Probing smooth and symplectic topology using maps to dimension two

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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.