Introduces fundamental heaps for surfaces, linking them to cocycle invariants.
problem Isotopy invariants of surface embeddings in 3-space.
method Definition of fundamental heaps using surface ribbons and heap operations.
result Fundamental heaps have a free part whose rank matches the number of connected components.
Defined by Joyce and Matveev, the fundamental quandle is a complete invariant of oriented classical knots. We consider invariants of knots defined from quotients of the fundamental quandle. In particular, we introduce the fundamental Latin Alexander quandle of a knot and consider its Gröbner basis-valued invariants, wh…
We find the complete set of fundamental invariants for systems of ordinary differential equations of order ≥4 under the group of point transformations generalizing similar results for contact invariants of a single ODE and point invariants of systems of the second and the third order. It turns out that starting fr…
The rho-invariant is an invariant of odd-dimensional manifolds with finite fundamental group, and lies in the representations modulo the regular representations (after tensoring with Q). It is a fundamental invariant that occurs in classifying lens spaces, their homotopy analogues, and is intimately related to the eta-…
Equivalent categories of groups and risandles for 3-manifolds.
problem Constructing invariants of 3-manifolds.
method Introducing risandles and showing equivalence with groups, categorifying coloring invariants.
result Equivalence of fundamental groups and risandles for infinitely many 3-manifolds.
We construct elements of the third quandle homology groups of knot quandles, which are called the shadow fundamental classes. They play the same roles for the shadow quandle cocycle invariants of knots as the fundamental classes of knot quandles does for the quandle cocycle invariants. As an application of the shadow f…
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.
The study identifies two sources of invariants in 2--nondegenerate CR geometries.
problem Characterizing fundamental invariants of 2--nondegenerate CR geometries.
method Analyzes the harmonic curvature and the difference in complex structures.
result Nontrivial examples of CR geometries can be obtained as deformations of models.
Study variational problems for integral invariants of maps between pseudo-Riemannian manifolds.
problem Understanding variational properties of integral invariants defined from the second fundamental form.
method Derive first variational formulae for integral invariants of degree two, show Euler-Lagrange equation for Chern-Federer energy, and provide examples of submanifolds.
result The Euler-Lagrange equation of the Chern-Federer energy functional reduces to a second order PDE.
We prove that on closed Riemannian manifolds with infinite abelian, but not cyclic, fundamental group, any isometry that is homotopic to the identity possesses infinitely many invariant geodesics. We conjecture that the result remains true if the fundamental group is infinite cyclic. We also formulate a generalization …
Study verifies asymptotic expansion for Reshetikhin-Turaev invariants of fundamental shadow link pairs.
problem Verifying the asymptotic expansion conjecture for Reshetikhin-Turaev invariants of fundamental shadow link pairs.
method Using logarithmic holonomies of meridians and hyperbolic cone structures, the study verifies the conjecture for pairs where M∖L is homeomorphic to a fundamental shadow link complement. result The asymptotic expansion conjecture is true for pairs (M,L) with sufficiently small cone angles and M∖L homeomorphic to a fundamental shadow link complement. Complete invariant for surfaces in 3-sphere derived from diagrams of fundamental groups.
problem Constructing a complete invariant for closed surfaces in the three-sphere.
method Diagram of fundamental groups, generalization of Kneser conjecture, extensions of Waldhausen's theorem.
result Proves the diagram of fundamental groups is a complete invariant for closed surfaces in the three-sphere.
We prove that the group-homological version of the generalized Goncharov invariant of finite-volume locally rank one symmetric spaces determines their generalized Neumann-Yang invariant, which is defined using ideal fundamental cycles.
Study proves higher-order conformal forms don't exist in odd dimensions.
problem Proving non-existence of higher-order conformal forms in odd dimensions.
method Analyzing conformal hypersurface embeddings and differential order invariants.
result General non-existence of higher-order conformal forms in odd dimensions.
Fundamental group of a manifold gives a deep effect on its underlying smooth structure. In this paper we introduce a new variant of the Donaldson invariant in Yang-Mills gauge theory from twisting by the Picard group of a four manifold in the case when the fundamental group is free abelian. We then generalize it to the…
Paper finds Hempel pairs distinguishable by Turaev-Viro invariants.
problem Surface bundles with non-isomorphic fundamental groups.
method Examines Hempel pairs and their Turaev-Viro invariants.
result Examples of Hempel pairs distinguishable and undistinguishable by Turaev-Viro invariants.
