Department of

February 2015 March 2015 April 2015 Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 8 9 10 11 12 13 14 8 9 10 11 12 13 14 5 6 7 8 9 10 11 15 16 17 18 19 20 21 15 16 17 18 19 20 21 12 13 14 15 16 17 18 22 23 24 25 26 27 28 22 23 24 25 26 27 28 19 20 21 22 23 24 25 29 30 31 26 27 28 29 30

Monday, March 2, 2015

**Abstract:** Let a circle act symplectically on a closed symplectic manifold $M$. If the action is Hamiltonian, we can pass to the reduced space; moreover, the fixed set largely determines the cohomology and Chern classes of $M$. In particular, symplectic circle actions with no fixed points are never Hamiltonian. This leads to the following important question: What conditions force a symplectic action with fixed points to be Hamiltonian? Frankel proved that Kahler circle actions with fixed points on Kahler manifolds are always Hamiltonian. In contrast, McDuff constructed a non-Hamiltonian symplectic circle action with fixed tori. Despite significant additional research, the following question is still open: Does there exists a non-Hamiltonian symplectic circle action with isolated fixed points? The main goal of this talk is to answer this question by constructing a non-Hamiltonian symplectic circle action with exactly 32 fixed points on a closed six-dimensional symplectic manifold. Based in part on joint work with J. Watts.