arXiv research

New concordance invariant from spectral sequence on Khovanov homology.

The spectral sequence's $E_k$-page is a link invariant for $k\ge 3$.

Study of invariants on manifolds with boundary involving equivariant spectral flow and η-invariants.

Study beta function for convex billiard maps, linking spectral invariants.

Researchers calculate spectral invariants from Dirichlet-to-Neumann map for Witten-Laplacian with potential.

Study spectral invariants over integers, discovering unboundedness and field-dependence.

Study of spectral invariants on CR contact manifolds with circle action.

GRID invariants block certain Lagrangian cobordisms in 3D.

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.

Spectral flow connects manifold geometry to rigidity criteria.

New spectral invariants recover Calabi invariant for surface dynamics.

New method calculates eta invariant without analytic continuation.

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…

Constructs equivariant spectral flow for Dirac-type operators on manifolds.

Paper studies spectral invariants and monopole Floer homology for rational homology three-spheres.

New spectral sequences define knot invariants.

The canonical trace and the Wodzicki residue on classical pseudodifferential operators on a closed manifold are characterised by their locality and shown to be preserved under lifting to the universal covering as a result of their local feature. As a consequence, we lift a class of spectral $ζ$-invariants using lifted …

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 approach connects 3D Chern-Simons theory to spectral networks.

Define mass invariant for asymptotically hyperbolic 3-manifolds with toroidal infinity.

Defines spectral varieties for non-simply connected manifolds and constructs conformal invariants.

New neural architectures invariant to sign flips and basis symmetries for graph representation learning.

The paper calculates spectral invariants of magnetic Steklov eigenvalues on Riemannian manifolds.

Defines spectral sequences for fiberwise Dirac operators and proves adiabatic limit formula.

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.

Study shows infinite Hofer diameter for Lagrangian orbits in cotangent bundles.

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…

New algorithm calculates $S$-invariants for links efficiently.

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…

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.

Optimal spectral estimators and AMP combine for efficient weak recovery in orthogonally invariant GLMs.

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…

Researchers study spectral asymmetry using pseudodifferential projections on the massless Dirac operator.

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…

In this short note, we prove that a bi-invariant Riemannian metric on $\mathrm{Sp}(n)$ is uniquely determined by the spectrum of its Laplace-Beltrami operator within the class of left-invariant metrics on $\mathrm{Sp}(n)$. In other words, on any of these compact simple Lie groups, every left-invariant metric which is n…

Spectral portfolio theory links neural networks to wealth dynamics via SGD weight matrices.

Computes a $ν$-invariant for Joyce's $G_2$-manifolds.

We associate a Taylor tower supplied by calculus of the embedding functor to the space of long knots and study its cohomology spectral sequence. The combinatorics of the spectral sequence along the line of total degree zero leads to chord diagrams with relations as in finite type knot theory. We show that the spectral …

New spectral sequence for $\mathcal{K}$-manifolds, computing cohomology and harmonic forms.

The SU(3)-Casson invariant for integral homology 3-spheres as studied by Boden-Herald possesses a 'spectral flow obstruction' to being an integer valued invariant which depends only on the non-degenerate (perturbed) moduli space of flat SU(3)-connections. This obstruction is the non-trivial spectral flow of a family of…

We review the concepts of the index of a Fredholm operator, the spectral flow of a curve of self-adjoint Fredholm operators, the Maslov index of a curve of Lagrangian subspaces in symplectic Hilbert space, and the eta invariant of operators of Dirac type on closed manifolds and manifolds with boundary. We emphasize var…

A new method for spectral positional encodings in directed graphs using Hermitian block Krylov subspaces.

The paper classifies K-contact forms on 3-manifolds and connects their orbits to spectral invariants.

Ozsvath and Szabo show that there is a spectral sequence whose E^2 term is the reduced Khovanov homology of L, and which converges to the Heegaard Floer homology of the (orientation reversed) branched double cover of S^3 along L. We prove that the E^k term of this spectral sequence is an invariant of the link L for all…