All Summer School and Conference talks will be in Room 328 of Boyd Research and Education Center. Schedule: Summer School Abstracts: Course 1: Pseudo-Anosov flows and the Thurston normMichael Landry, TA: Fran Herr 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. Course 2: Foliations and contact structures in dimension three Thomas Massoni, TA: Audrey Rosevear 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. Course 3: Low-dimensional symplectic dynamics: from periodic orbits to beyondDaniel Cristofaro-Gardiner, TA: Shaoyang Zhou Abstract: We will survey some recent progress in low-dimensional symplectic dynamics. The goal is to provide a user-friendly introduction to some of the powerful tools in this area, such as embedded contact homology, periodic Floer homology, quantitative Heegaard Floer theory, and feral curves, through stories about symplectic analogues of Weyl’s law and their applications. I will begin with some theorems about periodic orbits, and then move to further topics such as Reeb chords, groups of homeomorphisms, and general invariant sets. 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 C2-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 C0-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. Daniel Cristofaro-Gardiner: Low-action holomorphic curves and invariant setsAbstract: I will discuss a recent joint work with Rohil Prasad, about finding invariant sets via sequences of low-action pseudoholomorphic curves. We prove a compactness theorem, without imposing the typical assumption of uniformly bounded Hofer energy and valid in any dimension, extracting a family of closed Reeb-invariant subsets in the limit. At least in low-dimensions, we show that such curves exist in abundance. We obtain as applications: a new Le Calvez - Yoccoz type property of three-dimensional Reeb flows and area-preserving surface diffeomorphisms; the dense existence of non-dense geodesics on surfaces; a crossing energy theorem for SFT-style curves in symplectizations of any dimension; new progress on recognizing hyperbolicity through the periodic orbit set. Sergio Fenley: Exotic codimension one Anosov flowsAbstract: The Verjosky conjecture states that every codimension one Anosov flow indimensions 4 or higher is orbitally equivalent to a suspension Anosov flow. Codimension onemeans that either the weak stable or the weak unstable foliation of the flow has codimension one.This conjecture is 50 years old. In joint work with K. Mann and R. Potrie, we constructinfinitely many counterexamples to the conjecture in 4-manifolds. To achieve that, weneed a closed hyperbolic 3-manifold M, a faithful minimal representation of \pi_1(M) intoHomeo^+(S^1) (the circle), and a group equivariant Cannon-Thurston map f from S^1 to thesphere at infinity of hyperbolic 3-space. With this data we can construct a topological Anosovflow in dimension 4. If the set of non injective points f has image in the sphere at infinitywhich has measure, we show how to perturb the topological Anosov flow in a (smooth)Anosov flow, which is orbitally equivalent to it. We then show that there are infinitely manyexamples satisfying the data, producing counterexamples to the Verjovsky conjecture. 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. Ying Hu: The existence of taut foliations with zero Euler classAbstract: It is known that every oriented plane field on a closed 3-manifold is homotopic to an integrable one. However, this no longer holds if one requires the foliation to be taut. This leads naturally to the question of which second cohomology classes can arise as the Euler classes of co-oriented taut foliations on a given 3-manifold M. When M is a rational homology sphere, the second cohomology group is finite, and the zero class plays a distinguished role. In this talk, we present infinitely many rational homology 3-spheres, including small Seifert fibred, hyperbolic, and toroidal examples, that admit co-oriented taut foliations but do not admit any with vanishing Euler class. We will also discuss the implications of these examples in the context of the L-space conjecture. This is joint work with Steve Boyer, Cameron Gordon and Duncan McCoy. 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. 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.
