Onirban Islam (UP)
The Atiyah-Singer index theorem is one of the monumental results in Mathematics of the last century. Various extensions of this theorem in the context of Riemannian geometry are available but a Lorentzian generalisation was only reported a few years ago by Bär and Strohmaier [Amer. J. Math 141, 1421 (2019)]. They have proven that the Dirac operator on a spatially compact globally hyperbolic spacetime under suitable Atiyah-Patodi-Singer (APS) boundary conditions is Fredholm and obtained a formula for the Fredholm index. Since then, several authors have obtained various extensions of the Bär-Strohmaier index theorem. In this talk, I shall begin with a panorama of Lorentzian index theorems. Next, globally hyperbolic spacetimes with compact timelike boundary will be considered. Assuming the well-posedness of the Cauchy problem for a Dirac-type operator on such a spacetime under APS boundary conditions, I shall show that the Fredholm index is finite.