Pure Mathematics (Differential Geometry and Topology)

我的工作是辛拓扑，这是一个微分几何领域。流形上的辛结构是一个非退化的闭2-形式；它们存在于所有有向曲面、余切丛和光滑代数簇上。辛形式的几何与保体积几何有很大的不同，它融合了拓扑特征和刚性特征；后者是基于分析的全纯曲线不变量(“Floer理论”)来探索的。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，我关注的是一类称为Dehn扭曲的辛同构(即保持辛结构的微分同态)。Seidel在拉格朗日球面中广泛地研究了Dehn扭曲。我一直在研究实射影空间和复射影空间中的Dehn扭曲.我的第一个结果表明，在Stein流形(如仿射代数簇)中，不可能有这样的扭曲的乘积与恒等式相同，这推广了Seidel证明了球面上的扭转的相应定理的一个结果.该定理对射影空间扭曲和2-球面上Lefschetz原函数的自变量计数部分进行了单调刻画。我现在正在研究(I)上述结果可能推广到更一般的情形，例如闭流形而不是仿射簇，(Ii)由Ailsa Kating猜想的通过Lefschetz原函数的实射影平面扭曲的特定模型。这些结果应该能更深入地揭示辛映射类群中扭曲之间的可能关系。

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.