This paper studies torsion obstructions to complex sections on manifolds.
problem Torsion obstructions to finding complex sections on almost complex manifolds.
method Calculations using the Adams-Novikov spectral sequence for Thom spectra.
result Torsion obstructions for finding r complex sections of order p vanish for r<p2−p. The paper studies spectral sequences on Vaisman manifolds and finds cohomological obstructions.
problem Finding cohomological obstructions for foliations on Vaisman manifolds.
method Investigates spectral sequences associated with Riemannian foliations on Vaisman manifolds, using Betti numbers to establish lower bounds.
result Establishes cohomological obstructions for two-dimensional foliations to be induced from Vaisman structures, with quasi-regular foliations achieving the lower bound.
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…
New obstructions for periodic links from equivariant Khovanov homology.
problem Identifying periodic links.
method Equivariant Khovanov homology and spectral sequences.
result New obstructions for periodic links.
Develops obstruction theory for a specific 4-manifold index.
problem Computing the Z2-index of 4-manifolds with free involution. method Uses spectral sequences and cohomology with twisted coefficients.
result Computes the Z2-index for various examples. 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.
Incompatible operations affect Khovanov homology and spectral sequences.
problem Incompatibility between operations and spectral sequences.
method Observation of obstructions to integral lifting and spectrification.
result Lipshitz-Sarkar Steenrod operations are incompatible with Szabo's spectral sequence.
Analyzes complex structure deformations using cohomology contraction methods.
problem Deforming complex structures and identifying obstructions.
method Refined power series method for (p,q)-forms and complex structures, using Frölicher spectral sequence. result All obstruction classes lie in the kernel of contraction maps under natural vanishing conditions.
Study spectral flow for Callias operators to prove obstructions to positive scalar curvature.
problem Prove obstructions to positive scalar curvature on manifolds.
method Use spectral flow and odd K-cowaist to derive obstructions.
result Infinite odd K-cowaist is an obstruction to the existence of PSC metrics.
The paper studies obstructions to homotopy invariance of loop coproducts.
problem Characterizing obstructions to homotopy invariance of loop coproducts.
method Using a construction of Geoghegan and Nicas, the paper defines the Reidemeister trace and realizes the Goresky-Hingston coproduct as a map of spectra.
result The failure of a map to entwine spectral coproducts can be characterized by Chas-Sullivan multiplication with the Reidemeister trace.
Cappell and Shaneson pointed out in 1978 interesting properties of Browder - Livesay invariants which are similar to differentials in some spectral sequence. Such spectral sequence was constructed in 1991 by Hambleton and Kharshiladze. This spectral sequence is closely related to a problem of realization of elements of…
The paper studies Vafa-Witten equations on Kaehler manifolds and identifies obstructions to nontrivial solutions.
problem Analyzing solutions to Vafa-Witten equations over Kaehler manifolds.
method Identifying obstructions, gauge theoretical compactness, spectral covers, renormalized Higgs fields.
result Simpler proofs and new geometric interpretations for solutions to Vafa-Witten equations.
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 maps help understand deformations of modules over Lie algebroids.
problem Understanding deformations of modules over Lie algebroids.
method Introduce semiregularity maps and use DG-Lie algebra control.
result Semiregularity maps annihilate obstructions under certain conditions.
Optimizing quantum graphs yields geodesic nets on surfaces.
problem Finding optimal quantum graphs for geodesic nets.
method Optimizing functionals from spectral theory to find geodesic nets.
result Critical metrics for eigenvalues give rise to geodesic nets.
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…
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.
Researchers calculate exact moduli for type II flux backgrounds using spectral sequences.
problem Determining exact moduli of type II flux backgrounds in string theory.
method Using techniques from generalised geometry, they count infinitesimal deformations via a spectral sequence.
result The spectral sequence reproduces naïve expectations and shows all obstructions vanish, impacting the tadpole conjecture.
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.
