[Paper Review] Radon Transform for Sheaves

Honghao Gao|arXiv (Cornell University)|Dec 18, 2017

Advanced Numerical Analysis Techniques15 references3 citations

TL;DR

This paper introduces a sheaf-theoretic Radon transform functor on microlocal sheaf categories, proving it is an equivalence after microlocal localization. It establishes that sheaf categories microsupported along Legendrian submanifolds are invariant under the Radon transform, linking invariants of knots and links in R³ and S²×R via an equivalence that preserves simplicity and connects to the augmentation-sheaf correspondence.

ABSTRACT

We define the Radon transform functor for sheaves and prove that it is an equivalence after suitable microlocal localizations. As a result, the sheaf category associated to a Legendrian is invariant under the Radon transform. We also manage to place the Radon transform and other transforms in microlocal sheaf theory altogether in a diagram.

Motivation & Objective

• To define a Radon transform functor for sheaves in the context of microlocal sheaf theory.
• To prove that this transform induces an equivalence of microlocal sheaf categories after appropriate microlocal localization.
• To show that the sheaf category associated to a Legendrian submanifold is invariant under the Radon transform.
• To unify various transforms (Fourier-Sato, projective duality, spherical duality, Fourier-Tamarkin) within a single diagram using restriction and polarization.
• To establish a categorical equivalence between sheaf categories on R³ and S²×R for knot and link conormals, preserving simplicity and connecting to the augmentation-sheaf correspondence.

Proposed method

• Define the Radon transform functor using a kernel sheaf supported on the incidence variety of points and hyperplanes in Rⁿ×R.
• Use microlocal sheaf theory to analyze the singular support of sheaves and their behavior under the transform.
• Apply microlocal localization to restrict to relevant Lagrangian cones and Legendrian submanifolds in the cosphere bundle.
• Construct the Radon transform as a Fourier-type integral transform via derived pushforward and tensor product with a characteristic kernel sheaf.
• Establish relations between the Radon transform and other transforms (Fourier-Tamarkin, spherical duality) via restriction and polarization functors.
• Prove the equivalence of categories by showing the transform is fully faithful and essentially surjective after localization, using properties of the kernel and proper base change.

Experimental results

Research questions

• RQ1 Does a sheaf-theoretic version of the Radon transform exist that generalizes the classical integral transform?
• RQ2 Is the Radon transform functor an equivalence on microlocal sheaf categories after microlocal localization?
• RQ3 How does the Radon transform relate to other fundamental transforms in microlocal sheaf theory, such as Fourier-Sato and Fourier-Tamarkin?
• RQ4 Can the Radon transform be used to relate categorical invariants of Legendrian knots in different ambient spaces, such as R³ and S²×R?
• RQ5 Does the Radon transform preserve the simplicity of sheaves, and how does this relate to the augmentation-sheaf correspondence for Legendrian knots?

Key findings

• The Radon transform induces an equivalence of categories between microlocal sheaf categories on Rⁿ and Sⁿ⁻¹×R after microlocal localization.
• The sheaf category associated to a Legendrian submanifold is invariant under the Radon transform, providing a new categorical invariant.
• For a knot or link K in R³, the Radon transform induces an equivalence between the quotient categories Dᵇ_{Λ_K}(R³)/Loc(pt) and Dᵇ_{Λ′_K}(S²×R)/Loc(S²), preserving simplicity.
• The Radon transform is shown to be a restriction of the Fourier-Tamarkin transform, with an explicit kernel description involving the condition ⟨x,y⟩ + t − s ≥ 0.
• The Fourier-Tamarkin transform is identified as a restriction of the Radon transform, and the full diagram of transforms (Radon, Fourier-Sato, projective duality, spherical duality) is unified via restriction and polarization functors.
• The inverse of the Fourier-Tamarkin transform is given by convolution with the kernel k_{−⟨x,y⟩ + s − t ≥ 0}, confirming its invertibility and compatibility with the Radon framework.