We describe notions of tautness that arise in the study of $C^0$ foliations, $C^{1,0}$ or smoother foliations, and in geometry. We give examples to show that these notions are different, and discuss how these differences impact some classical foliation results. We construct examples of smoothly taut $C^{\infty,0}$ foliations that can be $C^0$ approximated by both weakly symplectically fillable, universally tight contact structures and by overtwisted contact structures.

Transverse 1-dimensional foliations play an important role in the study of codimension-one foliations. In Geom. Topol. Monogr. 19 (2015) 21-72, the authors introduced the notion of flow box decomposition of a 3-manifold M. This is a combinatorial decomposition of M that reflects both the structure of a given codimension-one foliation and that of a given transverse dimension-one foliation, and that is amenable to inductive strategies. In this paper, flow box decompositions are used to extend some classical foliation results to foliations that are not C-2. Enhancements of well-known results of Calegari on smoothing leaves, Dippolito on Denjoy blowup of leaves, and Tischler on approximations by fibrations are obtained. The methods developed are not intrinsically 3-dimensional techniques, and should generalize to prove corresponding results for codimension-one foliations in n-dimensional manifolds.

Suppose that $\mathcal F$ is a transversely oriented, codimension one foliation of a connected, closed, oriented 3-manifold. Suppose also that $\mathcal F$ has continuous tangent plane field and is {\sl taut}; that is, closed smooth transversals to $\mathcal F$ pass through every point of $M$. We show that if $\mathcal F$ is not the product foliation $S^1\times S^2$, then $\mathcal F$ can be $C^0$ approximated by weakly symplectically fillable, universally tight, contact structures. This extends work of Eliashberg-Thurston on approximations of taut, transversely oriented $C^2$ foliations to the class of foliations that often arise in branched surface constructions of foliations. This allows applications of contact topology and Floer theory beyond the category of $C^2$ foliated spaces.

We extend the Eliashberg-Thurston theorem on approximations of taut oriented $C^2$-foliations of 3-manifolds by both positive and negative contact structures to a large class of taut oriented $C^{1,0}$-foliations, where by $C^{1,0}$ foliation, we mean a foliation with continuous tangent plane field. These $C^{1,0}$-foliations can therefore be approximated by weakly symplectically fillable, universally tight, contact structures. This allows applications of $C^2$-foliation theory to contact topology and Floer theory to be generalized and extended to constructions of $C^{1,0}$-foliations.

Fractional Dehn twists give a measure of the difference between the relative isotopy class of a homeomorphism of a bordered surface and the Thurston representative of its free isotopy class. We show how to estimate and compute these invariants. We discuss the the relationship of our work to stabilization problems in classical knot theory, general open book decompositions, and contact topology. We include an elementary characterization of overtwistedness for contact structures described by open book decompositions.

We describe an invariant of a contact 3-manifold with convex boundary as an element of Juhász’s sutured Floer homology. Our invariant generalizes the contact invariant in Heegaard Floer homology in the closed case, due to Ozsváth and Szabó.

We present an alternate description of the Ozsvath-Szabo contact class in Heegaard Floer homology. Using our contact class, we prove that if a contact structure (M,ξ) has an adapted open book decomposition whose page S is a once-punctured torus, then the monodromy is right-veering if and only if the contact structure is tight.

