New concordance invariant from spectral sequence on Khovanov homology.
problem Concordance invariants from spectral sequences.
method Constructing from E(−1) spectral sequence on Khovanov homology. result Provides a bound on the nonorientable slice genus.
Unique metrics on symplectic groups identified by spectral data.
problem Identifying bi-invariant metrics on symplectic groups using spectral data.
method Proving spectral uniqueness using a strong spectral obstruction.
result Left-invariant metrics not right-invariant cannot be isospectral to bi-invariant metrics.
The spectral sequence's Ek-page is a link invariant for k≥3.
problem Proving the invariance of the spectral sequence pages.
method Analyzing the spectral sequence construction and its properties.
result The Ek-page is a link invariant for k≥3. Study of invariants on manifolds with boundary involving equivariant spectral flow and η-invariants.
problem Equivariant invariants on manifolds with boundary.
method Analysis of Dirac operators, winding numbers, spectral flow, Maslov indices, and η-invariants.
result Established relation between equivariant η-invariants and Maslov triple indices.
Researchers lift spectral ζ-invariants to coverings using defect formulae.
problem Lifting spectral ζ-invariants to coverings of closed manifolds.
method Using lifted defect formulae and Wodzicki residues.
result Derived integrals of Pontryagin and Chern forms for certain lifted spectral ζ-invariants.
Study beta function for convex billiard maps, linking spectral invariants.
problem Understanding spectral invariants of convex billiard maps.
method Birkhoff normal form via constructive generating functions, explicit beta function formula.
result Linked spectral invariants to beta function for convex billiard maps.
Researchers calculate spectral invariants from Dirichlet-to-Neumann map for Witten-Laplacian with potential.
problem Calculating spectral invariants from Dirichlet-to-Neumann map for Witten-Laplacian with potential.
method Established an effective procedure to calculate all coefficients of the spectral asymptotic formula of the Dirichlet-to-Neumann map.
result Explicitly provided the first four coefficients of the spectral asymptotic formula.
Study new spectral invariants for Laplace and Dirac operators on manifolds.
problem Understanding spectral invariants of elliptic operators on manifolds.
method Introduced new spectral invariants depending on eigenvalues and eigensections, computed asymptotic expansion.
result Computed first two coefficients of the asymptotic expansion of the new spectral invariant.
Study spectral invariants over integers, discovering unboundedness and field-dependence.
problem Dependence of spectral invariants on integer coefficients.
method Floer homology theory, focusing on Hamiltonian Floer homology.
result Spectral norm is unbounded over integers for complex projective spaces.
Study of spectral invariants on CR contact manifolds with circle action.
problem Analytic torsion and eta-like invariants on CR contact manifolds.
method Interpret spectral series topologically and dynamically using Reeb flow.
result Spectral series can be interpreted both topologically and dynamically.
GRID invariants block certain Lagrangian cobordisms in 3D.
problem Obstructing decomposable Lagrangian cobordisms in 3D.
method Filtered GRID invariants and link Floer homology.
result GRID invariants obstruct decomposable Lagrangian cobordisms.
New spectral invariants from two elliptic operators reveal manifold geometry.
problem Understanding geometric information from two elliptic operators on manifolds.
method Introducing and studying new relative spectral invariants, proving asymptotic expansions.
result Existence and computation of coefficients in the asymptotic expansion of new invariants.
Combining known spectral sequences with a new spectral sequence relating reduced and unreduced sl(N)-homology yields a relationship between the Homflypt-homology of a knot and its sl(N)-concordance invariants. As an application, some of the sl(N)-concordance invariants are shown to be linearly independent.
New spectral invariants distinguish Joyce orbifolds from other manifolds.
problem Distinguishing Joyce orbifolds from other G2-structures. method Introducing and computing two new spectral invariants for Joyce orbifolds.
result These invariants are more effective than existing invariants for distinguishing Joyce orbifolds.
Spectral flow connects manifold geometry to rigidity criteria.
problem Tackling rigidity of simply-connected closed manifolds.
method Spectral deformation flow and invariant-based approach.
result Spherical profile is the unique manifold-compatible asymptotic realization.
New spectral invariants recover Calabi invariant for surface dynamics.
problem Understanding Hamiltonian homeomorphisms and spectral invariants.
method Defining new spectral invariants for Lagrangian links in surfaces.
result Our invariants recover the Calabi invariant and resolve open questions.
