Zoom Link

Given an endperiodic surface homeomorphism f, I will explain how to find a periodic splitting sequence of train tracks carrying f's positive Handel-Miller lamination. When f is atoroidal, this allows us to build an unstable "veering branched surface" in the compactified mapping torus of f. (This is analogous to Agol's construction of layered veering triangulations, which I will describe.) From this unstable veering branched surface, one can build an associated stable branched surface such that the two form a "dynamic pair."

The talk will be relatively self-contained and won't require knowledge of pseudo-Anosov flows or dynamic pairs. However, it is the second in a series describing the Gabai-Mosher construction of pseudo-Anosov flows associated to a finite depth foliation F of an atoroidal 3-manifold M. The Gabai-Mosher construction is inductive, where the induction is over the length of a sutured manifold hierarchy. The results in this talk comprise the basis step.

This is joint work with Chi Cheuk Tsang.

Zoom Link

 In 1936 Ch. Ehresmann raised the question of ``classification'' of geometric structures modeled on a "classical geometry" (such as projective geometry) on a fixed topological manifold. An example is the classification of Euclidean structures (flat Riemannian metrics) on the n-torus, where the moduli space is naturally the biquotient GL(n,Z)\Gl(n,R)/O(n). By an observation of Thurston this problem intimately relates to the analogous problem of ``classifying'' representations of the fundamental group. This leads to interesting dynamical systems which are of interest in their own right. In this talk I will survey this subject with examples to illustrate the richness and depth of this theory.

This seminar series will take place on UC Berkeley campus, in Evans Hall room 939.

We will describe the construction of Alperin-Dicks-Porti of a sphere-filling curve which is invariant under the PSL(2,C) action of the fiber of the figure eight knot group. 

We present solutions to two problems on indefinite integral ternary quadratic forms. The first, highlighted by Margulis in 1990, concerns the distribution of the ternary Markoff spectrum associated with minima of forms. The second, initiated by Serre in 1990, is on the asymptotic growth of the number of isotropic forms of bounded height. To overcome difficulties involving high ramification, we develop a tool that proves useful for other distribution problems as well. Joint work with Alexander Gamburd, Amit Ghosh, and Peter Sarnak.

This seminar series will take place on UC Berkeley campus, in Evans Hall room 740.

The knot $10_6$ currently holds the distinction of being the `simplest’ knot whose unknotting number we do not know. It also holds a different distinction in that it is the key to identifying the `simplest’ counterexample to one of two conjectures on unknotting number (we just don’t know which…). We will discuss this knot, some of its useful properties, the search for its unknotting number, and the larger question of how to `efficiently’ compute unknotting numbers. This is joint work with Susan Hermiller.

Zoom Link

Zoom Link

In this talk, I will show how three notions a priori quite different in nature turn out to be equivalent in the context of smooth interval diffeomorphisms: the group theoretic property of being distorted (in the sense of Gromov), the topological group theoretic property of being quasi-reducible (that is: having conjugates arbitrarily close to the identity), and the dynamical property of being part of a C^1 flow without hyperbolic fixed points. This will be the occasion to gather some key facts about interval diffeomorphisms of various regularities which happen to play an important role in the study of codimension one foliations.

Zoom Link

We construct a curve graph consisting of non-peripheral, simple closed curves for surfaces of infinite type. We show that it is connected and often hyperbolic. We use the non-peripheral curve graph to show that a large set of big mapping class groups have at most quadratic divergence. A section application is that we prove a large set of big mapping class groups have infinite coarse rank. This talk is basd on joint work with Kasra Rafi and Assaf Bar-Natan. 

In this talk, we prove that any Radon measure on the horospherical foliation invariant under an Anosov subgroup is quasi-invariant under the translations by Jordan projections of elements of the subgroup. This is essentially the proof of the classification of horospherical invariant measures in joint work with Inhyeok Choi.

Many classical problems from knot theory and low-dimensional topology can be formulated as decision problems. Quite a few of them are now known to lie in complexity classes NP or co-NP, which is an upper bound on computational complexity. At the same time, only several such problems are known to be NP-hard, which is a lower bound. Even fewer such non-trivial problems are known to have a polynomial algorithm. We will discuss some such results related to 3-manifold triangulations and homeomorphism, unknotting number, and link equivalence. The proofs use tools from low-dimensional topology and knot theory, at times mixing them with algorithmic complexity theory and topological graph theory.

When $M$ is a hyperbolic $3$-manifold with $\partial M \neq \varnothing$, the geometric structure on $M$ can be deformed by varying the conformal structure on $\partial M$. If we further assume that the injectivity radius in the convex core of $M$ is bounded both above and below, then any such deformation decays exponentially fast inside the convex core. This phenomenon, known as \emph{geometric inflexibility}, is especially striking when $M$ is geometrically infinite: as one goes deeper into the degenerate end, the deformation becomes exponentially close to an isometry.

A beautiful application of this inflexibility, due to Brock and Bromberg, is the proof of convergence of the iterates of a pseudo-Anosov map on quasi-Fuchsian space. This, in turn, implies the hyperbolization theorem for $3$-manifolds that fiber over the circle with pseudo-Anosov monodromy. In this talk, I will explain the idea of geometric inflexibility and sketch the proof of this convergence result.

This seminar series will take place on UC Berkeley campus, in Evans Hall room 939.

Zoom Link

Zoom Link

This seminar series will take place on UC Berkeley campus, in Evans Hall room 939.

This seminar series will take place on UC Berkeley campus, in Evans Hall room 740.

Zoom Link

Zoom Link

Zoom Link

Zoom Link

This seminar series will take place on UC Berkeley campus, in Evans Hall room 939.

This seminar series will take place on UC Berkeley campus, in Evans Hall room 740.

Zoom Link

Zoom Link

Zoom Link

Zoom Link

This seminar series will take place on UC Berkeley campus, in Evans Hall room 939.

This seminar series will take place on UC Berkeley campus, in Evans Hall room 740.

Zoom Link

Zoom Link

Zoom Link

This seminar series will take place on UC Berkeley campus, in Evans Hall room 939.

This seminar series will take place on UC Berkeley campus, in Evans Hall room 740.

Zoom Link

Zoom Link

This seminar series will take place on UC Berkeley campus, in Evans Hall room 939.

This seminar series will take place on UC Berkeley campus, in Evans Hall room 740.

Zoom Link

Zoom Link

This seminar series will take place on UC Berkeley campus, in Evans Hall room 939.

This seminar series will take place on UC Berkeley campus, in Evans Hall room 740.

Zoom Link

Zoom Link

Zoom Link

Zoom Link

Zoom Link

This seminar series will take place on UC Berkeley campus, in Evans Hall room 939.

This seminar series will take place on UC Berkeley campus, in Evans Hall room 740.

Zoom Link

Zoom Link

Zoom Link

Zoom Link

This seminar series will take place on UC Berkeley campus, in Evans Hall room 740.

Zoom Link