Projectors defined in virtual Temperley-Lieb algebra with properties similar to Jones-Wenzl projectors.
problem Defining projectors in the virtual Temperley-Lieb algebra.
method Generalizing Jones-Wenzl projectors and defining new projectors with specific properties.
result Uniqueness of the projector fn and explicit formula for fn. 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.
New insights into contrastive learning reveal how projectors affect downstream performance.
problem Understanding how projectors in contrastive learning impact downstream linear classification accuracy.
method Identified and modeled two effects: expansion and shrinkage induced by contrastive loss.
result Linear projectors operating in the shrinkage regime hinder downstream classification accuracy.
We generalize categorified Jones-Wenzl projectors in odd Khovanov homology.
problem Categorify Jones-Wenzl projectors for odd Khovanov homology.
method Develop grading multicategories to replace grading categories, proving existence and uniqueness of categorified projectors.
result Existence and uniqueness of categorified Jones-Wenzl projectors in odd Khovanov homology.
Extremal weight projectors generalized for gl(N).
problem Categorify torus skein algebras for gl(N).
method Diagrammatic idempotents in affine extension of Temperley-Lieb category.
result Categorification of power-sum symmetric polynomials.
We construct a categorification of the quantum sl_3 projectors, the sl_3 analog of the Jones-Wenzl projectors, as the stable limit of the complexes assigned to k-twist torus braids (as k goes to infinity) in a suitably shifted version of Morrison and Nieh's geometric formulation of sl_3 link homology (math.GT/0612754).…
A sequence of Temperley-Lieb algebra elements corresponding to torus braids with growing twisting numbers converges to the Jones-Wenzl projector. We show that a sequence of categorification complexes of these braids also has a limit which may serve as a categorification of the Jones-Wenzl projector.
Study adiabatic limits of Calderon projector on manifolds with cylindrical ends.
problem Constructing Calderon projector in a specific geometric setting.
method Using resolvent estimates and b-calculus framework.
result Extend a result on adiabatic limits of Cauchy data spaces.
Projector-based approach quantifies uncertainties in sketched linear regression.
problem How sketching affects statistical properties of linear regression solutions.
method Projector-based approach to sketched linear regression that is exact and requires minimal assumptions.
result Derives key quantities from classic linear regression that account for combined uncertainties.
Embedding Projector visualizes and interprets embeddings interactively.
problem Exploring properties of embeddings in machine learning.
method Interactive visualization and interpretation tool for embeddings.
result Interactive exploration of embedding properties.
A GAN-based projector speeds up image recovery in linear inverse problems.
problem Efficiently solving linear inverse problems with convergence guarantees.
method A GAN-based projector trained with projected gradient descent (PGD).
result Guaranteed O(δ) reconstruction error in O(log(1/δ)) steps. We explicitly describe a relationship between the Lie theoretic and topological categorification of the Jones-Wenzl projector The two categorifications appear in arXiv:1007.4680 and arXiv:1005.5117 respectively.
Preprint proves quantum coideal Schur-Weyl duality and generalizes Jones-Wenzl projectors.
problem Quantum coideal Schur-Weyl duality and Jones-Wenzl projectors in type B/D.
method Combinatorial proofs and functional analytic arguments.
result Explicit proof of quantum coideal Schur-Weyl duality and generalization of Jones-Wenzl projectors.
Infinite braids' categorification proven using Khovanov homology.
problem Categorifying Jones-Wenzl projectors for infinite braids.
method Limiting Khovanov chain complex and homotopy types.
result Jones-Wenzl projectors categorified for infinite braids.
Calderón projector extended to fibred cusp operators.
problem Extending Calderón projector to non-compact manifolds with special fibred structure.
method φ-pseudodifferential calculus by Mazzeo and Melrose.
result Calderón projector for fibred cusp operators is a pseudodifferential operator.
