Pure Mathematics (Differential Geometry and Topology)

我研究辛拓扑，微分几何的一个领域。流形上的辛结构是非简并的闭2型；它们存在于所有有向曲面、共切束和光滑代数变体上。辛形式的几何形状与保体积几何形状有趣地不同，它混合了拓扑特征和刚性特征；后者是通过基于分析的全纯曲线不变量（“花理论”）来探讨的。我的主要研究问题是拉格朗日子流形和辛流形的对称性。目前，我关注的是一类辛形态（即保持辛结构的微分同态），称为Dehn扭转。塞德尔广泛地研究了拉格朗日球中的德恩扭转。我一直在研究真实和复杂投影空间中的Dehn扭曲。我的第一个结果表明，在Stein流形（类似仿射代数变体）中，不可能有这样的扭转同位素与恒等的乘积，这扩展了Seidel的结果，Seidel证明了球体中扭转的相应定理。该定理使用了对射影空间扭曲的单项式描述和对2球上的Lefschetz纤振的截面进行计数的论证。我现在正在研究(I)上述结果在更一般情况下的可能扩展，例如闭合流形而不是仿射变体，以及（ii）由Ailsa Keating推测的通过Lefschetz纤振的真实投影平面扭曲的特定模型。这些应该能让我们更深入地了解辛映射类组中的曲折之间可能存在的关系。

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.