Heap theory applied to framed links yields new invariants.
problem Developing invariants for framed links using heap theory.
method Introducing fundamental heap, defining cocycle invariant using ternary cohomology.
result Found cocycles and computed invariants for specific link families.
Introducing the notion of stabilized fundamental group for the complement of a branch curve in CP2, we define effectively computable invariants of symplectic 4-manifolds that generalize those previously introduced by Moishezon and Teicher for complex projective surfaces. Moreover, we study the structure of these inv…
New method classifies spin 4-manifolds using Kervaire-Milnor invariant.
problem Classifying spin 4-manifolds up to stabilisation.
method Using Kervaire-Milnor invariant to compute Arf invariants and classify.
result New stable classification of spin 4-manifolds with 2-dimensional fundamental groups.
We show that on a closed Riemannian manifold with fundamental group isomorphic to Z, other than the circle, every isometry that is homotopic to the identity possesses infinitely many invariant geodesics. This completes a recent result of the second author.
We define the extremal length of elements of the fundamental group of the twice punctured complex plane and give upper and lower bounds for this invariant. The bounds differ by a multiplicative constant. The main motivation comes from 3-braid invariants and their application.
New bounds on Euler characteristics for certain manifolds with finite groups.
problem Estimating Euler characteristics of specific topological manifolds.
method Defining a new invariant and using cohomological invariants of fundamental groups.
result Established new bounds on minimal Euler characteristics for 4-manifolds.
The paper defines and analyzes quandles of knotoids and linkoids.
problem Tackling the invariant properties of knotoids and linkoids.
method Defining fundamental quandles and pointed quandles, proving invariance, and introducing new invariants.
result Fundamental pointed quandles enhance the fundamental quandle and distinguish 1-linkoids.
A singular point of a smooth map F: M -> N of manifolds is a point in M at which the rank of the differential dF is less than the minimum of dimensions of M and N. The classical invariant of the set S of singular points of F of a given type is defined by taking the fundamental class [\bar{S}]\in H_*(M) of the closure o…
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. We consider the classical problem of a position of n-dimensional manifold M in R^{n+2}. We show that we can define the fundamental (n+1)-cycle and the shadow fundamental (n+2)-cycle for a fundamental quandle of a knotting M to R^{n+2}. In particular, we show that for any fixed quandle, quandle coloring, and shadow quan…
Complete surgery obstructions for manifolds with finite fundamental group, disproving a conjecture.
problem Disproving the Oozing Conjecture for manifolds with finite fundamental group.
method Description and calculation of surgery obstructions up to homotopy equivalence.
result New obstructions found for Arf invariant product formulas in codimensions ≥ 4, disproving the Oozing Conjecture.
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.
We describe a presentation for the augmented fundamental rack of a link in the lens space L(p,1). Using this presentation, the (enhanced) counting rack invariants that have been defined for the classical links are applied to the links in L(p,1). In this case, the counting rack invariants also include the informatio…
New cohomology theory shows compact Lie group actions are Morita invariant.
problem Establishing Morita invariance for cohomology of compact Lie group actions.
method Using bibundles to transfer coefficient systems between Morita equivalent groupoids.
result Twisted Bredon-Illman cohomology is Morita invariant for compact Lie group actions.
Classifies stable diffeomorphism of spin 4-manifolds with specific fundamental groups.
problem Classifying stable diffeomorphism of spin 4-manifolds with given fundamental groups.
method Formulated conjectural relationships between algebraic invariants and obstructions, proved for specific groups.
result Proved conjectures for specific fundamental groups, providing complete algebraic stable classification.
The Yamabe invariant is an invariant of a closed smooth manifold defined using conformal geometry and the scalar curvature. Recently, Petean showed that the Yamabe invariant is non-negative for all closed simply connected manifolds of dimension ≥5. We extend this to show that Yamabe invariant is non-negative for a…
Paper provides a condition to distinguish links up to 4-moves.
problem Distinguishing links up to 4-moves is generally difficult.
method Defined a quotient of the fundamental group invariant preserved by 4-moves.
result Provides a necessary condition for triviality up to 4-moves.