Categorifies Hecke algebra subalgebra using sheaves and bimodules.
problem Categorify maximal commutative subalgebra of type A Hecke algebra.
method Construct monoidal functor from coherent sheaves on flag Hilbert scheme to Soergel bimodules, match Hochschild homology with sheaf Euler characteristics.
result Categorified projectors correspond to dg algebras of functions on affine charts of flag Hilbert schemes, conjecturally matching homology of glN projectors. The Jones-Wenzl projectors play a central role in quantum topology, underlying the construction of SU(2) topological quantum field theories and quantum spin networks. We construct chain complexes whose graded Euler characteristic is the "classical" projector in the Temperley-Lieb algebra. We show that they are homotopy…
We compute the Khovanov lasagna module of S²×S², confirming a conjecture.
problem Computing the Khovanov lasagna module of S²×S².
method Interpreting Manolescu-Neithalath's formula as a homotopy colimit, using categorified projectors.
result The Khovanov lasagna module of S²×S² is trivial.
For a Dirac operator Dgˉ over a spin compact Riemannian manifold with boundary (Xˉ,gˉ), we give a natural construction of the Calderón projector and of the associated Bergman projector on the space of harmonic spinors on Xˉ, and we analyze their Schwartz kernels. Our approach is based on th…
New identities link Frobenius elements to Jones-Wenzl projectors at roots of unity.
problem Understanding relationships between Frobenius elements and Jones-Wenzl projectors at roots of unity.
method Obtained skein identities relating Frobenius elements to Jones-Wenzl projectors in the Kauffman bracket skein module.
result Skein identities provide new proofs of the existence of the Chebyshev-Frobenius homomorphism.
Unified theories for colored sl(2) knot homology.
problem Equivalence of different models for colored sl(2) knot homology.
method Conceptualized properties into a Chebyshev system and proved its uniqueness.
result Equivalence of Khovanov and Cooper-Krushkal models for the unknot.
In the spirit of noncommutative geometry we construct all inequivalent vector bundles over the (2,2)-dimensional supersphere S2,2 by means of global projectors p via equivariant maps. Each projector determines the projective module of finite type of sections of the corresponding `rank 1' supervector bundle ove…
New categorification method for infinite braids.
problem Categorifying highest-weight projectors for infinite braids.
method Limiting colored Khovanov-Rozansky homology of infinite braids.
result Partial isomorphism between HOMFLY-PT homology of braids and infinite torus knots.
Let (M, g) be a compact smooth Riemannian manifold. We obtain new off-diagonal estimates as λ tend to infinity for the remainder in the pointwise Weyl Law for the kernel of the spectral projector of the Laplacian onto functions with frequency at most λ. A corollary is that, when rescaled around a non self-focal point, …
The study improves norms of spectral projectors on specific surfaces.
problem Improving the L2oL∞ norm of spectral projectors on certain surfaces. method Quantum Integrability, joint basis of eigenfunctions, Lagrangian oscillatory functions, caustics, BKW decay.
result Polynomial improvement on the L2oL∞ norm for generic simple spheres of revolution and the Euclidean disk. The paper studies elliptic operators on manifolds with boundary.
problem Characterizing boundary conditions for elliptic operators on manifolds.
method Using Calderón projectors and mixed order Sobolev spaces, the paper describes the space of boundary values and characterizes Fredholm and regular realisations.
result Characterization of boundary conditions for elliptic operators leading to Fredholm and regular realisations.
The existence and continuity for the Calderon projector of the perturbed odd signature operator on a 3-manifold is established. As an application we give a new proof of a result of Taubes relating the mod 2 spectral flow of a family of operators on a homology 3-sphere with the difference in local intersection numbers o…
New geometric structures derived from Hessian metrics and tensors.
problem Constructing geometric structures from Hessian metrics and tensors.
method Introduced a family of Hessian metrics and constructed a (1,1)-tensor field. result Derived golden and metallic structures from the tensor field.
The study reveals scaling limits of spectral projectors on Riemannian manifolds.
problem Understanding spectral function scaling limits on Riemannian manifolds.
method New off-diagonal estimates and Weyl law application.
result Scaling limit of spectral projector is a normalized Bessel function.
We consider a compact Riemannian manifold with a Hermitian line bundle whose curvature is non-degenerate. The Laplacian acting on high tensor powers (the semiclassical regime) of the bundle exhibits a cluster of low-energy states. We demonstrate that the orthogonal projectors onto these states are the Fourier component…
We describe surfaces in R^{N^2-1} generated by the holomorphic solutions of the supersymmetric CP^{N-1} model. We show that these surfaces are described by the fundamental projector constructed out of the solutions of this model and that in the CP^{N-1} case the corresponding surface is a sphere. Although the coordinat…
New homology for infinite multi-colored braids, completing previous work.
problem Categorification of highest-weight projectors for infinite braids.
method Defining limiting Khovanov-Rozansky homology for semi-infinite braids.
result Categorifies highest-weight projectors for a large class of braids.
