University of Birmingham > Talks@bham > Analysis Seminar > Index theory and boundary value problems for general first-order elliptic differential operators

Index theory and boundary value problems for general first-order elliptic differential operators

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If you have a question about this talk, please contact Andrew Morris.

Connections between index theory and boundary value problems are an old topic, dating back to the seminal work of Atiyah-Patodi-Singer in the mid-70s where they proved the famed APS Index Theorem for Dirac-type operators. From relative index theory arising in the study of positive scalar curvature metrics to a rigorous understanding of the chiral anomaly for the electron in particle physics, this index theorem has been a central tool to many aspects of modern mathematics.

APS showed that local boundary conditions are topologically obstructed for index theory. Therefore, a central theme emerging from the work of APS is the significance of non-local boundary conditions for first-order elliptic differential operators. An important contribution from APS was to demonstrate how their crucial non-local boundary condition for the index theorem could be obtained by a spectral projection associated to a so-called adapted boundary operator. In their application, this was a self-adjoint first-order elliptic differential operator.

The work of APS generated tremendous amount of activity in the topic from the mid-70s onwards, culminating with the Bär-Ballmann framework in 2010. This is a comprehensive machine useful to study elliptic boundary value problems for first-order elliptic operators on measured manifolds with compact and smooth boundary. It also featured an alternative and conceptual reformulation of the famous relative index theorem from the point of view of boundary value problems. However, as with other generalisations, a fundamental assumption in their work was that an adapted boundary operator can always be chosen self-adjoint.

Many operators, including all Dirac-type operators, satisfy this requirement. In particular, this includes the Hodge-Dirac operator as well as the Atiyah-Singer Dirac operator. Recently, there has been a desire to study more general first-order elliptic operators, with the quintessential example being the Rarita-Schwinger operator on 3/2-spinors. This operator has physical significance, arising in the study of the delta baryon, analogous to the way in which the Atiyah-Singer Dirac operator arises in the study of the electron. However, not only does the Rarita-Schwinger operator fail to be of Dirac-type, it can be shown that outside of trivial geometric situations, this operator can never admit a self-adjoint adapted boundary operator.

In this talk, I will present work with Bär where we extend the theory for first-order elliptic differential operators to full generality. That is, we make no assumptions on the spectral theory of the adapted boundary operator. The ellipticity of the original operator allows us to show that, modulo a lower order additive perturbation, the adapted boundary operator is in fact bi-sectorial. Identifying the spectral theory makes the problem tractable, although departure from self-adjointness significantly complicates the analysis. Therefore, we employ a mixture of methods coming from pseudo-differential operator theory, bounded holomorphic functional calculus, semi-group theory, and maximal regularity to extend the Bär-Ballman framework to the fully general situation.

Time permitting, I will also talk about recent work on the relative index theorem for general first-order elliptic differential operators, possible harmonic-analytic perspectives of the APS index theorem, as well as recent developments in the study of noncompact boundary, Lipschitz boundary, and problems in L^p.

This talk is part of the Analysis Seminar series.

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