Singular Chern Classes of Schubert Varieties via Small Resolution

We discuss a method for calculating the Chern-Schwartz-MacPherson (CSM) class of a Schubert variety in the Grassmannian using small resolutions introduced by Zelevinsky. As a consequence, we show how to compute the Chern-Mather class and local Euler obstructions using small resolutions instead of the Nash blowup. The algorithm obtained for CSM classes also allows us to prove new cases of a positivity conjecture of Aluffi and Mihalcea.Comment: Addressed referee's comments, Section 6.2 contains new material; 35 pages, 3 figures, and 2 table

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Quadratic Tangles in Planar Algebras

In planar algebras, we show how to project certain simple "quadratic" tangles onto the linear space spanned by "linear" and "constant" tangles. We obtain some corollaries about the principal graphs and annular structure of subfactors

Conceptual Problems in Scattering from Localized non-Hermitian Potentials

We highlight the conceptual issues that arise when one applies the quasi-Hermitian framework to analyze scattering from localized non-Hermitian potentials, in particular complex square-wells or delta-functions. When treated in the framework of conventional quantum mechanics, these potentials are generally considered as effective theories, in which probability is not conserved because of processes that have been ignored. However, if they are treated as fundamental theories, the Hilbert-space metric must be changed. In order for the newly-defined probability to be conserved, it must differ from the standard one, even at asymptotically large distances from the scattering centre, and the mechanism for this is the non-locality of the new metric, as we show in detail in the model of a single complex delta function. However, properties of distant bound-state systems, which do not interact physically with the non-Hermitian scattering potential, should not be affected. We analyze a model Hamiltonian that supports this contention.Comment: The emphasis has been changed from v1, recognizing that it makes physical sense that the wave functions of scattering states are fundamentally changed in the quasi-Hermitian framework. In contrast, bound states should not be significantly affected by the introduction of a distant non-Hermitian scattering potentia