The Hodge spectra help distinguish orbifolds from manifolds with singularities.
problem Distinguishing orbifolds from manifolds based on their singular sets.
method Computing heat invariants of Hodge Laplacians on p-forms. result The Hodge spectra of 0- and 1-forms distinguish orbifolds from manifolds with singularities. The Hodge spectra can distinguish orbifolds from manifolds, especially in low dimensions.
problem Distinguishing orbifolds from manifolds using Hodge spectra.
method Computing heat invariants of Hodge Laplacians on p-forms. result The Hodge spectra, particularly the 0- and 1-spectra, can distinguish orbifolds from manifolds in low dimensions. We define notions of higher order spectra of a complex quasi-projective manifold with an action of a finite group G and with a G-equivariant automorphism of finite order, some of their refinements and give Macdonald type equations for them.
Let M be a 2m-dimensional compact Riemannian manifold. We show that the spectrum of the Hodge Laplacian acting on m-forms does not determine whether the manifold has boundary, nor does it determine the lengths of the closed geodesics. Among the many examples are a projective space and a hemisphere that have the s…
The article studies spinor and tensor fields on curved spaces, deriving formulas and spectra.
problem Understanding spinor and tensor fields on curved spaces.
method Weitzenböck-type formulas, explicit factorization of Laplace operator, representation theory.
result Explicit factorization of the Laplace operator and spectra calculation on constant curvature spaces.
We use topological methods to prove a semicontinuity property of the Hodge spectra for analytic germs defined on an isolated surface singularity. For this we introduce an analogue of the Seifert matrix (the fractured Seifert matrix), and of the Levine--Tristram signatures associated with it, defined for null-homologous…
Extends plate problems to differential forms on manifolds.
problem Eigenvalue problems for buckling and clamped plates on differential forms.
method Characterizes smallest eigenvalues, proves spectra equivalence, obtains estimates.
result Spectra of plate problems on forms coincide with functions in bounded domains.
New method generalizes eigenvalue inequality to surfaces with boundaries.
problem Eigenvalue inequality for surfaces with boundaries.
method Generalized Rohleder's approach to differential forms, presenting Hodge-Laplacian spectrum.
result Obtained inequality for eigenvalues of Hodge-Laplacian and Dirichlet problems.
For a closed Riemannian orbifold O, we compare the spectra of the Laplacian, acting on functions or differential forms, to the Neumann spectra of the orbifold with boundary given by a domain U in O whose boundary is a smooth manifold. Generalizing results of several authors, we prove that the metric of O can be…
Unified method for analyzing evolving manifolds using de Rham-Hodge theory.
problem Analysis of evolving geometric and topological properties of manifolds.
method Evolutionary de Rham-Hodge method applied to filtration-induced families of de Rham complexes.
result Three sets of topology-preserving singular spectra reveal topological persistence and geometric progression.
To every n-dimensional lens space L, we associate a congruence lattice L in Zm, with n=2m−1 and we prove a formula relating the multiplicities of Hodge-Laplace eigenvalues on L with the number of lattice elements of a given ∥⋅∥1-length in L. As a consequence, we show th…
The paper proves wave operator existence and completeness for Hodge Laplacians.
problem Proving the existence and completeness of wave operators for Hodge Laplacians.
method Integral criterion, probabilistic Bismut-type formulae, heat semigroup, local curvature bounds.
result Absolutely continuous spectra of Hodge Laplacians coincide under quasi-isometry.
Calculates Laplacian spectra on Calabi-Yau hypersurfaces.
problem Computing the spectrum of the Laplacian on complex manifolds.
method Numerical computation of eigenvalues and eigenmodes for line bundles.
result Agreement with exact results for P3 and a torus, first numerical results for Fermat quintic. Results on symplectic spinors and their higher spin versions, concerning representation theory and cohomology properties are presented. Exterior forms with values in the symplectic spinors are decomposed into irreducible modules including finding the hidden symmetry (Schur--Weyl--Howe type duality) given by a represent…
New spectral sequence connects to topological Hochschild homology.
problem Connecting spectral sequences to topological Hochschild homology.
method Developed a spectral sequence and applied Tate diagonal techniques.
result Spectral sequence converges to localized topological Hochschild homology.