Characterizes Schrödinger operator boundedness on weighted Riemannian manifolds.
problem Classifying functions V for bounded Schrödinger operator Δ−V. method Investigates weighted L2-boundedness of Hodge projector. result Characterizes function V for Schrödinger operator boundedness. Proves L2 Frölicher inequality on complex manifolds.
problem Calculating L2 Betti and Hodge numbers. method Uses spectral projectors of (Dh)2 to build an injection. result New proof of classical Frölicher inequality.
Extends Khovanov cohomology to colored links using stable homotopy types.
problem Extending Khovanov cohomology to colored links.
method Defining a stable homotopy type X_col(L_c) for colored links L, using categorified Jones-Wenzl projectors and infinite torus braids.
result Computes stable homotopy types for specific colored links and makes a conjecture for others.
In the present paper we discuss the cabling procedure for the colored HOMFLY polynomial. We describe how it can be used and how one can find all the quantities such as projectors and R-matrices, which are needed in this procedure. The constructed matrix forms of the projectors and the fundamental $\mathcal{…
Categorifies weight-preserving maps in sl(2) representations.
problem Categorifying weight-preserving linear maps between sl(2) representations.
method Introduces a quotient of the affine Temperley-Lieb category and studies diagrammatic idempotents.
result Categorifies the Kauffman bracket skein algebra of the annulus and Chebyshev polynomials of the first kind.
We give a construction of homotopy algebras based on ``higher derived brackets''. More precisely, the data include a Lie superalgebra with a projector on an Abelian subalgebra satisfying a certain axiom, and an odd element Δ. Given this, we introduce an infinite sequence of higher brackets on the image of the project…
Researchers create functors to match colored homologies of knots and links.
problem Equivalence of colored HOMFLYPT homologies for links and knots.
method Constructing functors on singular Soergel bimodules to identify homologies.
result Established parity results for intrinsic column-colored homology of positive torus knots.
We show that the superconformal symmetries of the (1,1) sigma model decompose into a set of more refined symmetries when the target space admits projectors P±, and the orthogonal complements Q±, covariantly constant with respect to the two natural torsionful connections ∇(±) that arise in the …
Quantum propagation studied for Berezin-Toeplitz operators.
problem Asymptotic behavior of quantum propagators and spectral projectors.
method Geometric analysis of Hamiltonian flows and Maslov indices.
result Introduction of quantum states associated with Lagrangian submanifolds.
Equivalence of norms on manifolds with curvature bounds established.
problem Establishing equivalence of norms on manifolds with bounded sectional curvature.
method Using spectral projector and thickness condition for subsets.
result Constant in equivalence depends only on manifold dimension, curvature bounds, and frequency threshold.
Consider a dataset of vector-valued observations that consists of noisy inliers, which are explained well by a low-dimensional subspace, along with some number of outliers. This work describes a convex optimization problem, called REAPER, that can reliably fit a low-dimensional model to this type of data. This approach…
We study de Rham cohomology for various differential calculi on finite groups G up to order 8. These include the permutation group S_3, the dihedral group D_4 and the quaternion group Q. Poincare' duality holds in every case, and under some assumptions (essentially the existence of a top form) we find that it must hold…
Proposes a method to enforce nestedness in subspace learning methods.
problem Consistency between data representations when choosing different subspaces dimensions.
method Lifts Grassmannian optimization criteria to flag manifolds via nested projectors.
result Successfully addresses the nestedness issue in several machine learning methods.
Maps asymptotically embed conic transforms from circle bundles.
problem Embedding conic transforms from circle bundles.
method Asymptotic embeddings using equivariant Szegő projectors.
result Maps embed conic transforms from circle bundles.
We prove that the trace of the logarithmic term of the Toeplitz kernel on a contact manifold is a contact invariant, generalizing K. Hirachi's invariant for the Szego kernel on a CR manifold. When the base manifold is the three-sphere, this vanishes identically.
By applying the symplectic cutting operation to cotangent bundles, one can construct a large number of interesting symplectic cones. In this paper we show how to attach algebras of pseudodifferential operators to such cones and describe the symbolic properties of the algebras.