We answer the natural question: when are a regular Poisson structure along with a complex structure transverse to its symplectic leaves induced by generalized complex structure? The leafwise symplectic form and transverse complex structure determine an obstruction class in a certain cohomology, which vanishes if and on…
The paper extends Khovanov homology results to sl(n) homologies and provides bounds on knot properties.
problem Extending Khovanov homology results to sl(n) homologies and knot properties. method Spectral sequence arguments and Levine-Zemke's ribbon concordance obstruction.
result Bounds on the alternation number and Turaev genus of knots.
The paper solves obstructions to the Fredholm perturbation property for manifolds with corners.
problem Obstructions to the Fredholm perturbation property for compact connected manifolds with corners.
method Introduces a topological space whose singular cohomology is canonically isomorphic to conormal homology and whose K-theory is naturally isomorphic to the K-theory groups of the algebra K_b(X).
result Provides a rational isomorphism between K-theory groups and periodic conormal homology groups, solving obstructions to the Fredholm perturbation property.
Minimal Morse functions on Poincaré dodecahedral space are selected via spectral properties.
problem Identifying minimal Morse functions on the Poincaré dodecahedral space.
method Spectral selection property P, obstruction principle, conformal variations, finite dimensional reduction.
result Restoration of minimal Morse selection on the Poincaré dodecahedral space via spectral mechanisms.
New method deforms function algebras on manifolds using spectral decomposition.
problem Deforming function algebras on compact Riemannian manifolds.
method Introducing a bilinear product on the finite spectral core of smooth functions using unimodular phases.
result The product extends to a Sobolev algebra and admits iteration under certain conditions.
Let (F,u)\to P\to N be a symplectic fibration in math.SG/0503268 McDuff has defined a subgroup Ham^s(F,u) of the group of symplectic automorphisms of(F,u). She has shown that the cohomology class [u] of u can be extended to P if and only if the symplectic fibration has an Ham^s reduction. To show this result, she const…
We construct a braid conjugacy class invariant κ by refining Plamenevskaya's transverse element ψ in Khovanov homology via the annular grading. While κ is not an invariant of transverse links, it distinguishes some braids whose closures share the same classical invariants but are not transversely isotopic. Using …
Due to spectral obstructions, a scattering theory in the Lax-Phillips sense for the wave equation for differential p-forms on H^{n+1} cannot be developed. As a consequence, Huygens' principle for the wave equation in this context does not hold. If we restrict the class of forms and we consider the case of coclosed p-fo…
Smooth approximations of Kähler-Ricci solitons found using quantized metrics and Futaki invariants.
problem Finding smooth approximations of Kähler-Ricci solitons on Fano manifolds.
method Using semiclassical estimates and quantized Futaki invariants to extend a strategy from Donaldson and Tian-Zhu.
result Smooth approximations of Kähler-Ricci solitons can be found as quantized metrics.
Proves uniqueness of certain spacetime solutions with extremal horizons.
problem Proving uniqueness of extremal Schwarzschild de Sitter spacetime solutions.
method Analytic proof in four and higher dimensions, spectral problem for hyperbolic surfaces.
result Proves extremal Schwarzschild de Sitter solutions are unique up to identifications.
We show that elliptic complexes of (pseudo)differential operators on smooth compact manifolds with boundary can always be complemented to a Fredholm problem by boundary conditions involving global pseudodifferential projections on the boundary (similarly as the spectral boundary conditions of Atiyah, Patodi and Singer …
Global stability proved for Navier-Stokes equations on hyperbolic space.
problem Stability of the Navier-Stokes equations on hyperbolic space.
method Proved global stability with exponential decay rate for small initial data.
result Exponential decay rate of $μλ_\Def^{(3)}$ for Navier-Stokes equations on hyperbolic space.
Researchers found counterexamples to conjectures about optimal transport maps on curved spaces.
problem Extending Caffarelli's contraction theorem to curved spaces.
method Constructing counterexamples to precise conjectures.
result Found counterexamples to Milman's conjectures about optimal transport maps on curved spaces.
