Taut co-orientable foliations are associated with non-trivial elements of Heegard-Floer homology, hence, if a 3-manifold admits a taut, co-oriented foliation, it is not an L-space (Kronheimer-Mrowka-Ozsváth-Szabó). Conjecturally (Boyer-Gordon-Watson, Juhász), the converse is also true for irreducible manifolds. Thus far, the evidence from Dehn surgery on knots in S3 is consistent with this conjecture. We consider the L-space Knot Conjecture: if a knot has no reducible or L-space surgeries, then it is persistently foliar, meaning that for each boundary slope there is a taut, co-oriented foliation meeting the boundary of the knot complement in curves of that slope. For rational slopes, these foliations may be capped off by disks to obtain a taut, co-oriented foliation in every manifold obtained by Dehn surgery on that knot. I will describe an approach, applicable in a variety of settings, to constructing families of foliations realizing all boundary slopes. Recalling the work of Ghiggini, Hedden, Ni, Ozsváth-Szabó (and more recently, Juhász and Baldwin-Sivek) revealed that Dehn surgery on a knot in S3 can yield an L-space only if the knot is fibered and strongly quasipositive, we note that this approach seems to apply more easily when the knot is far from being fibered. As applications of this approach, we find that among the alternating and Montesinos knots, all those without reducible or L-space surgeries are persistently foliar. In addition, we find that any connected sum of alternating knots, Montesinos knots, or fibered knots is persistently foliar. Furthermore, any composite knot with a persistently foliar summand is easily shown to be persistently foliar. This work is joint with Charles Delman.[-]

In these lectures, we will review what it means for a 3-manifold to have a hyperbolic structure, and give tools to show that a manifold is hyperbolic. We will also discuss how to decompose examples of 3-manifolds, such as knot complements, into simpler pieces. We give conditions that allow us to use these simpler pieces to determine information about the hyperbolic geometry of the original manifold. Most of the tools we present were developed in the 1970s, 80s, and 90s, but continue to have modern applications.[-]

Complex projective structures, or PSL( $2, \mathbb{C})$-opers, play a central role in the theory of uniformization of Riemann surfaces. A very natural generalization of this notion is to consider complex projective structures with ramification points. This gives rise to the notion of branched projective structure, which is much more flexible in many aspects. For example, any representation of a surface group with values in $\operatorname{PSL}(2, \mathbb{C})$ is obtained as the holonomy of a branched projective structure. We will show that one of the central properties of complex projective structures, namely the complex analytic structure of their moduli spaces, extends to the branched case.[-]

Various surgery operations on dimension four begin with a 4–manifold $X$ and an embedded surface $S$, then remove a neighborhood of $S$ and replace it with something else to produce an interesting new 4–manifold. In a few standard surgery constructions, especially the Gluck twist operation, I will show how, given a trisection diagram of $X$ with decorations that describe the embedded surface $S$, to produce a trisection diagram for the new 4–manifold.
This is joint work with Jeff Meier.[-]

In this talk, we will develop the theory of generalized bridge trisections for smoothly embedded closed surfaces in smooth, closed four-manifolds. The main result is that any such surface can be isotoped to lie in bridge trisected position with respect to a given trisection of the ambient four-manifold. In the setting of knotted surfaces in the four-sphere, this gives a diagrammatic calculus that offers a promising new approach to four-dimensional knot theory. However, the theory extends to other ambient four-manifolds, and we will pay particular attention to the setting of complex curves in simple complex surfaces, where the theory produces surprisingly satisfying pictures and leads to interesting results about trisections of complex surfaces.
This talk is based on various joint works with Dave Gay, Peter Lambert-Cole, and Alex Zupan.[-]

In these lectures, we will review what it means for a 3-manifold to have a hyperbolic structure, and give tools to show that a manifold is hyperbolic. We will also discuss how to decompose examples of 3-manifolds, such as knot complements, into simpler pieces. We give conditions that allow us to use these simpler pieces to determine information about the hyperbolic geometry of the original manifold. Most of the tools we present were developed in the 1970s, 80s, and 90s, but continue to have modern applications.[-]

In these lectures, we will review what it means for a 3-manifold to have a hyperbolic structure, and give tools to show that a manifold is hyperbolic. We will also discuss how to decompose examples of 3-manifolds, such as knot complements, into simpler pieces. We give conditions that allow us to use these simpler pieces to determine information about the hyperbolic geometry of the original manifold. Most of the tools we present were developed in the 1970s, 80s, and 90s, but continue to have modern applications.[-]