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Thursday, April 29, 2021

**Abstract:** Consider a compact symplectic manifold with nonzero first Chern class. Its index is defined to be the largest integer dividing the first Chern class in the second cohomology group (modulo torsion). In the algebraic geometry setting there is a question of relating the index to the Betti numbers of the manifold in the case in which the manifold is a Fano variety, which we refer to as a ``positivity condition on the first Chern class''. This question is not fully answered and there are still open conjectures about it, for instance the Mukai conjecture. In the first part of this talk I will introduce the audience to results which are already known in the symplectic setting, relating the index to the Betti numbers for manifolds admitting some special Hamiltonian circle action. In the second part of the talk I will introduce elliptic genera and show how their behaviour can be used to deduce relations between the Betti numbers and the index without assuming the aforementioned positivity condition on the first Chern class.