Toward a dynamical theory of Thurston's norm

Spaces are often better understood by cutting them into pieces of lower dimension. For example, on a topographical map of a mountainous area, the 1-dimensional lines of constant altitude provide valuable information about a landscape. The breaks in the smooth pattern of contour lines indicate significant locations---high points, low points, and mountain passes. Moreover, the lines tell us how water will flow over the landscape. The pattern formed by these lines is known in mathematics as a "foliation," and the paths taken by water determine a "flow." This project focuses on flows associated to foliations, but in one dimension higher than the map example: the foliations are comprised of 2-dimensional pieces inside a 3-dimensional space, similar to the universe in which we live. The PI will address longstanding open questions relating the fields of geometry, topology, and dynamics in 3-dimensions. In addition to original research, this project will involve the training of graduate students and support a mathematics periodical run by undergraduates and faculty at the PI's institution. The project's overarching goal is to better understand the relationship between taut foliations and pseudo-Anosov flows in 3-manifolds, and to leverage this relationship to better understand the structure of the Thurston norm. A first concrete goal is to prove, with C.C. Tsang, a strengthening of a famous unpublished theorem of Gabai and Mosher: every taut finite depth foliation of a compact, irreducible, atoroidal 3-manifold is almost transverse to a pseudo-Anosov flow. The strengthening consists of characterizing when the construction yields a pseudo-Anosov flow “without perfect fits,” a property relevant to the study of the Thurston norm. A second goal builds on the first, and is to relate the collection of pseudo-Anosov flows almost transverse to a given taut finite depth foliation to the collection of universal circles for that foliation. The PI aims to link these two families using the Gabai-Mosher construction, giving an approach to Mosher's Transverse Finiteness Conjecture. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.