Yunhyung Cho

On the f-vectors of Gelfand-Cetlin polytopes

A Gelfand-Cetlin polytope is a convex polytope obtained as an image of certain completely integrable system on a partial flag variety. In this paper, we give an equivalent description of the face structure of a GC-polytope in terms of so called the face structure of a ladder diagram. Using our description, we obtain a partial differential equation whose solution is the exponential generating function of f-vectors of GC-polytopes. This solves the open problem (2) posed by Gusev, Kritchenko, and Timorin in [GKT].

Embedded Surfaces for Symplectic Circle Actions

The purpose of this article is to characterize symplectic and Hamiltonian circle actions on symplectic manifolds in terms of symplectic embeddings of Riemann surfaces. More precisely, it is shown that (1) if (M, ω) admits a Hamiltonian S1-action, then there exists a two-sphere S in M with positive symplectic area satisfying ‹c1(M, ω), [S]› > 0, and (2) if the action is non-Hamiltonian, then there exists an S1-invariant symplectic 2-torus T in (M, ω) such that ‹c1(M, ω), [T]› = 0. As applications, the authors give a very simple proof of the following well-known theorem which was proved by Atiyah-Bott, Lupton-Oprea, and Ono: Suppose that (M, ω) is a smooth closed symplectic manifold satisfying c1(M, ω) = λ·[ω] for some λ ∈ R and G is a compact connected Lie group acting effectively on M preserving ω. Then (1) if λ < 0, then G must be trivial, (2) if λ = 0, then the G-action is non-Hamiltonian, and (3) if λ > 0, then the G-action is Hamiltonian.