Lefschetz fibrations in symplectic topology and applications to mirror symmetry

denis Auroux的研究项目旨在使用Lefschetz纤维、分支覆盖和它们的单不变量（映射类群或编织群分解）来研究辛4流形的拓扑。特别是，Auroux正在研究奇异辛曲线的同位素和非同位素现象，复杂投影曲面与辛4流形之间的关系，以及Luttinger手术在拉格朗日环面中的作用。这使他探索了一些单不变量的算法方面，最著名的是用于操纵辫子和辫子分解的算法，以及Hurwitz问题。他还计划研究圆盘上Lefschetzfibrations的枚举不变量，以及具有开卷结构的接触流形的接触同调与其单形的花同调之间的关系。在一个不同的方向上，Auroux正在探索kontsevich的同构镜像对称猜想及其一些推广，建立在最近与L. Katzarkov和d。奥洛夫。主要内容是对朗道-金兹堡模型及其辛几何的研究，以便从同调镜像对称的角度来理解一般类型的各种镜像对称，并探索代数几何中的各种构造。辛流形是具有特殊结构的几何空间（允许面积测量，但不允许距离测量）。虽然它们最初出现在经典力学的哈密顿公式中，但数学家们最近对它们的几何和拓扑（它们内在的“形状”）非常感兴趣，部分原因是理论物理学（弦理论）的激励问题。该项目旨在研究辛流形的拓扑结构，使用的方法首先由S. donaldson和随后由Auroux开发，该方法包括将辛流形投影到更简单的流形上，并研究该投影“折叠”的点。这产生了通过组合数据的完整描述，将许多几何问题简化为纯粹的算法考虑。该项目的主要目标之一是将辛流形的拓扑特征与复杂代数流形（一种更特殊、更容易理解的几何空间类别）的拓扑特征联系起来。此外，Auroux也在研究镜像对称现象，通过研究空间的辛几何，这些空间是一些众所周知的复杂流形族的“镜像”；这是数学和理论物理交界的一个重要问题。

DMS-060148Denis AurouxDenis Auroux's research project aims to use Lefschetz fibrations, branchedcoverings, and their monodromy invariants (mapping class group or braidgroup factorizations) to study the topology of symplectic 4-manifolds.In particular, Auroux is studying isotopy and non-isotopy phenomena forsingular symplectic curves, the relationship between complex projectivesurfaces and symplectic 4-manifolds, and the role of Luttinger surgeryalong Lagrangian tori in this context. This leads him to explore somealgorithmic aspects of monodromy invariants, most notably algorithms formanipulating braids and braid factorizations, and the Hurwitz problem.He also plans to investigate enumerative invariants for Lefschetzfibrations over the disc, and the relation between the contact homologyof a contact manifold equipped with an open book structure and the Floerhomology of its monodromy. In a different direction, Auroux is exploringKontsevich's homological mirror symmetry conjecture and some of itsgeneralizations, building upon recent joint work with L. Katzarkov andD. Orlov. The main ingredient is the study of Landau-Ginzburg models andtheir symplectic geometry, in order to understand mirror symmetry for someexamples of varieties of general type and explore various constructionsin algebraic geometry from the perspective of homological mirror symmetry. Symplectic manifolds are geometric spaces with special structures (allowingarea measurements, but not distance measurements). While they first arosein the Hamiltonian formulation of classical mechanics, mathematicians haverecently become very interested in their geometry and topology (theirintrinsic "shape"), in part due to motivating questions from theoreticalphysics (string theory). This project aims to study the topology ofsymplectic manifolds using an approach developped first by S. Donaldsonand subsequently by Auroux, which consists in projecting them onto simplermanifolds and studying the points where this projection is "folded".This yields a complete description by combinatorial data, reducing muchof the geometry to purely algorithmic considerations. One of the maingoals of the project is to relate the topological features of symplecticmanifolds with those of complex algebraic manifolds (a more special, muchbetter understood class of geometric spaces). In addition, Auroux is alsoinvestigating the phenomenon of mirror symmetry, by studying the symplecticgeometry of spaces that are "mirror" to some well-understood families ofcomplex manifolds; this is an important question at the interface betweenmathematics and theoretical physics.