for ball quotient fundamental groups.
problem Proving nonvanishing cohomology for ball quotient fundamental groups.
method Using arithmetic lattices and profinite completions, the paper constructs an open subgroup with nontrivial cohomology.
result The virtual cohomological dimension of the fundamental group is at least 2n. We prove that the fundamental group of any Seifert 3-manifold is conjugacy separable. That is, conjugates may be distinguished in finite quotients or, equivalently, conjugacy classes are closed in the pro-finite topology.
This part 2 discusses virtual fundamental chain and cycle technique for K-systems.
problem Foundation of virtual fundamental chain and cycle technique for K-systems.
method Consider a system of spaces with Kuranishi structures and their simultaneous perturbations.
result Discuss the virtual fundamental chain and cycle technique for K-systems.
Paper proves fundamental groups of certain 3+ manifold are virtually free.
problem Understanding fundamental groups of manifolds with specific curvature properties.
method Analyzing the fundamental group of closed manifolds with (n-1)-positive Ricci curvature.
result Fundamental groups of closed manifolds with 2-positive Ricci curvature are virtually free.
In this paper, we show that a nontrivial compact graph manifold is nonpositively curved if and only if its fundamental group virtually embeds into a right-angled Artin group. As a consequence, nonpositively curved graph manifolds have linear fundamental groups.
Finite stature proven for cube complexes with cyclonormal edges.
problem Understanding the structure of cube complexes with specific edge properties.
method Analyzing the fundamental groups of edge and vertex spaces, showing cyclonormality and virtual specialness.
result The fundamental group of a cube complex has finite stature with respect to vertex groups.
Cancellation theorem for 4-manifolds with virtually abelian fundamental group.
problem Understanding the relationship between stable homeomorphism and homeomorphism for 4-manifolds with virtually abelian fundamental groups.
method Analyzing the fundamental group and using the virtually abelian property to reduce the required number of summands in the cancellation theorem.
result For virtually abelian fundamental groups, the number of summands needed for stable homeomorphism to homeomorphism can be significantly reduced.
We prove that rationally essential manifolds with suitably large fundamental groups do not admit any maps of non-zero degree from products of closed manifolds of positive dimension. Particular examples include all manifolds of non-positive sectional curvature of rank one and all irreducible locally symmetric spaces of …
A new method for virtual homotopy classes of virtual strings.
problem Addressing an open problem of Turaev regarding virtual homotopy classes of virtual strings.
method Generalizing Turaev's method to multistring based matrices for virtual n-strings.
result Construction of similar invariants for virtual homotopy classes of virtual n-strings.
This paper studies an algebraic invariant of virtual knots called the biquandle. The biquandle generalizes the fundamental group and the quandle of virtual knots. The approach taken in this paper to the biquandle emphasizes understanding its structure in terms of compositions of morphisms, where elementary morphisms ar…
New proof for complex 3D shapes.
problem Complex 3D shapes with infinite fundamental groups.
method Direct proof using finite covers.
result Every shape has a simpler version.
Characterizes holonomies of convex projective cusps.
problem Understanding holonomies in strictly convex projective geometry.
method Complete characterization of holonomies for strictly convex and round cusps, building families of generalized cusps.
result Produces the first example of generalized cusps with non-virtually nilpotent fundamental group.
Proves strong version of Hopf problem for hyperbolic circle bundles.
problem Strong version of Hopf problem for circle bundles over aspherical manifolds with hyperbolic fundamental groups.
method Analyzes circle bundles over aspherical manifolds with hyperbolic fundamental groups.
result Every self-map of non-zero degree is either homotopic to a homeomorphism or a non-trivial covering and the bundle is virtually trivial.
Defines kappa classes on KSBA spaces, generalizing classes on curves.
problem Generalizing Miller-Morita-Mumford classes to KSBA stable varieties and pairs.
method Defining kappa classes on KSBA moduli spaces and computing them in specific cases.
result Computed kappa classes on Campedelli surfaces, including finding the Chow ring of a specific GIT quotient.
Classifies virtual links up to a specific move.
problem Characterizing 2-component virtual links using Ξ-moves. method Refined odd writhe and linking numbers.
result Classifies 2-component virtual links up to Ξ-moves. The paper defines and analyzes configuration Lie groupoids and orbifold braid groups.
problem Understanding the structure and properties of orbifold braid groups.
method Definitions and proofs of fibration theorems, short exact sequences, and poly-virtually free structures.
result The pure orbifold braid groups have poly-virtually free structure, generalizing classical braid groups.
The writhe polynomial invariant is proven for virtual knots via shell moves.
problem Determining equivalence of oriented virtual knots using writhe polynomials.
method Introducing shell moves to prove equivalence of writhe polynomials.
result Two virtual knots are equivalent if they can be transformed by shell moves.
Virtual knot theory is a generalization (discovered by the author in 1996) of knot theory to the study of all oriented Gauss codes. (Classical knot theory is a study of planar Gauss codes.) Graph theory studies non-planar graphs via graphical diagrams with virtual crossings. Virtual knot theory studies non-planar Gauss…
Survey on quasi-isometric rigidity for piecewise geometric manifolds.
problem Understanding quasi-isometric rigidity for groups of piecewise geometric manifolds.
method Analysis of geometric group actions and quasi-actions on Riemannian manifolds and trees of spaces via asymptotic cones.
result Quasi-isometric rigidity results for fundamental groups of 3-manifolds and higher-dimensional graph manifolds.
Study on the topology of leaves in singular Riemannian foliations.
problem Characterizing the topology of leaves in singular Riemannian foliations.
method Analyzing the fundamental groups and using nilpotent spaces.
result Leaves of singular Riemannian foliations are finitely covered by nilpotent spaces when M is simply connected. In 2007 Agol showed that if N is an aspherical compact 3-manifold with empty or toroidal boundary such that its fundamental group is virtually RFRS, then N is virtually fibered. We give a largely self-contained proof of Agol's theorem using complexities of sutured manifolds.
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.
A new invariant for links in lens spaces with advantages over traditional methods.
problem Invariants for links in lens spaces.
method Constructing a virtual quandle for links in lens spaces L(p,1). result The virtual quandle is an essential invariant and its presentation can be easily written from a link's band diagram.
Let M be a compact oriented irreducible 3-manifold which is neither a graph manifold nor a hyperbolic manifold. We prove that the fundamental group of M is virtually special.
Let M be a complete hyperbolic 3-manifold of finite volume that admits a decomposition into right-angled ideal polyhedra. We show that M has a deformation retraction that is a virtually special square complex, in the sense of Haglund and Wise and deduce that such manifolds are virtually fibered. We generalise a theorem…
Open manifolds with nonnegative Ricci curvature and Euclidean volume growth have finitely generated fundamental groups.
problem Understanding fundamental groups of open manifolds with specific curvature properties.
method Analyzing universal covers with Euclidean volume growth and applying topological group theory.
result Fundamental groups are finitely generated and virtually abelian under given conditions.
We characterise the virtually abelian groups which are fundamental groups of compact Kähler manifolds and of smooth projective varieties. We show that a virtually abelian group is Kähler if and only if it is projective. In particular, this allows to describe the Kähler condition for such groups in terms of integral sym…
Study calculates virtually-cyclic dimension for specific mapping class groups.
problem Calculating virtually-cyclic dimension for mapping class groups of specific surfaces.
method Analyzes mapping class groups of punctured spheres with up to six punctures.
result Determines virtually-cyclic dimension for specific surfaces.