On the index of Dirac operators on arithmetic quotients

On the index of Dirac operators on arithmetic quotients

Using the Arthur-Selberg trace formula we express the index of a Dirac operator on an arithmetic quotient manifold as the integral over the index form plus a sum of orbital integrals. For the Euler operator these orbital integrals are shown to vanish for products of certain rank one spaces. In this case the index theorem looks exactly as in the compact case.

Lectures on Dirac Operators and Index Theory

survey on the rule of the due & hindering relying on the sheikh ansaris ideas

قاعده مقتضی و مانع در متون فقهی کم و بیش مستند احکام قرار گرفته و مورد مناقشه فقهاء و اصولیین می باشد و مشهور معتقند مقتضی و مانع، قاعده نیست بلکه یکی از مسائل ذیل استصحاب است لذا نگارنده بر آن شد تا پیرامون این قاعده پژوهش جامعی انجام دهد. به عقیده ما مقتضی دارای حیثیت مستقلی است و هر گاه می گوییم مقتضی احراز شد یعنی با ماهیت مستقل خودش محرز گشته و قطعا اقتضاء خود را خواهد داشت مانند نکاح که ...

On the Cobordism Invariance of the Index of Dirac Operators

We describe a “tunneling” proof of the cobordism invariance of the index of Dirac operators. The goal of this note is to present a very short proof of the cobordism invariance of the index. More precisely, if D̂ is a Dirac operator on an odd dimensional manifold M̂ with boundary ∂M̂ = M then we show that the index of its restriction D to M is zero. The novelty of this proof consists in the fact th...

Analysis on arithmetic quotients

For (b), let Θ be any subset of Γ, x in its complement, U as in (a). The neighbourhood xU of x contains at most one element of Γ. There exists a neighbourhood of x contained in xU and not containing any element of Θ. For (c), let V = U ·U, which is compact. The intersection of Γ with V is compact, and covered by disjoint neighbourhoods of each of its points. This intersection must therefore be ...