New method calculates eta invariant without analytic continuation.
problem Spectral asymmetry of non-semibounded systems.
method Direct pseudodifferential technique for curl operator.
result Eta invariant can be traced as spectral projection difference.
New spectral sequences link knot homology to instantons, revealing concordance invariants.
problem Understanding the relationship between knot homology and instantons.
method Established spectral sequences connecting Bar-Natan homology to instanton homology.
result Derived concordance invariants from instanton homology groups.
In this paper we study some new von Neumann spectral invariants associated to the Laplacian acting on L^2 differential forms on the universal cover of a closed manifold. These invariants coincide with the Novikov-Shubin invariants whenever there is no spectral gap in the spectrum of the Laplacian, and are homotopy inva…
Study spectral invariants of singularities using Schrödinger operators.
problem Understanding the Milnor number of quasi-homogeneous singularities.
method Local index theory of Schrödinger operators and heat kernel expansions.
result Define torsion type invariants to study singularities.
Constructs equivariant spectral flow for Dirac-type operators on manifolds.
problem Calculating spectral flow for Dirac-type operators on manifolds with group actions.
method Equivariant spectral flow construction for paths of Dirac-type operators on manifolds.
result Relates delocalised η-invariants and ρ-invariants for different positive scalar curvature metrics.
Paper studies spectral invariants and monopole Floer homology for rational homology three-spheres.
problem Tackles the existence of positive scalar curvature metrics on ribbon homology cobordisms.
method Defines an R-filtration on the equivariant complex of monopole Floer homology via Chern-Simons-Dirac functional, leading to a spectral invariant.
result Shows that the spectral invariant provides an obstruction to the existence of positive scalar curvature metrics on ribbon homology cobordisms.
New spectral sequences define knot invariants.
problem Understanding strongly invertible knots.
method Two spectral sequences in knot Floer homology.
result Numerical invariant defined for strongly invertible knots.
Spectral sequence links knot Floer homology to HOMFLY-PT polynomial.
problem Connecting knot Floer homology to HOMFLY-PT polynomial.
method Constructing a spectral sequence from cube of resolutions.
result Spectral sequence converges to knot Floer homology and ranks HOMFLY-PT homology.
Seidel-Smith and Hendricks used equivariant Floer cohomology to define some spectral sequences from symplectic Khovanov homology and Heegaard Floer homology. These spectral sequences give rise to Smith-type inequalities. Similar-looking spectral sequences have been defined by Lee, Bar-Natan, Ozsváth-Szabó, Lipshitz-Tre…
New invariants detect a specific graph in spatial webs.
problem Detecting specific graphs in spatial webs.
method Introduced new invariants and used spectral sequences.
result Proved invariants detect the planar theta graph.
New approach connects 3D Chern-Simons theory to spectral networks.
problem Understanding Chern-Simons invariants in 3D manifolds.
method Constructing equivalences between bundles and spectral networks.
result New formulas for Chern-Simons invariants of 3D manifolds.
Defines spectral varieties for non-simply connected manifolds and constructs conformal invariants.
problem Analyzing spectra of magnetic Laplacians on non-simply connected manifolds.
method Definition of spectral varieties and construction of conformal invariants.
result New conformal invariants of immersions of surfaces into 3- and 4-dimensional spaces.
Define mass invariant for asymptotically hyperbolic 3-manifolds with toroidal infinity.
problem Define mass invariant for asymptotically hyperbolic 3-manifolds with toroidal infinity.
method Define mass invariant for asymptotically hyperbolic 3-manifolds with toroidal infinity.
result Define mass invariant for asymptotically hyperbolic 3-manifolds with toroidal infinity.
New neural architectures invariant to sign flips and basis symmetries for graph representation learning.
problem Learning invariant graph representations from eigenvectors.
method SignNet and BasisNet neural architectures that are invariant to sign flips and basis symmetries.
result Proven to be universal, approximating any continuous function of eigenvectors with desired invariances.
New annular link invariants from spectral sequence filtration.
problem Defining invariants for links in annuli.
method Sarkar-Seed-Szabó spectral sequence filtration.
result Necessary and sufficient conditions for braid properties.
The paper calculates spectral invariants of magnetic Steklov eigenvalues on Riemannian manifolds.
problem Investigating spectral invariants of magnetic Steklov eigenvalues on Riemannian manifolds.
method Established an effective procedure to calculate all coefficients of the heat trace asymptotic expansion.
result Explicitly provided expressions for the first four coefficients of the heat trace asymptotic expansion.
