Mathematics Colloquia and Seminars

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We associate a closed Lagrangian submanifold L of C^n to each Delzant polytope. We prove that L is monotone if and only if the polytope P is Fano. The Lagrangian L is diffeomorphic to the total space of fiber bundle over T^k, where the fiber is the so-called real moment-angle manifold associated to P. In some cases real moment angle manifolds are diffeomorphic to connected sums of sphere products. Similar theorems can be proved for Lagrangians of CP^n and (CP^n)^k. Using this method we can construct a huge number of monotone Lagrangian submanifolds. Many of constructed monotone Lagrangians are smoothly isotopic, but they are not Hamiltonian isotopic. Also, we will discuss restrictions on Maslov class of monotone Lagrangian submanifolds of C^n. We will show that in certain cases our examples realize all possible minimal Maslov numbers. If time permits, we will discuss applications to monotone Lagrangian cobordisms.

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