arXiv research

Study adiabatic limits of Calderon projector on manifolds with cylindrical ends.

Calderón projector extended to fibred cusp operators.

For a Dirac operator $D_{\bar{g}}$ over a spin compact Riemannian manifold with boundary $(\bar{X},\bar{g})$, we give a natural construction of the Calderón projector and of the associated Bergman projector on the space of harmonic spinors on $\bar{X}$, and we analyze their Schwartz kernels. Our approach is based on th…

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.

We compute the index of the Dirac operator on spin Riemannian manifolds with conical singularities, acting from $L^p(Σ^+)$ to $L^q(Σ^-)$ with $p,q>1$. When $1+\frac{n}{p}-\frac{n}{q}>0$ we obtain the usual Atiyah-Patodi-Singer formula, but with a spectral cut at $\frac{n+1}{2}-\frac{n}{q}$ instead of 0 in the definitio…

Establishes Calderón-Zygmund inequalities on evolving Riemannian manifolds.

Projectors defined in virtual Temperley-Lieb algebra with properties similar to Jones-Wenzl projectors.

I review the milestones of the mathematical work of Krzysztof P. Wojciechowski. This will at the same time be a tour of Analysis and Geometry of Boundary Value Problems. Starting in the 80s I will discuss the spectral flow and the general linear conjugation problem, the Calderon projector and the topology of space of e…

Uniqueness proof for Calderón's problem on real-analytic vector bundles.

Jones-Wenzl projectors lifted to Khovanov spectra, proving knot conjectures.

New insights into contrastive learning reveal how projectors affect downstream performance.

We solve the partial data Calderón problem for the connection Laplacian on Riemann surfaces.

We generalize categorified Jones-Wenzl projectors in odd Khovanov homology.

The Stone-Weierstrass theorem aids in solving inverse problems on specific manifolds.

The paper addresses geometric analysis on non-compact Riemannian manifolds, proving Calderón-Zygmund inequalities.

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.

Derives properties of heat kernel for Rumin complex on Heisenberg groups.

We outline an approach to the inverse problem of Calderón that highlights the role of microlocal normal forms and propagation of singularities and extends a number of earlier results also in the anisotropic case. The main result states that from the boundary measurements it is possible to recover integrals of the unkno…

Researchers solve a nonlocal parabolic equation on manifolds using source-to-solution maps.

New example shows non-compact manifolds can lack $L^p$-Calderón-Zygmund inequalities.

Study improves Calderón problem for systems, uniquely determining connections and potentials.

Study magnetic perturbations in Riemannian and Lorentzian Calderón problems.

Researchers solve the Calderón problem for fractional Dirac operators.

The study shows that close hypersurfaces have uniformly bounded inequalities.

Method solves Calderón problem for surfaces near disks.

Two different spacetimes can mimic each other's boundary measurements.

Being motivated by the problem of deducing $L^p$-bounds on the second fundamental form of an isometric immersion from $L^p$-bounds on its mean curvature vector field, we prove a (nonlinear) Calderón-Zygmund inequality for maps between complete (possibly noncompact) Riemannian manifolds.

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.

We construct the Calderon projection on the space of Cauchy datas for a twisted Dirac operator in the Mischenko--Fomenko pseudodifferential calculus for operators acting on bundles of finitely generated $C^*$--Hilbert modules on a compact manifold with boundary. In particular an invertible double is constructed general…

Researchers solve a formally determined inverse problem in Lorentzian geometry.

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…

Sharp threshold found for metric uniqueness in Riemannian Calderón-type problems.

We consider a conformally invariant version of the Calderón problem, where the objective is to determine the conformal class of a Riemannian manifold with boundary from the Dirichlet-to-Neumann map for the conformal Laplacian. The main result states that a locally conformally real-analytic manifold in dimensions $\geq …

Based on a construction due to B. Güneysu and S. Pigola (\textit{Adv. Math.} \textbf{281} (2015), pp.353--393), for each $p \in [1,\infty]$ and $m \in \mathbb{Z}_{\geq 2}$, we exhibit an $m$-dimensional Riemannian open manifold $\mathcal{M}$ on which the $L^p$-Calderón--Zygmund estimate \begin{equation*} \|\nabla \nabl…

In this note we show that on any compact subdomain of a Kähler manifold that admits sufficiently many global holomorphic functions, the products of harmonic functions form a complete set. This gives a positive answer to the linearized anisotropic Calderón problem on a class of complex manifolds that includes compact su…

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…

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…

Anisotropic metric on manifolds uniquely determined by boundary data.

Several proofs have been published of the Mod Z gluing formula for the eta-invariant of a Dirac operator. However, so far the integer contribution to the gluing formula for the eta-invariant is left obscure in the literature. In this article we present a gluing formula for the eta-invariant which expresses the integer …

We compute the Khovanov lasagna module of S²×S², confirming a conjecture.

New identities link Frobenius elements to Jones-Wenzl projectors at roots of unity.

We consider an arbitrary linear elliptic first--order differential operator A with smooth coefficients acting between sections of complex vector bundles E,F over a compact smooth manifold M with smooth boundary N. We describe the analytic and topological properties of A in a collar neighborhood U of N and analyze vario…

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 $(2,2)$-dimensional supersphere $S^{2,2}$ 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…