We apply spectral sequences to derive both an obstruction to the existence of n-fold prolongations and a topological classification. Prolongations have been used in the literature in an attempt to prove that every Engel structure on M×S1 with characteristic line field tangent to the fibers is determi…
We review a cochain-free treatment of the classical van Kampen obstruction θto embeddability of an n-polyhedron into R^{2n} and consider several analogues and generalizations of θ, including an extraordinary lift of θwhich in the manifold case has been studied by J.-P. Dax. The following results are obtained. - The mod…
Geometric Arbitrage Theory reformulates a generic asset model possibly allowing for arbitrage by packaging all assets and their forwards dynamics into a stochastic principal fibre bundle, with a connection whose parallel transport encodes discounting and portfolio rebalancing, and whose curvature measures, in this geom…
The paper presents a new method to create exotic 4-manifolds and surfaces that remain exotic after stabilization.
problem Stabilization of exotic 4-dimensional phenomena and knotted surfaces.
method Elementary approach to constructing exotic 4-manifolds and surfaces, including examples in closed, simply connected 4-manifolds.
result The construction yields exotic surfaces in the 4-ball that remain exotic after stabilization, detected by Khovanov homology.
We use tools from generalized complex geometry to develop the theory of SKT (a.k.a. pluriclosed Hermitian) manifolds and more generally manifolds with special holonomy with respect to a metric connection with closed skew-symmetric torsion. We develop Hodge theory on such manifolds showing how the reduction of the holon…
Researchers approximate spectral targets on manifolds with constant negative curvature.
problem Prescribing an arbitrary finite portion of the Laplace-Beltrami spectrum on manifolds of constant negative curvature.
method Constructing macroscopically heterogeneous hyperbolic covering manifolds in d≥3 and using discrete spectral limit theorems in d=2. result Any finite strictly increasing target list can be approximated to arbitrary precision by a closed manifold of constant negative curvature.
Describes Wall's finiteness obstruction.
problem Finiteness obstruction in algebraic topology.
method Self-contained description of Wall's finiteness obstruction.
result Comprehensive explanation of Wall's finiteness obstruction.
New obstructions for knots in high dimensions prevent double sliceness.
problem Preventing double sliceness of knots in high dimensions.
method Developed L(2)-signature obstructions for metabelian knot groups. result Found infinite families of knots not obstructed by previous invariants.
The paper explores higher fixed point theorems for foliations with applications to rigidity and integrality.
problem Understanding the topological and geometric properties of foliations.
method Applications of higher Lefschetz theorems for foliations, involving Haefliger cohomology.
result The non-triviality of the higher A-hat genus of the foliation in Haefliger cohomology can be an obstruction to the existence of non-trivial leaf-preserving compact connected group actions.
Fewer obstructions for small graphs in knotless embedding.
problem Finite obstructions for knotless embedding of small graphs.
method Analysis of graph minor obstructions and knotless embedding.
result There are only three obstructions for size 23 graphs, significantly fewer than previously known.
3028 obstructions found for embedding without knots.
problem Finding obstructions for knotless embedding.
method Surveying recent work, updating obstructions, and proposing new questions.
result New obstructions with μ=6 and insights into connectivity.
Obstructing deformations of vector forms on Kähler manifolds.
problem Deformations of vector forms on compact Kähler manifolds.
method Annihilating obstruction classes by cohomology classes.
result Obstruction classes are annihilated by cohomology classes.
Study calculates Euler obstruction for determinantal varieties with ICIS singular set.
problem Calculating Euler obstruction for determinantal varieties with ICIS singular set.
method Obtained formulas to calculate Euler obstruction for determinantal varieties with ICIS singular set.
result Formulas to calculate the Euler obstruction for determinantal varieties with ICIS singular set.
The paper develops obstructions for embedding 2D complexes into 4D space.
problem Embedding 2D complexes into 4D space and understanding obstructions.
method Uses Goodwillie-Weiss calculus and intersections of Whitney disks.
result Two approaches to obstructions lead to the same result.
Global obstructions found for conformally Einstein metrics in 6D.
problem Obstructing the existence of conformally Einstein metrics in six dimensions.
method Presentation of global conformal invariants.
result Nontrivial global conformal invariant obstructing conformally Einstein metrics.
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.