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.
The paper extends Laplacian spectra approximations to vector bundles.
problem Approximating the spectrum of the connection Laplacian.
method Extending the graph connection Laplacian to vector bundles and proving spectrum approximation.
result The spectrum of the extended operator approximates the spectrum of the connection Laplacian.
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…
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.
Unified method to compute Laplace spectra on homogeneous principal bundles.
problem Computing the Laplace-Beltrami spectrum on homogeneous principal bundles.
method Unified representation-theoretic approach using generalized canonical variations and spectral branching criterion.
result Explicit formulas for the full spectra of several geometric families.
Constructs AHSS for twisted differential generalized cohomology theories.
problem Generalizing AHSS for twisted settings and differential cohomology.
method Builds on previous work, uses bundles of spectra with flat connections.
result Establishes twisted differential spectra as bundles of spectra with flat connections.
This research connects quantum spectra of flag bundles to prime factorization of integers.
problem Understanding the quantum spectra of flag bundles and their relation to prime numbers.
method Functorial and inductive properties of vertical quantum cohomology, relating to analytic number theory.
result The degeneracy of the small vertical quantum spectrum of a Grassmann bundle is controlled by the prime factorization of ranks.
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. Paper proposes a new method to handle spectral variability in hyperspectral unmixing.
problem Spectral variability within endmember classes affects unmixing performance.
method Adaptive bundles and double sparsity to promote sparsity on spectra and classes.
result Successfully determines variable number of classes and estimates their abundances.
The study finds effective lower bounds for spectra of random surfaces and bundles.
problem Determining the spectrum of Laplacian on random surfaces and bundles.
method Analysis of random covering surfaces and unitary bundles over finite-area non-compact hyperbolic surfaces.
result With high probability, the spectrum of random surfaces and bundles has no eigenvalues below a certain threshold.
To every Hermitian vector bundle with connection over a compact Riemannian manifold M one can associate a corresponding connection Laplacian acting on the sections of the bundle. We define analogous combinatorial metric dependent Laplacians associated to triangulations of M and prove that their spectra converge, as…
We develop differential algebraic K-theory for rings of integers in number fields and we construct a cycle map from geometrized bundles of modules over such a ring to the differential algebraic K-theory. We also treat some of the foundational aspects of differential cohomology, including differential function spectra a…
Researchers compute differential K-theory for moduli stacks.
problem Computing differential K-theory for moduli stacks of principal G-bundles.
method Using homotopy theory of presheaves of spaces and spectra, they formulate results in terms of invariant polynomials and representation rings.
result They successfully compute the connective differential K-theory and differential cohomology of moduli stacks.
Exponential localization of eigensections for Bochner-Schrödinger operator.
problem Understanding spectral properties of Bochner-Schrödinger operator on high tensor powers of Hermitian line bundles.
method Approximation of operator by model Schrödinger operator with constant magnetic field, analysis of spectrum.
result Spectrum of Bochner-Schrödinger operator in gaps is discrete and eigensections decay exponentially.
Let M be a real 2m-torus equipped with a translation-invariant metric h and a translation-invariant symplectic form w; the latter we interpret as a magnetic field on M. The Hamiltonian flow of half the norm-squared function induced by h on T^*M (the "kinetic energy") with respect to the twisted symplectic form w_{T^*M}…
Connects 4-manifold topology to topological modular forms.
problem Understanding the topology of 4-manifolds.
method Topologically twisted compactification of 6d (1,0) theories on 4-manifolds.
result New invariant of 4-manifolds using equivariant weakly holomorphic modular forms.
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.
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.
Generalizes Floer homotopy via Morse-Bott theory.
problem Constructing equivariant models in Floer theory.
method Morse-Bott theory, flow categories, stable homotopy types.
result Equivalence of Borel equivariant spectra for certain Lagrangians.
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 how bottom of spectra changes with Riemannian coverings.
problem Behavior of bottom of spectra under Riemannian coverings.
method Analysis of scalar Schrödinger operators on Riemannian manifolds.
result Changes in the bottom of spectra observed under coverings.
We study cubical sets without degeneracies, which we call square sets. These sets arise naturally in a number of settings and they have a beautiful intrinsic geometry; in particular a square set C has an infinite family of associated square sets J^i(C), i=1,2,..., which we call James complexes. There are mock bundle pr…
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 …
Using certain Thom spectra appearing in the study of cobordism categories, we show that the odd half of the Miller-Morita-Mumford classes on the mappping class group of a surface with negative Euler characteristic vanish in integral cohomology when restricted to the handlebody subgroup. This is a special case of a more…
Graphs approximate Laplacian spectra on manifolds.
problem Approximating Laplacian spectra on complex manifolds.
method Graph Laplacians on proximity graphs.
result Spectra of graph Laplacians approximate the Laplacian spectra of 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.
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.
Covering spectra match if the covering is amenable, with conditions on curvature.
problem Matching spectra of Riemannian coverings under amenability conditions.
method Analyzing spectra of Riemannian manifolds and their coverings under completeness and curvature constraints.
result Spectra match if the covering is amenable, with conditions on curvature.
CRBM extracts speech features from complex spectra directly.
problem Speech coding ignores phase information in complex spectra.
method CRBM learns relationships between visible and hidden units from complex-valued spectra.
result CRBM outperforms conventional methods in speech coding.
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.
Bounds on spectral gaps of hyperbolic 3-manifolds and orbifolds.
problem Constraining the spectra of Laplace operators on hyperbolic manifolds and orbifolds.
method Linear programming and spectral identities derived from the conformal bootstrap and Selberg trace formula.
result Upper bounds on the first and second Laplacian eigenvalues, and spectral gaps of hyperbolic 3-manifolds and orbifolds.
Continuous Hausdorff dimension of dynamical spectra under small perturbations.
problem Stability of Hausdorff dimension in dynamical systems.
method Generic perturbations of metrics and functions on surfaces.
result Hausdorff dimension varies continuously and is preserved across Markov spectra.
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…
Study compares spectra of differential operators on Riemannian coverings.
problem Comparing spectra of differential operators on Riemannian coverings.
method Analyzes the spectrum of differential operators on bundles under Riemannian coverings.
result Spectrum of D1 is contained in the essential spectrum of D2 under certain conditions. 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.
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.
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…
Wavelet scattering spectra model non-Gaussian time-series, proving scale invariance for self-similar processes.
problem Modeling non-Gaussian time-series with stationary increments.
method Complex wavelet transform for scale variations, joint correlation matrix for scale dependencies, second wavelet transform for diagonalization, maximum entropy models conditioned by scattering spectra coefficients.
result Scattering spectra of self-similar processes are scale invariant, allowing statistical testing and generation of new time-series.
Constructs Khovanov spectra for periodic links, proving rank inequalities.
problem Understanding Khovanov homology for periodic links.
method Equivariant Khovanov spectra using Burnside functor construction.
result Rank inequalities for Khovanov homologies and annular filtrations of prime-periodic links.