All Summer School and Conference talks will be in Room 328 of Boyd Research and Education Center. Schedule: Summer School Abstracts: Michael Landry: Pseudo-Anosov flows and the Thurston norm Abstract: I will begin by surveying the theory of fibered faces via flows, a rich picture developed by Thurston, Fried, Mosher, and others. This will allow us to touch on the Thurston norm, pseudo-Anosov flows, automorphisms of surfaces (finite and infinite type), and taut foliations. It is tempting to look for a unified flow-theoretic framework for the Thurston norm that generalizes the fibered picture. To do this requires some way of producing pseudo-Anosov flows which are not suspensions, so I will survey the Gabai-Mosher construction of pseudo-Anosov flows from sutured manifold hierarchies that is currently being exposited by Chi Cheuk Tsang and myself. I will also discuss some examples of pseudo-Anosov flows that have interesting behavior with respect to the Thurston norm. Thomas Massoni: Foliations and contact structures in dimension three Abstract: While foliations and contact structures are by definition very different objects, they exhibit deep and surprising connections in dimension three. The goal of this course is to present some recent developments in the study of their interactions. After recalling the basics of foliation theory and contact geometry in dimension three (Lecture I), I will focus on the Eliashberg–Thurston theorem on approximating foliations by contact structures (Lecture II). Finally, I will present converse results on constructing foliations from suitable pairs of contact structures (Lecture III). This material provides the technical background for my research talk "Ziggurats and taut foliations" at the Conference. Conference Abstracts: Julian Chaidez: Convex hypersurfaces and robust heterodimensional dynamics Abstract: A closed oriented hypersurface in a contact manifold is called robustly non-convex if it cannot be approximated by convex hypersurfaces in the $C^2$-topology. In recent work, I constructed the first examples of such hypersurfaces in standard contact 2n+1-space using tools from partially hyperbolic dynamics and blenders. In this talk, I will explain a proof that every closed oriented hypersurface in a contact manifold of dimension five and above is isotopic to a robustly non-convex hypersurface by an arbitrarily $C^2$-small isotopy. The proof uses a robust and local obstruction to convexity built using blenders, and reveals an intriguing connection between convexity and the well-known Palis Conjecture in dynamics. Surena Hozoori: Strongly adapted contact geometry of Anosov 3-flows Abstract: We will discuss some recent developments in the contact geometric theory of Anosov 3-flows, whose roots go back to the works of Mitsumatsu and Eliashberg-Thurston in the mid 1990s. In particular, we provide a contact geometric characterization of Anosov 3-flows based on interactions with Reeb dynamics, as well as investigate the basic properties of the resulting geometries. Time permitting, we will discuss how these results allow one to re-approach some classical questions in Anosov dynamics. Michael Landry: From quasigeodesic to pseudo-Anosov flows Abstract: Calegari conjectured that every quasigeodesic flow in a closed hyperbolic 3-manifold can be deformed to a pseudo-Anosov flow. I will outline a proof of this conjecture, joint with Steven Frankel. Robert Lipshitz: Towards Legendrian contact homotopy Abstract: The Chekanov-Eliashberg dga, or Legendrian contact homology, was the first modern invariant of Legendrian knots in R^3. This talk is a progress report on a project to give a stable homotopy refinement of Legendrian contact homology, inducing operations like Steenrod squares on linearized Legendrian contact homology. In general, contact homology is defined by counting J-holomorphic curves, but in this case those counts reduce to the Riemann Mapping Theorem and the invariant is described purely combinatorially, from an appropriate knot diagram, and our refinement has a similarly combinatorial flavor. After recalling the basics of Legendrian knot theory and Legendrian contact homology, we will outline our program to refine it, sketch its status, and perhaps describe some examples. This is joint with Lenhard Ng and Sucharit Sarkar. Thomas Massoni: Ziggurats and taut foliations Abstract: It is expected that taut foliations on 3-manifolds typically admit (almost) transverse pseudo-Anosov flows. This motivates a natural question: given a pseudo-Anosov flow $\phi$ on a 3-manifold $M$ and a suitable link $L \subset M$ of closed orbits of $\phi$, for which Dehn surgery multislopes along $L$ does the surgered manifold admit a taut foliation transverse to the induced flow? The set of such multislopes has a remarkable staircase structure with corners at rational multislopes that accumulate at very specific points. These sets can be algorithmically computed in many small examples. In work in preparation with Jonathan Zung, we explain some key features of these 'ziggurat' sets and justify their name, using tools from contact geometry. Agustin Moreno: Cone structures from a dynamical and probabilistic viewpoint Abstract: The goal of this talk is to explore, from a geometric and probabilistic point of view, the dynamics of cone structures (i.e. distributions of conical sets) adapted to open book decompositions. This is inspired by the picture which arises in the study of the circular restricted three body problem (CR3BP). A problem that inspired this investigation is that of understanding, for the CR3BP, which points in configuration space can be reached from a predetermined point. Diego Santoro: The set of L-space surgeries on two-bridge linksAbstract: An L-space is a rational homology 3-sphere whose Heegaard Floer homology is minimal. Given a link in the 3-sphere, one can study the set of slopes on the boundary of its exterior for which the corresponding Dehn surgeries are L-spaces, and compare it with analogous sets associated to conjecturally equivalent properties, namely the non-existence of taut foliations and the non-left-orderability of the fundamental group.For knots in the 3-sphere, and more generally in rational homology spheres, the set of L-space surgery slopes is now well understood. In contrast, much less is known for links with more than one component, and only few examples are currently available.In this talk, I will present a joint work with Hugo Zhou, where we address this problem for two-bridge links, providing a classification of their L-space surgery sets. As a consequence of our results, we obtain an optimal uniform bound on the volume of any hyperbolic L-space that is surgery on a two-bridge link, and a classification of L-space satellite knots whose associated two-component pattern link is a two-bridge link.
