George Petroulakis | Searcharxiv

arXiv2026

Localization of Singularities and Universal Geometric Rank Bounds in the Satake Correspondence

This article introduces a framework for the localization and isolation of singularities in the affine Grassmannian. Our primary result is a structural factorization of the transition matrix $C$ between the Mirković--Vilonen (MV) basis and the convolution basis into $C = P \cdot M \cdot A \cdot Q^{-1}$, where the four factors represent: equivariant localization ($Q$), fusion via nearby cycles ($A$), local intersection cohomology stalks ($M$), and diagonal normalization ($P$). Utilizing this factorization, and by introducing the Geometric Efficiency metric ($η$) we establish a Universal Geometric Rank Bound, proving that the rank of the transition matrix $C$ is bounded by the dimension of the local Braden--MacPherson (BMP) stalks.

math.AG↗

arXiv2020

Nontrivial linear projections of the Grassmannian Gr_3(C^6)

A typical linear projection of the Grassmannian in its Plucker embedding is injective, unless its image is a projective space. A notable exception are self-adjoint linear projections, which have even degree. We consider linear projections of Gr_3(C^6) with low-dimensional centers of projection. When the center has dimension less than five, we show that the projection has degree 1. When the center has dimension five and the projection has degree greater than 1, we show that it is self-adjoint.

math.AG↗