Special Lecture: The Atiyah-Singer Index Theorem and Almost Complex Spheres


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April 12, 2024 10:30 am - 10:55 am
Science Center 507
Address: 1 Oxford Street, Cambridge, MA 02138 USA

Dhruv Goel - Harvard AB 2024

When is a real smooth manifold secretly a complex manifold? For this, it is necessary, but not sufficient, for the manifold’s tangent bundle to be a complex vector bundle, a condition called being “almost complex”. In this talk, I will give several examples of complex, almost complex, and (orientable, even-dimensional) not-even-almost complex manifolds. I will then discuss how the Atiyah-Singer Index Theorem can be used to show that certain smooth manifolds are not almost complex, focusing on the case of the twisted Dirac operator on spinor bundles on spheres.