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We report on work with David Gay and Daniel Hartman with application to diffeormophisms of the 4-sphere.

We will give an overview during the first 50 minutes or so, then take a brief break enabling people to leave, then about 30 minutes of some details. We get kicked out of the room by 5PM.

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A nearly hundred-year old question by Fatou asks for a synthesis of the following two kinds of holomorphic dynamical systems under a common framework of holomorphic correspondences on the Riemann sphere: (a) Kleinian groups acting on the Riemann sphere (b) iteration of complex polynomials on the Riemann sphere. Sullivan's dictionary gave us a way of translating techniques from one of these fields to give results in the other. In a relatively recent development, building on Sullivan's dictionary, a bridge has been built between these two classes in the spirit of Bers' simultaneous uniformization theorem. New holomorphic dynamical systems on the Riemann sphere have thus been discovered that arise as combinations or matings of Kleinian groups and polynomials. In some cases, these single valued matings give rise to multi-valued algebraic correspondences on the Riemann sphere, partially fulfilling Fatou's dream. A particular consequence of these constructions is an analog of the compactness theorem for Bers slices of punctured sphere groups. In 1982, Thurston posed a number of questions that guided the development of the theory of Kleinian groups for the next 3 decades. With the above analog of Bers compactness in place, many of these questions reincarnate themselves in this new context. We will survey some of these developments and questions. This is joint work with Yusheng Luo and Sabyasachi Mukherjee.

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I will describe recent work with Sergio Fenley and Rafael Potrie to construct (infinitely many) new examples of smooth Anosov flows on 4-manifolds.  This will be essentially independent from Sergio's earlier talk on the subject (in particular that talk is not a prerequisite), focusing on the smooth setting rather than topological.  

You are cordially invited to attend our End-of-Semester potluck on Wednesday, May 13th from 5:00-7:00pm. We encourage you to bring your family or a guest---the more the merrier!

For those of you who do not know, a 'potluck' party is one in which the food is provided by YOU.  You prepare a dish, such as a salad, a main dish, or a dessert, bringing enough for yourself, your family/guests, and 4 more.  This way, everyone can have a little of each dish.  SLMath will provide beverages.

George will formally introduce the projection maps from the maximal hyperbolic space to the quasi-trees and quasi-lines.

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I will give an overview of how one gets from the "Lipschitz model map" to a bi-Lipschitz homeomorphism. The main things that can go wrong are entanglements of surfaces and Margulis tubes, and the way one shows these things do not happen is by a detailed analysis of the geometric limits of sequences of model maps and their manifolds. I will focus mostly on illustrative examples, so as to not get entangled ourselves.

I will discuss some future directions coming out of my papers with Antonio Alfieri. 

I will walk through the website for the census of veering triangulations with up to 16 tetrahedra (https://math.okstate.edu/people/segerman/veering.html, joint work with Andreas Giannopoulos and Saul Schleimer) and demonstrate some of the functionality of the veering code base (https://github.com/henryseg/Veering, joint work with Anna Parlak and Saul Schleimer).

During my PhD, I developed a program to construct (taut) foliations from pairs of (tight) contact structures, providing a converse to Eliashberg--Thurston's celebrated approximation theorem. While this technology works well at producing new taut foliations near old ones, I will present some failed attempts at using these tools to construct many more taut foliations. I will discuss the subtleties and obstructions in more detail in my subsequent talk in the 'Getting dirty with foliations' seminar.

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Foliations and contact structures are by definition very different objects, lying on opposite extremes of the integrability spectrum. However, Eliashberg and Thurston proved in their breakthrough work that aspherical foliations in dimension 3 can always be approximated by contact structures; think of it as maximally crumbling puff pastry. During my PhD, I proved a converse result on the construction of foliations from suitable (pairs of) contact structures. This is like reconstructing the puff pastry from its contact crumbs! In this talk, I will present some key ideas behind this construction, and discuss some resulting flexibility phenomena for taut foliations.

Speaker: Ellis Buckminster     Title: Foliations forcing closed orbits

Abstract: Knowing topological and geometric properties about foliations on 3-manifolds often yields dynamical information about transverse flows, and vice versa. In this vein, we address the following question: What properties of a foliation force all transverse flows to have closed orbits? We fully resolve this question in the case of depth one foliations on closed hyperbolic 3-manifolds, showing roughly that foliations of “intermediate complexity” admit transverse flows without closed orbits, while the most simple and most complicated foliations force closed orbits. This is joint work with Audrey Rosevear.

Speaker: Ross Griebenow        Title: Hyperbolization of fibered 3-manifolds

Abstract: We sketch Thurston's proof that a 3-manifold fibering over the circle admits a hyperbolic structure if and only if the manifold is atoroidal.

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Brandis will discuss the combinatorial HHS construction used to show that $\Gamma$ is an HHG.

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