We survey various Alexander-type invariants of plane curve complements, with an emphasis on obstructions on the type of groups that can arise as fundamental groups of complements to complex plane curves. Also included are some new computations of higher-order degrees of curves, which are invariants defined in a previou…
Paper proves Legendrian knots with same GL-rack have similar invariants.
problem Classifying Legendrian knots based on their invariants.
method Examined fundamental GL-racks and their relationship to Thurston-Bennequin and rotation numbers.
result Two Legendrian knots with isomorphic fundamental GL-racks have similar invariants.
This paper explores tradeoffs between invariance and sensitivity in adversarial examples.
problem Understanding the limitations of existing adversarial defenses.
method Study of invariance-based adversarial examples and their impact on model accuracy.
result Adversarial defenses against sensitivity-based attacks can harm invariance-based attacks, necessitating new approaches.
Generalizes Toledo invariant to singular klt varieties, proving Milnor-Wood inequality.
problem Proving Milnor-Wood inequality for singular klt varieties.
method Generalized Toledo invariant to singular klt varieties and proved Milnor-Wood inequality.
result Milnor-Wood inequality for singular klt varieties.
New exotic 4-manifolds with even b2+ and Z/2Z fundamental group.
problem Creating new exotic smooth structures on 4-manifolds with specific fundamental groups.
method Using double node surgery and rational blowdown constructions on elliptic fibrations with a free involution.
result Construction of infinitely many irreducible exotic smooth structures.
Proves conditions for Willmore surfaces to have finite ends or finite total curvature.
problem Conditions for Willmore surfaces to have finite ends or finite total curvature.
method Analyzes scale-invariant second fundamental form near infinity.
result Proves conditions for Willmore surfaces to have finite ends or finite total curvature.
New methods distinguish exotic 4-manifolds using Heegaard Floer homology.
problem Distinguishing exotic 4-manifolds from standard ones.
method Using Heegaard Floer homology to define new invariants.
result New invariants remain distinct in covers and can distinguish exotic manifolds.
Study topological 4-manifolds with specific fundamental groups.
problem Classify topological 4-manifolds with 4-dimensional fundamental group.
method Use algebraic topology, Poincaré duality, and Kirby-Siebenmann invariant.
result Two manifolds are homeomorphic if they are s-cobordant and have same Kirby-Siebenmann invariant.
Given an affine isometry of R3 with hyperbolic linear part, its Margulis invariant measures signed Lorentzian displacement along an invariant spacelike line. In order for a group generated by hyperbolic isometries to act properly on R3, the sign of the Margulis invariant must be constant over the group. We show…
Infinitesimal calculations link fundamental groups to Lie algebras.
problem Calculating logarithm maps in fundamental groups.
method Hopf invariants defined by Harrison cohomology of commutative cochains.
result Zeroth Harrison cohomology is a universal dual to Malcev Lie algebra.
Conformal invariants of manifolds of non-positive scalar curvature are studied in association with growth in volume and fundamental group.
The paper connects arithmetic invariants of hyperbolic 3-manifolds.
problem Understanding the arithmetic properties of hyperbolic 3-manifolds.
method Analyzes profinite completions and algebraic invariants of fundamental groups.
result Uniform lattices with isomorphic profinite completions have identical arithmetic properties.
The paper introduces a new method to characterize cosmological models using observer-based invariants.
problem Equivalence problem for cosmological models in four-dimensional gravity theories.
method Modified Cartan-Karlhede algorithm adapted to fundamental observers, including derivatives of the time-like vector field.
result A list of invariants that completely characterize cosmological models, independent of coordinates.
We define the fundamental quandle of a spatial graph and several invariants derived from it. In the category of graph tangles, we define an invariant based on the walks in the graph and cocycles from nonabelian quandle cohomology.
Proves conjecture on deformation invariance of big fundamental groups.
problem Stability of big fundamental groups under small deformations.
method Deformation regularity of equivariant pluriharmonic maps and techniques from Shafarevich conjectures.
result Deformation openness of big fundamental groups for varieties with big complex local systems.