Geometry and Dynamics in the Teichmüller space and the Outer space.

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We propose to study various aspects of geometry and dynamics of Teichmüller space and the closely related Outer space. Below is the list of my active projects and the names of collaborators that are involved in each of these projects. ******1-- Geometry of Teichmüller space equipped with Thurston Metric (with David Dumas, Babak Modami, Anna Lenzhen, Jing Tao and Fanny Kassel) ***Thurston introduced a metric on Teichmüller space that uses the hyperbolic geometry, rather than conformal geometry, to define the distance between two points. This metric is more natural in many ways, in particular, in the way it interacts with the Thurston boundary of Teichmüller space. Recent studies have shown that it equips Teichmüller space with rich structure. However, its geometry remains larger unexamined. ******2-- Geometry of Teichmüller Space Equipped with Teichmüller Metric (with Maxime Fortier Bourque, Misha Kapovich and Robert Young) ***This is an old topic, however with many open problems. We examine convexity properties of Teichmüller space as well as the behavior of Dehn functions in Teichmüller space.******3- Translation Surfaces of Finite and Infinite Type (with Anja Randecker and Howard Masur) ***These problems are natural extensions of recent progress in the field of flat surfaces (including the work of Eskin-Mirzakhani). We examine which directions in a translation surface of infinite type define a uniquely ergodic foliation, the same question asked by Veech in the case of surfaces of finite type. ******4- Random Walks in Mapping class group (with Alex Eskin) ***Can the Lebesgue measure in the boundary of Teichmüller space be obtained as the stationary measure of a random walk in the mapping class group?******5- Geometry of Outer space and related complexes (with Mladen Bestvina and Yulan Qing) ***Outer space is a space constructed as a direct analogy with Teichmüller space. However, there are considerably fewer tools available. For example, is there an analogue of the distance formula for Out(F_n)? I the free factor complex uniformly hyperbolic? Is the free splitting complex uniformly hyperbolic? ******6-- Counting problems in Teichmüller space (with Juan Souto) ***This is following and extending the work of Mirzakhani. Considering points in Teichmüller space as geodesic currents provides a new point of view where, using analogies with the symmetric space, many difficult counting problems become approachable.******7- Big Mapping Class Group (with Juliette Juliette Bavard and Spencer Dowdall) ***This is a new field and even the most basic problems are open. We ask if these large mapping class group have the strong distortion property in the sense of Schreier. ******8- Shape of Moduli Space (with Maxime Fortier Bourque and Robert Young) ***The shape of moduli space remains mysterious. For example, what is the Cheeger constant of moduli space? Is it coarse, homogenous almost everywhere? Does it resemble an expander graph?**