Lefschetz fibrations in symplectic topology and applications to mirror symmetry

Denis Auroux的研究项目旨在使用Lefschetz纤维化，分支覆盖及其单值不变量（映射类群或辫子群分解）来研究辛4-流形的拓扑，特别是Auroux正在研究奇异辛曲线的合痕与非合痕现象，复射影曲面与辛4-流形的关系，and the role作用of Luttinger卢廷格surgeryalong沿Lagrangian拉格朗日tori环面in this context上下文.这使他探索一些算法方面的monodromy不变量，最显着的算法formanipulating辫子和辫子factorizations，和Hurwitz问题。他还计划调查枚举不变量的Lefschetzfibrations在光盘，和接触homologyof一个接触流形配备了一个开放的书结构和Floerhomologyof monodromy之间的关系。在另一个方向，Auroux正在探索Kontsevich的同调镜像对称猜想和它的一些推广，建立在最近与L。Katzarkov和D.奥尔洛夫。主要内容是对Landau-Ginzburg模型及其辛几何的研究，目的是为了理解一般类型簇的镜像对称，并从同调镜像对称的角度探讨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（它们的内在“形状”）非常感兴趣，部分原因是理论物理学（弦理论）的激发问题。本项目的目的是利用S. Donaldsonand随后由Auroux，其中包括在投影到simplermanifold和研究点，这个投影是“folded”.这产生了一个完整的描述组合数据，减少了much的几何纯算法的考虑.该项目的主要目标之一是将辛流形的拓扑特征与复代数流形（一种更特殊、更容易理解的几何空间）的拓扑特征联系起来。此外，Auroux也是调查现象的镜像对称，通过研究symplecticgeometry的空间是“镜像”的一些很好理解的家庭ofcomplex流形;这是一个重要的问题之间的接口betweenmathematics和理论物理。

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.