Defines spectral sequences for fiberwise Dirac operators and proves adiabatic limit formula.
problem Calculating eta invariants for fibrations.
method Heat kernel method and analytic localization techniques.
result Extends remainder terms of eta invariants in fibrations.
The paper connects group extensions, cochains, and spectral sequences.
problem Understanding the relationship between group extensions and spectral sequences.
method Using connection cochains, the paper derives a formula for the extension class.
result A formula clarifies the relation among connection cochains, extension classes, and the LHS spectral sequence.
We introduce three spectral sequences which give some expressions of colored Jones polynomials. Each spectral sequence contains a Khovanov-type homology groups. Two of them are derived from a bicomplex of the colored Jones polynomial. The other is the spectral sequence that deduces a colored Rasmussen invariant of link…
Study the spectral flow of Dirac operators on spinor bundles.
problem Understanding the asymptotic behavior of spectral flow for Dirac operators.
method Variation of eta invariant and local index theory technique.
result Established a uniform estimate of the eta invariant for large parameter values.
Study shows infinite Hofer diameter for Lagrangian orbits in cotangent bundles.
problem Determining boundedness of spectral metric on Lagrangian orbit spaces.
method Utilized wrapped Floer cohomology to define spectral invariant and pseudo-metric.
result Proved infinite Hofer diameter for Lagrangian orbits in cotangent bundles.
Study spectral invariants for polygons and orbisurfaces.
problem Compute spectral invariants for polygons and orbisurfaces.
method Study the heat kernel and asymptotic expansion of the heat trace.
result Explicit formulas for heat invariants of polygons and orbisurfaces.
A proper etale Lie groupoid is modelled as a (noncommutative) spectral geometric space. The spectral triple is built on the algebra of smooth functions on the groupoid base which are invariant under the groupoid action. Stiefel-Whitney classes in Lie groupoid cohomology are introduced to measure the orientability of th…
We study the relationship between Bar-Natan's perturbation in Khovanov homology and Szabo's geometric spectral sequence, and construct a link invariant that generalizes both into a common theory. We study a few properties of the new invariant, and introduce a family of s-invariants from the new theory in the same spiri…
We study the eta-invariant, defined by Atiyah-Patodi-Singer a real valued invariant of an oriented odd-dimensional Riemannian manifold equipped with a unitary representation of its fundamental group. When the representation varies analytically, the corresponding eta-invariant may have an integral jump, known also as th…
New algorithm calculates S-invariants for links efficiently.
problem Cannot be done by standard spectral sequence methods.
method Developed an algorithm for S-invariants of links. result Demonstrated efficiency and applicability to sl(3)-link homology.
We introduce the notion of a Khovanov-Floer theory. Roughly, such a theory assigns a filtered chain complex over Z/2 to a link diagram such that (1) the E_2 page of the resulting spectral sequence is naturally isomorphic to the Khovanov homology of the link; (2) this filtered complex behaves nicely under planar isotopy…
Optimal spectral estimators and AMP combine for efficient weak recovery in orthogonally invariant GLMs.
problem Parameter estimation from generalized linear models with complex correlation structures.
method Spectral initialization and approximate message passing (AMP) algorithm.
result Established rigorous performance guarantees for spectral initialization and AMP.
We show that the SU(3) Casson invariant for spliced sums along certain torus knots equals 16 times the product of their SU(2) Casson knot invariants. The key step is a splitting formula for su(n) spectral flow for closed 3-manifolds split along a torus.
Researchers study spectral asymmetry using pseudodifferential projections on the massless Dirac operator.
problem Understanding spectral asymmetry for the massless Dirac operator.
method Constructing a negative order pseudodifferential asymmetry operator from spectral projections.
result Computed the principal symbol of the asymmetry operator, accounting for gauge invariance.
This paper describes a topological method to compute the spectral flow of a family of twisted Dirac operators, it includes two detailed examples. Briefly, a formula of Atiyah, Patodi and Singer expresses the spectral flow in terms of Chern-Simons invariants and rho invariants. The first step is to construct a flat cobo…
I review the milestones of the mathematical work of Krzysztof P. Wojciechowski. This will at the same time be a tour of Analysis and Geometry of Boundary Value Problems. Starting in the 80s I will discuss the spectral flow and the general linear conjugation problem, the Calderon projector and the topology of space of e…