Investigates point spectra of vector fields and their properties.
problem Understanding the point spectra of vector fields.
method Define and study point spectra, prove properties under isometries, and analyze compactly supported fields.
result Point spectra are well-behaved under isometries and trivial for compactly supported fields.
Khovanov spectra are shown to be functorial under certain conditions.
problem Understanding functoriality of Khovanov spectra.
method Proving functoriality up to homotopy and sign for Khovanov spectra.
result Khovanov spectra are functorial under specific conditions.
We give a simple sufficient condition for Quinn's "bordism-type spectra" to be weakly equivalent to strictly associative ring spectra. We also show that Poincare bordism and symmetric L-theory are naturally weakly equivalent to monoidal functors. Part of the proof of these statements involves showing that Quinn's funct…
Method calculates spectra of Rarita-Schwinger operator on symmetric spaces.
problem Calculating spectra of the Rarita-Schwinger operator on compact symmetric spaces.
method Using Weitzenböck formulas, Laplace operator, Casimir operator, Freudenthal's formula, and branching rules.
result Obtained spectra on the sphere, complex projective space, and quaternionic projective space.
Proves spectra equivalence for Riemannian manifolds.
problem Equivalence of Almgren-Pitts and phase-transition half-volume spectra.
method Proof of spectra equivalence for Riemannian manifolds.
result Confirms conjecture about spectra equivalence.
We compute the bridge spectra of cables of 2-bridge knots. We also give some results about bridge spectra and distance of Montesinos knots.
Proves Hodge structures on modular functors, providing formulas for Hodge numbers.
problem Existence and uniqueness of Hodge structures on modular functors.
method Non-Abelian Hodge correspondence and Ocneanu rigidity.
result Explicit formulas for Hodge numbers in modular functors of level 2 times an odd number.
We give hodge structures on quasitoric orbifolds. We define orbifold hodge numbers and show a correspondence of orbifold hodge numbers for crepant resolutions of quasitoric orbifolds. In short we extend hodge structures to a non complex setting .
Study calculates spectra of minimal hypersurfaces in hyperbolic space.
problem Computing Laplacian spectra of minimal hypersurfaces.
method Analyzes hypersurfaces in hyperbolic space with specific asymptotic data.
result Obtains spectra and extremal properties of the bottom of the spectrum.
Paper resolves decades-old problem about L-spectra.
problem Identifying L-spectra local information with geometric data. method Proved equivalence of L-orientations and characteristic classes. result Levitt-Ranicki's theory equivalent to Brumfiel-Morgan's classes.
New metrics compare rational spectra using optimal transport.
problem Comparing rational spectra efficiently and accurately.
method Optimal transport and linear-systems theory.
result Established connection to Wasserstein distance.
Study on Hodge theory for almost complex manifolds.
problem Determining Hodge numbers for almost complex manifolds.
method Review and analysis of recent developments in Hodge theory for almost complex manifolds.
result Hodge numbers are almost complex, almost Kähler, or birational invariants in dimension four.
We determine the structure of the Hodge ring, a natural object encoding the Hodge numbers of all compact Kaehler manifolds. As a consequence of this structure, there are no unexpected relations among the Hodge numbers, and no essential differences between the Hodge numbers of smooth complex projective varieties and tho…
In this paper, we numerically investigate the length spectra and the low-lying eigenvalue spectra of the Laplace-Beltrami operator for a large number of small compact(closed) hyperbolic (CH) 3-manifolds. The first non-zero eigenvalues have been successfully computed using the periodic orbit sum method, which are compar…
Tomova, along with results of Bachman and Schleimer, showed that any high distance knot has a stair-step bridge spectrum. In this paper, we compute the bridge spectra and distance of generalized Montesinos knots. In particular, we produce the first example of a class of knots which attain the stair-step bridge spectra …
We refine the Morgan's work on mixed Hodge structures on Sullivan's 1--minimal models by using non-abelian Hodge theory. As an application, we give explicit representatives of real unipotent variations of mixed Hodge structures over compact K"ahler manifolds.
New ICA method for sources with mixed spectra.
problem Inaccurate separation of sources with temporal autocorrelations and mixed spectra.
method Estimates spectral density functions and line spectra using cubic splines and indicator functions, then maximizes the Whittle likelihood function.
result Outperforms existing ICA methods in simulations and EEG data applications.
