Big Fiber Theorems, Ideal-Valued Measures, and Symplectic Topology

I will discuss an adaptation of Gromov's ideal-valued measures to symplectic topology. It leads to a unified viewpoint at three "big fiber theorems": the Centerpoint Theorem in combinatorial geometry, the Maximal Fiber Inequality in topology, and the Non-displaceable Fiber Theorem in symplectic topology, and yields applications to symplectic rigidity. Joint work with Adi Dickstein, Yaniv Ganor, and Frol Zapolsky.