Pure Mathematics (Differential Geometry and Topology)

我的工作在辛拓扑，微分几何的一个领域。一个辛结构的流形是一个非退化的封闭2-形式;这些存在于所有定向表面，余切丛，并顺利代数簇。辛形式的几何有趣地不同于保体积几何，并且融合了拓扑特征和刚性特征;后者由基于分析的全纯曲线不变量（“Floer理论”）探索。我的主要研究问题涉及拉格朗日子流形和辛流形的对称性。目前，我专注于一类辛同胚（即保持辛结构的同胚）称为德恩扭曲。塞德尔广泛研究了拉格朗日球中的德恩扭曲。我一直在研究德恩扭曲在真实的和复杂的射影空间。我的第一个结果表明，在斯坦流形（如仿射代数簇）不能有一个产品的这种扭曲同位素的身份，这扩展了结果塞德尔谁证明了相应的定理扭曲领域。该定理使用一个monodromy描述射影空间扭曲和一个参数计数部分的莱夫谢茨纤维化的2 sever.I现在正在研究（i）可能的扩展上述结果更一般的情况下，例如封闭的流形，而不是仿射品种，和（ii）一个特定的模型为一个真实的射影平面扭曲通过莱夫谢茨纤维化，由Ailsa Keating，prostretured。这些应该给更多的洞察辛映射类群之间的扭曲可能的关系。

I work in symplectic topology, an area of differential geometry.A symplectic structure on a manifold is a non-degenerate closed 2-form; these exist on all oriented surfaces, on cotangent bundles, and on smooth algebraic varieties. The geometry of symplectic forms is interestingly different to volume-preserving geometry and blends topological features and rigidity features; the latter are explored by holomorphic curve invariants ("Floer theory"), which are based on analysis. My main research questions concern Lagrangian submanifolds, and symmetries of symplectic manifolds. At the moment I am focusing on a class of symplectomorphisms (i.e diffeomorphisms that preserve the symplectic structure) called Dehn twists.Seidel has extensively studied Dehn twists in Lagrangian spheres. I have been studying Dehn twists in real and complex projective spaces.My first result shows that in Stein manifolds (like affine algebraic varieties) one cannot have a product of such twists isotopic to the identity, which extends a result of Seidel who proved the corresponding theorem for twists in spheres. The theorem uses a monodromy description for projective space twists and an argument counting sections of Lefschetz fibrations over the 2-sphere.I am now studying (i) possible extensions of the above result to more general situations, e.g. closed manifolds rather than affine varieties, and (ii) a particular model for a real projective plane twist via Lefschetz fibrations, conjectured by Ailsa Keating. These should give more insight into the possible relations amongst twists in symplectic mapping class groups.