Probing smooth and symplectic topology using maps to dimension two

私家侦探建议使用映射到2维空间作为探针的光滑和辛拓扑的流形，在类比使用的莫尔斯功能，这是映射到1维空间。私家侦探建议考虑一般稳定映射到2-流形（莫尔斯2-函数），Lefschetz纤维化，开卷分解和环面和局部环面纤维化，并提出了几个问题，建议如何统一研究这些不同类型的功能。研究莫尔斯2-功能，在合作与罗比，应导致新的不变量的光滑流形，特别是在4维的兴趣。与托马斯马克合作的莱夫谢茨纤维化和开书分解的研究围绕着辛凸性和4维辛外科手术，因此旨在辛4流形的新构造。复曲面几何形状在P.I.的合作与Olguta Buse，着眼于3维接触类似的辛嵌入和包装问题。最后，这三条线都被私家侦探绑在了一起。的计划，以发展一个一般性的理论，顺利4-流形与局部复曲面纤维化的积分仿射2-复体（如在热带几何），这样的图片是隐含在某些方面的PI？我们与Symington，Stipsicz和Mark一起工作，并将在本程序中适当地开发。光滑流形是像我们生活的宇宙一样的空间。通常我们认为它是一个三维空间，但如果我们包括时间，它实际上是四维的。维度只是你需要给出的指定你的位置的数字的数量（例如纬度，经度和海拔）。统计社会科学家在许多方面工作，例如，当他们在调查中为每个人记录许多数字时。理解不同维度空间的整体结构（拓扑结构）在从机器人到统计学到生物学的应用中非常重要，并且其自身的美也很吸引人。这是私家侦探合作者将通过在二维表面投射这些空间的阴影来理解这种结构，就像我们通过投射到二维视网膜上的图像来理解我们周围的三维世界一样。为了接触更广泛的受众，鼓励对数学的热情萌芽，P.I.将与欧几里得实验室合作，利用虚拟学习环境为高中生运行一个数学研究项目，研究与该项目主要研究项目有关的问题。

The P.I. proposes to use maps to 2-dimensional spaces as a probe of the smooth and symplectic topology of manifolds, in analogy with the use of Morse functions, which are maps to 1-dimensional spaces. The P.I. proposes to consider generic stable maps to 2-manifolds (Morse 2-functions), Lefschetz fibrations, open book decompositions and toric and locally toric fibrations, and proposes several problems that suggest how to unify the study of these different types of functions. The study of Morse 2-functions, in collaboration with Robion Kirby, should lead to new invariants of smooth manifolds, especially of interest in dimension 4. The study of Lefschetz fibrations and open book decompositions, in collaboration with Thomas Mark, centers around symplectic convexity and 4-dimensional symplectic surgeries and is thus aimed at new constructions of symplectic 4-manifolds. Toric geometry plays a role in the P.I.'s collaboration with Olguta Buse, which looks at 3-dimensional contact analogs of symplectic embedding and packing problems. Finally, all three threads are tied together by the P.I.?s program to develop a general theory of smooth 4-manifolds with locally toric fibrations over integral affine 2-complexes (as in tropical geometry); such a picture is implicit in certain aspects of the P.I.?s work with Symington, Stipsicz and Mark and will be developed properly in this program.Smooth manifolds are spaces like the one we live in, the universe. Normally we think of it as a 3-dimensional space, but if we include time it is really 4-dimensional. The dimension is just the number of numbers you need to give to specify your location (latitude, longitude and elevation, for example). Statistical social scientists work in many dimensions, for example, when they record many numbers for each individual in a survey. Understanding the overall structure (the topology) of spaces of various dimensions is important in applications ranging from robotics to statistics to biology and is also appealing for its own beauty. Here the P.I. and collaborators will work to understand this structure by casting shadows of these spaces on 2-dimensional surfaces, in much the same way that we understand the 3-dimensional world around us through images projected onto our 2-dimensional retinas. To reach out to a broader audience and encourage budding enthusiasm for mathematics, the P.I. will work with Euclid Lab to run a mathematics research program for high school students utilizing a virtual learning environment, studying questions pertaining to the main research program of this project.