- Danny Calegari: Universal Circles (minicourse)
Abstract: We explain the construction of Thurston's
universal circle for taut foliations.
Vincent Colin: Contact structures, branched surfaces and foliations (minicourse)
Abstract: In dimension three, contact structures are interacting with foliations in a non trivial way. In particular, they benifited from Gabai's theory of sutured manifolds and taut foliations. On the other hand, contact structures come with rich Floer type invariants that are already meaningful to foliations.
The goal of the lecture series is to describe some of these results.
Sergio Fenley: Free Seifert fibered pieces of pseudo-Anosov flows
Abstract: We prove a structure theorem for pseudo-Anosov flows
restricted to Seifert fibered pieces of three manifolds.
The piece is called periodic if there is a Seifert fibration so
that a regular fiber is freely homotopic, up to powers, to a closed
orbit of the flow. A non periodic Seifert fibered piece is called
free. In this talk we consider free Seifert pieces. We show that,
in a carefully defined neighborhood of the free piece, the
pseudo-Anosov flow is orbitally equivalent to a hyperbolic blow
up of a geodesic flow piece. A geodesic flow piece is a finite
cover of the geodesic flow on a compact hyperbolic surface, usually
with boundary. We introduce almost k-convergence groups,
and an associated convergence theorem. We also introduce an
alternative model for the geodesic flow of a hyperbolic surface
that is suitable to prove these results, and we define
what is a hyperbolic blow up. This is joint work with Thierry
Steven Frankel: Flows, planes, and circles/Quasigeodesic and pseudo-Anosov flows (2 talks)
Abstract: First Talk: One can often gain insight into a dynamical system by understanding the topology and geometry of its underlying phase space. For example, given a generic flow on a surface, the Euler characteristic of the surface determines the sum of the indices of stationary points of the flow. One can also use this idea in the opposite direction, using a flow to understand the topology of a surface.
In this talk, we will discuss the analogous idea one dimension up, seeing how a flow can be used to understand the geometry and topology of a 3-manifold. We will concentrate on quasigeodesic and pseudo-Anosov flows, which have a special relationship with fibrations and foliations. In addition, we will see that one can study these flows "at infinity," where their continuous 3-dimensional dynamics is reflected in a discrete 1-dimensional dynamical system called the universal circle.
Second Talk: A flow is called quasigeodesic when its flowlines are coarsely comparable to geodesics. A flow is called pseudo-Anosov when it has a hyperbolic structure in the transverse direction. We will prove Calegari's conjecture, that every quasigeodesic flow on a closed hyperbolic 3-manifold can be turned into a pseudo-Anosov flow.
David Gabai: Negatively curved branched surfaces
Abstract: In two remarkable papers about 10 years ago Tao Li, using
branched surfaces, proved the strong Waldhausen conjecture:That a non
Haken hyperbolic 3-manifold has only finitely many irreducible Heegaard
splittings. Using minimal surface theory and negatively curved branched
surfaces we gave an effective version. We discuss this result and mention
how the solution to a purely combinatorial branched surface conjecture
would also give a solution. This is joint work with Toby Colding.
Will Kazez: Tautness of foliations
Abstract: I will give a brief overview of the role tautness plays in the study of foliations of 3-manifolds. Elementary examples will be constructed to show that notions of tautness that are equivalent for fairly smooth foliations are not equivalent in the world of less smooth foliations. These examples of "phantom tori” have implications in the study of approximations of taut foliations by contact structures. This is joint work with Rachel Roberts.
Tao Li: Laminations, foliations and branched surfaces/Tunnel numbers of satellite knots (2 talks)
Abstract: First Talk: We discuss how to use branched surfaces to construct essential laminations and taut foliations.
Second Talk: We prove that the tunnel number of a satellite knot is no smaller than the tunnel numbers of its companion and its pattern knot. We also discuss a relation between Heegaard genus and degree-one maps.
Kathryn Mann: Orderable groups and actions on 1-manifolds
Abstract: Group actions on 1-manifolds appear often in the theory of codimension one foliations (through foliated circle bundles, R-covered foliations, Thurston's universal circle construction, etc.) and a useful way to capture such an action is through the algebraic data of a left or circular order on a group.
In this talk, I explain new work connecting the topology of the space of left or circular orders on a group G with the space of actions of G on the line or circle. We'll see that isolated circular orders correspond precisely to a strong rigidity phenomenon for group actions, give a number of examples, and as a particular application disprove a conjecture that free groups do not admit isolated circular orders. This is joint work with Cristobal Rivas.
Shigenori Matsumoto: Weak form of equidistribution theorem for harmonic measures of foliations by hyperbolic surfaces
Rachel Roberts:From spines to branched surfaces
Abstract: Call a knot K in the 3-sphere persistently foliar if every
manifold obtained by rational (non-trivial) Dehn surgery along K admits a
taut, co-orientable foliation. I will describe a construction of laminar
branched surfaces that is useful for demonstrating that a knot is
persistently foliar. This construction, which may be viewed as a
generalization of Gabai's disk decomposition, begins with a spine built
from the boundary of a tubular neighbourhood of the knot, a (not
necessarily orientable) spanning surface, and decomposing disks. This is
joint work with Charles Delman.
Saul Schleimer: Veering triangulations - constructions, conjectures, and counterexamples (2 talks)
Abstract: In the first lecture I will give an overview (with many pictures!) of some of the flavors of triangulations of three-manifolds: material, ideal, geometric, angled, taut, and veering. These are roughly arranged from more flexible to less. As we shall see, veering triangulations have no ``local moves'' and so are the most rigid.
In the second lecture we'll explore the natural conjecture following from the lack of local moves: any three-manifold $M$ has only finitely many veering triangulations. I will give several families of examples and, in particular, show that the most direct approach to the finiteness conjecture fails. This is joint work with David Futer and Henry Segerman.