Counselor: Please tell the Court your name. Expert Witness: My name is Will Kazez Counselor: No, no, no! Your name is... This is not a good start. I am not naturally a nervous person. I have survived teaching calculus to a large class that included the entire freshman football team of the University of Pennsylvania, but I've never been an Expert Witness. Even though I'm confident of the mathematics, I'm not sure I like the idea of being cross-examined. But still, I'm just rehearsing my testimony with the lawyer, and even if I've got my own name a little wrong, what's the worry? At any rate, lawyers do not like being interrupted. Counselor: No, no, no! Your name is DOCTOR William H. Kazez. Expert Witness: O.K. My name is DOCTOR William H. Kazez. Counselor: And how are you currently employed, Dr.~Kazez? Expert Witness: I am a Lecturer in the Department of Mathematics at Cornell University. Counselor: And tell the Court, Dr. Kazez, are you familiar with the theorem of Pythagoras? Expert Witness: Well your Honor, I don't mean to brag, but yes, I am familiar with the theorem of Pythagoras. Now this is good! I have rehearsed the last line in my mind many times, and I say bring it on. I'm ready for any cross-examination by any lawyer or judge. Let them take their best shot. But first, maybe I should tell you about the case? My next-door neighbor at the time was Barry Strom, Director of Cornell's Legal Services. He had a client come to him for help with a problem in elementary geometry. The client was living in a trailer that he kept parked next to one of the boundary lines of his property. One night, his neighbor approached him, with shotgun in hand, and told him to move the trailer, because, in the neighbor's humble opinion, it was parked over the boundary line. We mathematicians like to

We describe the natural gluing map on sutured Floer homology which is induced by the inclusion of one sutured manifold (M',Γ') into a larger sutured manifold (M,Γ), together with a contact structure on M-M'. As an application of this gluing map, we produce a (1+1)-dimensional TQFT by dimensional reduction and study its properties.

We initiate the study of the monoid of right-veering diffeomorphisms on a compact oriented surface with nonempty boundary. The monoid strictly contains the monoid of products of positive Dehn twists. We explain the relationship to tight contact structures and open book decompositions.

We give a necessary and sufficient condition for the addition of a collection of disjoint bypasses to a convex surface to be universally tight -- namely the nonexistence of a polygonal region which we call a virtual pinwheel.

Contact structures on 3-manifolds are analyzed by decomposing the manifold along convex surfaces. Background results of Giroux, Eliashberg, Colin, and Honda are discussed with an emphasis on examples. Convex decompositions are then used to give a new proof of the Gabai-Eliashberg-Thurston Theorem on the existence of universally tight contact structures and also to study the contact topology of a space in the presence or absence of tori. Classification of tight contact structures on fibred manifolds and related open questions are also discussed. This paper is based on a series of talks given at the Tokyo Institute of Technology from Jun 3-7, 2002.

We present a new, completely three-dimensional proof of thefact, due to the combined work of Gabai and Eliashberg-Thurston,that every closed, oriented, connected, irreducible 3-manifoldwith nonzero second homology carries a universally tight contactstructure.

As a first step towards understanding the relationship between foliations and tight contact structures on hyperbolic 3-manifolds, we classify "extremal" tight contact structures on a surface bundle M over the circle with pseudo- Anosov monodromy. More specifically, there is exactly one tight contact structure (up to isotopy) whose Euler class, when evaluated on the fiber, equals the Euler characteristic of the fiber. This rigidity theorem is a consequence of properties of the action of pseudo-Anosov maps on the complex of curves of the fiber and a remarkable flexibility property of convex surfaces in M. Indeed, this flexibility can already be seen in surface bundles over the interval, where an analogous classification theorem is also established.

As a first step towards understanding the relationship between foliations and tight contact structures on hyperbolic 3-manifolds, we classify "extremal" tight contact structures on a surface bundle M over the circle with pseudo-Anosov monodromy. More specifically, there is exactly one tight contact structure (up to isotopy) whose Euler class, when evaluated on the fiber, equals the Euler characteristic of the fiber. This rigidity theorem is a consequence of properties of the action of pseudo-Anosov maps on the complex of curves of the fiber and a remarkable flexibility property of convex surfaces in M. Indeed, this flexibility can already be seen in surface bundles over the interval, where an analogous classification theorem is also established.

We use convex decomposition theory to (1) reprove the existence of a universally tight contact structure on every irreducible 3-manifold with nonempty boundary, and (2) prove that every toroidal 3-manifold carries infinitely many nonisotopic, nonisomorphic tight contact structures.