The cosmetic surgery conjecture, posed by Cameron Gordon in 1990, is a uniqueness statement that (essentially) says a knot in an arbitrary 3-manifold is determined by its complement N. In the past three decades, this conjecture has been extensively studied, especially in the setting where the knot complement N embeds into the 3-sphere. Many different invariants of knots and 3-manifolds have been applied to this problem. After surveying some of these obstructions, I will describe some recent work (joint with Jessica Purcell and Saul Schleimer) that uses hyperbolic methods to reduce the cosmetic surgery conjecture for any particular cusped manifold N to a finite computer search.

Our work is implemented in code, available on Github, which takes as input a given cusped 3-manifold and tests it for cosmetic fillings. The talk will include a code demonstration. Among our computational results are that the cosmetic surgery conjecture holds for all knots in S^3 up to 19 crossings and all one-cusped manifolds in the SnapPy census up to 9 tetrahedra.

I will sketch the proof of a theorem of Gabai on the existence of a taut, depth one foliation on link complements that are disk decomposable

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A discrete surface subgroup of GL(4,R) acts on 3-dimensional projective space, and the groups that act nicely on a properly convex domain form a stable class. We are interested in a) what kinds of topological and geometrical invariants classify these objects and b) how do they degenerate? These are both questions that have been answered for Kleinian surface subgroups of SO(3,1)<GL(4,R).

In this talk, I will focus on a special class of subgroups that fix a point in RP^3 and identify the degenerations of convex cocompact surface subgroups within this class.  The goal will be to describe a beautiful connection between the limit sets of such “degenerate” surface groups and the geometry of the stable norm on homology with respect to a certain asymmetric metric on the surface introduced by Danciger—Stecker.  There is, in particular, an oriented geodesic lamination of maximal stretch whose positive endpoints are all crushed to the same point in the limit set.  

My hope is that this talk, based on joint work with Marit Bobb, will be interesting and accessible to members of both programs.  

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Mladen Bestvina has kindly agreed to give us an introduction to projection complex machinery

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In the Riemannian symmetric space X = SL(3,R)/SO(3), consider a totally geodesic hyperbolic plane Y arising as an SO(2,1)-orbit. Given a geodesic L contained in Y, the higher-rank geometry allows one to translate Y along a transverse maximal flat, producing a one-parameter family of “floating” geodesic planes Y_{L,t}.

 

In this talk, I will describe the geometric mechanism behind the following result: as the height parameter t tends to infinity, the nearest-point projection of Y_{L,t} onto the reference plane Y converges to the geodesic L. This is a key ingredient in the construction of floating geodesic planes whose projections to quotients of X by Zariski-dense discrete subgroups of SL(3,R) have fractal closures.

A distinctive feature of the picture is that the fibers of the nearest-point projection contain entire maximal flats, leading to parallel families of geodesics and more intricate asymptotic configurations. The proof combines a detailed analysis of Busemann functions and the interaction with the visual boundary of the symmetric space.

This talk will take place on campus, in Evans Hall room 891.

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Anosov representations form a stable and rich class of discrete subgroups of semisimple Lie groups that today is recognized as the correct higher rank generalization of rank one convex cocompact subgroups of Lie groups. In this talk, I will discuss joint work with Subhadip Dey on constructing new classes of Anosov groups arising from amalgamations and HNN extensions of Anosov representations; many of these Anosov groups provide new examples of hyperbolic groups that fail to embed as discrete subgroups of rank 1 Lie groups.

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.

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Partially hyperbolic diffeomorphisms form an open set of diffeomorphisms of manifolds and arise naturally in smooth dynamics in relation with problems of stable ergodicity and robust transitivity. They contain as examples time one maps of Anosov flows as well as other algebraic constructions and their perturbations. In this talk I want to discuss the problem of topological classification of these systems, mention how they relate to dynamical problems, but mostly explain how this classification can be reduced to a problem about transverse pairs of foliations by Gromov hyperbolic leaves in 3-manifolds. This is joint work with Sergio Fenley.

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 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.

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

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

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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

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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

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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

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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

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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

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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

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

Zoom Link