Probing smooth and symplectic topology using maps to dimension two

P.I.建议使用到2维空间的映射来探索流形的光滑和辛拓扑，类似于使用Morse函数，后者是到1维空间的映射。P.I.建议考虑到2-流形(Morse 2-函数)的一般稳定映射、Lefschetz函数、开卷分解以及Toric和局部Toric函数，并提出了几个问题，建议如何统一对这些不同类型函数的研究。与Robion Kirby合作对Morse 2-函数的研究将产生光滑流形的新的不变量，特别是在4维方面。与Thomas Mark合作研究Lefschetz纤颤和开卷分解围绕辛凸性和4维辛手术展开，从而旨在构造新的辛4-流形。环面几何在P.I.S与Olguta Buse的合作中扮演了一个角色，该合作着眼于辛嵌入和填充问题的三维接触类比。最后，P.I.S程序将这三条线索联系在一起，发展了积分仿射2-复形(如在热带几何中)上具有局部环面纤颤的光滑4-流形的一般理论；这样的图景隐含在P.I.？S与Symington、Stipsicz和Mark合作的某些方面，并将在本程序中得到适当的发展。光滑流形是像我们生活的空间-宇宙。通常我们认为它是一个3维空间，但如果我们包括时间，它实际上是4维的。维度只是您指定位置所需提供的数字的数量(例如，纬度、经度和海拔)。例如，当统计社会科学家为一项调查中的每个人记录了许多数字时，他们在许多方面进行了研究。了解不同维度空间的整体结构(拓扑)在从机器人学到统计学再到生物学的各种应用中都很重要，而且它本身也很吸引人。在这里，P.I.和合作者将通过在二维表面上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。为了接触到更广泛的受众，鼓励人们对数学萌生热情，P.I.将与欧几里德实验室合作，为高中生开展一个数学研究项目，利用虚拟学习环境，研究与该项目主要研究项目相关的问题。

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.