Special Year 2015-16: Geometric Structures on 3-manifolds

The non-orientable genus (a.k.a crosscap number) of a knot is the smallest genus over all non-orientable surfaces spanned by the knot. In this talk, I’ll describe joint work with Christine Lee, in which we obtain two-sided linear bound of the...

We will discuss methods of decomposing knot and link complements into polyhedra. Using hyperbolic geometry, angled structures, and normal surface theory, we analyze geometric and topological properties of knots and links.

Given a trivalent graph embedded in 3-space, we associate to it an instanton homology group, which is a finite-dimensional $\mathbf{Z}/2$ vector space. The main result about the instanton homology is a non-vanishing theorem, proved using techniques...

Given a trivalent graph embedded in 3-space, we associate to it an instanton homology group, which is a finite-dimensional $\mathbf{Z}/2$ vector space. The main result about the instanton homology is a non-vanishing theorem, proved using techniques...

In this talk, I present my proof of the conjecture that there are only a finite number of hyperbolic 3-manifolds of bounded volume and trace field degree.