Jones-Wenzl projectors lifted to Khovanov spectra, proving knot conjectures.
problem Understanding Jones-Wenzl projectors in Khovanov spectra.
method Constructing and studying lifted projectors via maps and polynomial actions.
result Complete computation of 3-colored Khovanov spectrum of the unknot, proving conjectures.
We introduce a framework, twisted parametrized stable homotopy theory, for describing semi-infinite homotopy types. A twisted parametrized spectrum is a section of a bundle whose fibre is the category of spectra. We define these bundles in terms of modules over a stack of parametrized spectra and in terms of diagrams o…
Functor decomposes Khovanov spectra for non-alternating diagrams.
problem Computing Khovanov spectra for diagrams without alternating pairs.
method Functor from cube to Burnside 2-category, decomposition into simplicial complexes.
result Homotopy type of almost-extreme Khovanov spectra computed.
Novel Hodge theory developed for Carrollian bundles, addressing black hole event horizons.
problem Developing Hodge theory on Carrollian bundles with degenerate metrics.
method Constructing Hodge star and Laplacian operators on Carrollian Rimes-bundles. result Hodge--de Rham Laplacian defined on event horizons of Schwarzschild black holes.
Novel construction of Bauer--Furuta invariant using sheaves of spectra.
problem Constructing the Bauer--Furuta invariant without finite-dimensional approximations.
method Using sheaves of spectra and Borel--Moore homology, avoiding approximations.
result Defines the shriek functors and Thom spectra for index calculations.
Study on sine-cones' spectra and stability under Ricci-de Turck flow.
problem Analyzing stability and rigidity of sine-cones.
method Computed spectra of specific operators on sine-cones.
result Conditions for sine-cones' dynamic stability and rigidity.
We prove explicit upper and lower bounds for the L1-moment spectra for the Brownian motion exit time from extrinsic metric balls of submanifolds Pm in ambient Riemannian spaces Nn. We assume that P and N both have controlled radial curvatures (mean curvature and sectional curvature, respectively) as view…
Geometric model for Hodge filtered complex cobordism constructed.
problem Constructing a geometric model for Hodge filtered complex cobordism.
method Refinement of Pontryagin-Thom construction to create an explicit isomorphism.
result Explicit isomorphism between geometric and abstract models for complex manifolds.
Introduces a new Hodge theory using vector fields on manifolds.
problem Developing a new Hodge theory for manifolds with vector fields.
method Defines a vector field induced Hodge L2-inner product, codifferential, and Laplacian. result Established de Rham-Hodge theory for closed and boundary manifolds.
The paper describes correlations of spectra for higher rank Anosov representations.
problem Understanding correlations of spectra for Anosov representations of higher rank groups.
method Relates correlation problem to counting projections in truncated hypertubes.
result Extends previous work on rank one representations to higher rank.
Formulas for spectra of higher spin operators on sphere subbundles.
problem Finding spectra of higher spin operators on specific subbundles of spinor-valued tensors.
method Explicit formulas derived for spectra in both even and odd dimensions.
result Spectra formulas for higher spin operators and their squares.
Study bottom of spectra on orbifolds via coverings.
problem Behavior of bottom of spectra under orbifold coverings.
method Analysis of scalar Schrödinger operators on orbifolds.
result Results apply to geometrically finite and conformally compact orbifolds.
Resolves conjectures on non-abelian Hodge loci for quasi-projective varieties.
problem Understanding non-abelian Hodge loci for quasi-projective varieties.
method Analyzes Z-local systems and polarized variations of Hodge structures. result Proves algebraicity of non-abelian Hodge loci for Q-anisotropic monodromy. Proves a tropical version of Clemens-Schmid sequence for tropical varieties.
problem Existence of tropical cycles with specific cohomology classes.
method Tropical Hodge theory and Clemens-Schmid sequence analogy.
result Proves tropical Hodge conjecture for rationally triangulable varieties.
Integrally splits L-spectra of integers into simpler components.
problem Understanding the homotopy type of L-spectra of integers.
method Using Anderson duality and splitting into simpler spectra.
result Splits L-spectra of integers into simpler components.
We view strict ring spectra as generalized rings. The study of their algebraic K-theory is motivated by its applications to the automorphism groups of compact manifolds. Partial calculations of algebraic K-theory for the sphere spectrum are available at regular primes, but we seek more conceptual answers in terms of lo…