Geometric Structures on 3-manifolds

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...

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...

(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...

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...

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...

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...

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...

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...

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...

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...