Geometric Structures on 3-manifolds

NonLERFness of groups of certain mixed 3-manifolds and arithmetic hyperbolic $n$-manifolds

Hongbin Sun
University of California, Berkeley
May 3, 2016
I will show that the groups of mixed 3-manifolds containing arithmetic hyperbolic pieces and the groups of certain noncompact arithmetic hyperbolic $n$-manifolds ($n > 3$) are not LERF. The main ingredient is a study of the set of virtual fibered boundary slopes for cusped hyperbolic 3-manifolds, and some specialty of Bianchi manifolds.

Meridional essential surfaces of unbounded Euler characteristics in knot complements

João Nogueira
University of Coimbra
April 20, 2016
In this talk we will discuss further the existence of knot complements with essential surfaces of unbounded Euler characteristics. More precisely, we show the existence of a knot with an essential tangle decomposition for any number of strings. We also show the existence of knots where each complement contains meridional essential surfaces of simultaneously unbounded genus and number of boundary components. In particular, we construct examples of knot complements each of which having all possible compact surfaces embedded as meridional essential surfaces.

Veering Dehn surgery

Saul Schleimer
University of Warwick
April 12, 2016
(Joint with Henry Segerman.) It is a theorem of Moise that every three-manifold admits a triangulation, and thus infinitely many. Thus, it can be difficult to learn anything really interesting about the three-manifold from any given triangulation. Thurston introduced ``ideal triangulations'' for studying manifolds with torus boundary; Lackenby introduced ``taut ideal triangulations'' for studying the Thurston norm ball; Agol introduced ``veering triangulations'' for studying punctured surface bundles over the circle.

Collapsing hyperbolic structures: from rigity to flexibility and back

Steve Kerckhoff
Stanford University
April 5, 2016
This talk will be about some phenomena that occur as (singular) hyperbolic structures on 3-manifolds collapse to and transition through other geometric structures. Typically, the collapsed structures are much more flexible than the hyperbolic structures, leading to the question of which structures arise as limits of hyperbolic structures.

The solution to the sphere packing problem in 24 dimensions via modular forms

Stephen Miller
Rutgers University
April 4, 2016
Maryna Viazovska recently made a stunning breakthrough on sphere packing by showing the E8 root lattice gives the densest packing of spheres in 8 dimensional space [arxiv:1603.04246]. This is the first result of its kind for dimensions $> 3$, and follows an approach suggested by Cohn-Elkies from 1999 via harmonic analysis.

Slowly converging pseudo-Anosovs

Mark Bell
University of Illinois, Urbana-Champaign
March 22, 2016
A classical property of pseudo-Anosov mapping classes is that they act on the space of projective measured laminations with north-south dynamics. This means that under iteration of such a mapping class, laminations converge exponentially quickly towards its stable lamination. We will discuss a new construction (joint with Saul Schleimer) of pseudo-Anosovs where this exponential convergence has base arbitrarily close to one and so is arbitrarily slow.

Counting closed orbits of Anosov flows in free homotopy classes

Sergio Fenley
Florida State University; Visitor, School of Mathematics
March 18, 2016
This is joint work with Thomas Barthelme of Penn State University. There are Anosov and pseudo-Anosov flows so that some orbits are freely homotopic to infinitely many other orbits. An Anosov flow is $R$-covered if either the stable or unstable foliations lift to foliations in the universal cover with leaf space homeomorphic to the reals. These are extremely common. A free homotopy class is a maximal collection of closed orbits of the flow that are pairwise freely homotopic to each other.

Proper affine actions of right angled Coxeter groups

Jeffrey Danciger
University of Texas, Austin
March 8, 2016
We prove that any right-angled Coxeter group on $k$ generators admits a proper action by affine transformations on $\mathbb R^{k(k-1)/2}$. As a corollary, many interesting groups admit proper affine actions including surface groups, hyperbolic three-manifold groups, and Gromov hyperbolic groups of arbitrarily large virtual cohomological dimension. Joint work with Francois Gueritaud and Fanny Kassel.

Morse index and multiplicity of min-max minimal hypersurfaces

Fernando Codá Marques
Princeton University
March 1, 2016
The Min-max Theory for the area functional, started by Almgren in the early 1960s and greatly improved by Pitts in 1981, was left incomplete because it gave no Morse index estimate for the min-max minimal hypersurface. Nothing was said also about the fundamental problem of multiplicity. In this talk I will describe our current efforts to develop the theory further. I will discuss the first general Morse index bounds for minimal hypersurfaces produced by the theory. We also settle the multiplicity problem for the classical case of one-parameter sweepouts.

Free group Cayley graph and measure decompositions

Yong Hou
Princeton University; Visitor, School of Mathematics
February 23, 2016
I will talk about convex-cocompact representations of finitely generated free group $F_g$ into $\mathrm{PSL}(2,\mathbb C)$. First I will talk about Schottky criterion. There are many ways of characterizes Schottky group. In particular, convex hull entropy criterion, Hausdorff dimension criterion. In addition we can also construct measure decomposition on Cayley graph, which is a generalization of the Culler-Shalen decomposition, gives criterion on primitive sets. And, I will discuss primitive curves in hyperbolic Handlebody that are Schottky.