A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
We generalize categorified Jones-Wenzl projectors in odd Khovanov homology.
We compute the Khovanov lasagna module of S²×S², confirming a conjecture.
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.
In previous work, we have constructed diagrammatic idempotents in an affine extension of the Temperley-Lieb category, which describe extremal weight projectors for sl(2), and which categorify Chebyshev polynomials of the first kind. In this paper, we generalize the construction of extremal weight projectors to the case…
We introduce a graphical calculus for computing morphism spaces between the categorified spin networks of Cooper and Krushkal. The calculus, phrased in terms of planar compositions of categorified Jones-Wenzl projectors and their duals, is then used to study the module structure of spin networks over the colored unknot…
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.
We introduce a quotient of the affine Temperley-Lieb category that encodes all weight-preserving linear maps between finite-dimensional sl(2)-representations. We study the diagrammatic idempotents that correspond to projections onto extremal weight spaces and find that they satisfy similar properties as Jones-Wenzl pro…
We show that the limiting unicolored Khovanov-Rozansky chain complex of any infinite positive braid categorifies a highest-weight projector. This result extends an earlier result of Cautis categorifying highest-weight projectors using the limiting complex of infinite torus braids. Additionally, we sh…
We study the representation theory of the smallest quantum group and its categorification. The first part of the paper contains an easy visualization of the 3j-symbols in terms of weighted signed line arrangements in a fixed triangle and new binomial expressions for the 3j-symbols. All these formulas are realized as gr…
Preprint proves quantum coideal Schur-Weyl duality and generalizes Jones-Wenzl projectors.
We construct a categorification of the maximal commutative subalgebra of the type Hecke algebra. Specifically, we propose a monoidal functor from the (symmetric) monoidal category of coherent sheaves on the flag Hilbert scheme to the (non-symmetric) monoidal category of Soergel bimodules. The adjoint of this functo…
We show that the limiting Khovanov chain complex of any infinite positive braid categorifies the Jones-Wenzl projector. This result extends Lev Rozansky's categorification of the Jones-Wenzl projectors using the limiting complex of infinite torus braids. We also show a similar result for the limiting Lipshitz-Sarkar-Kh…
Categorifies Jones polynomial using Lie theory.
We show that Khovanov homology (and its sl(3) variant) can be understood in the context of higher representation theory. Specifically, we show that the combinatorially defined foam constructions of these theories arise as a family of 2-representations of categorified quantum sl(m) via categorical skew Howe duality. Uti…
Constructs dg categories from surfaces using Khovanov homology.
We construct an action of a polynomial ring on the colored sl(2) link homology of Cooper-Krushkal, over which this homology is finitely generated. We define a new, related link homology which is finite dimensional, extends to tangles, and categorifies a scalar-multiple of the sl(2) Reshetikhin-Turaev invariant. We expe…
We define a limiting Khovanov-Rozansky homology for semi-infinite positive multi-colored braids, and we show that this limiting homology categorifies a highest-weight projector for a large class of such braids. This effectively completes the extension of Cautis' similar result for infinite twist braid…
The stable Khovanov-Rozansky homology of torus knots has been conjecturally described as the Koszul homology of an explicit non-regular sequence of polynomials. We verify this conjecture against newly available computational data for sl(3)-homology. Special attention is paid to torsion. In addition, explicit conjectura…
We extend Lipshitz-Sarkar's definition of a stable homotopy type associated to a link L whose cohomology recovers the Khovanov cohomology of L. Given an assignment c (called a coloring) of positive integer to each component of a link L, we define a stable homotopy type X_col(L_c) whose cohomology recovers the c-colored…
We discuss a new perspective on Khovanov homology, using categorifications of tensor products. While in many ways more technically demanding than Khovanov's approach (and its extension by Bar-Natan), this has distinct advantage of directly connecting Khovanov homology to a categorification of \$(\mathbb{C}^2)^{\otimes …
Projectors defined in virtual Temperley-Lieb algebra with properties similar to Jones-Wenzl projectors.
Jones-Wenzl projectors lifted to Khovanov spectra, proving knot conjectures.
New insights into contrastive learning reveal how projectors affect downstream performance.
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).…
Study adiabatic limits of Calderon projector on manifolds with cylindrical ends.
Calderón projector extended to fibred cusp operators.
Linear regression is a classic method of data analysis. In recent years, sketching -- a method of dimension reduction using random sampling, random projections, or both -- has gained popularity as an effective computational approximation when the number of observations greatly exceeds the number of variables. In this p…
Witt algebra acts on categorified quantum groups in type A.
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…
New identities link Frobenius elements to Jones-Wenzl projectors at roots of unity.
For a Dirac operator over a spin compact Riemannian manifold with boundary , we give a natural construction of the Calderón projector and of the associated Bergman projector on the space of harmonic spinors on , and we analyze their Schwartz kernels. Our approach is based on th…
Coloring numbers are one of the simplest combinatorial invariants of knots and links to describe. And with Joyce's introduction of quandles, we can understand them more algebraically. But can we extend these invariants to tangles -- knots and links with free ends? Indeed we can, once we categorify. Starting from the de…
Embeddings are ubiquitous in machine learning, appearing in recommender systems, NLP, and many other applications. Researchers and developers often need to explore the properties of a specific embedding, and one way to analyze embeddings is to visualize them. We present the Embedding Projector, a tool for interactive v…
Unified theories for colored sl(2) knot homology.
In the spirit of noncommutative geometry we construct all inequivalent vector bundles over the -dimensional supersphere by means of global projectors via equivariant maps. Each projector determines the projective module of finite type of sections of the corresponding `rank 1' supervector bundle ove…
A Generative Adversarial Network (GAN) with generator trained to model the prior of images has been shown to perform better than sparsity-based regularizers in ill-posed inverse problems. Here, we propose a new method of deploying a GAN-based prior to solve linear inverse problems using projected gradient descent (…
Proves a categorified relation in Khovanov homology.
We construct complexes of Soergel bimodules which categorify the Young idempotents corresponding to one-column partitions. A beautiful recent conjecture of Gorsky-Rasmussen relates the Hochschild homology of categorified Young idempotents with the flag Hilbert scheme. We prove this conjecture for an…
We investigate two "categorified" braid conjugacy class invariants, one coming from Khovanov homology and the other from Heegaard Floer homology. We prove that each yields a solution to the word problem but not the conjugacy problem in the braid group.
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.
We develop a categorical index calculus for elliptic symbol families. The categorified index problems we consider are a secondary version of the traditional problem of expressing the index class in K-theory in terms of differential-topological data. They include orientation problems for moduli spaces as well as similar…
New link homologies categorify Jones polynomial at odd prime powers.
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…
The paper studies elliptic operators on manifolds with boundary.
New algorithm classifies and generates genomic sequences using RG-flow categorifier.
New geometric structures derived from Hessian metrics and tensors.
We use categorical skew Howe duality to find recursion rules that compute categorified sl(N) invariants of rational tangles colored by exterior powers of the standard representation. Further, we offer a geometric interpretation of these rules which suggests a connection to Floer theory. Along the way we make progress t…