Members Seminar

One way to construct new 3-manifolds is by surgery on a knot in the 3-sphere; that is, we remove a neighborhood of a knot, and reglue it in a different way. What 3-manifolds can be obtained in this manner? We provide obstructions using the Heegaard...

In this talk I will discuss some classical and new applications of the abc conjecture, its relation to conjectures about elliptic curves, and some (admittedly weak) unconditional partial results.

*Please excuse the first 45seconds of video, due to technical issues, the video breaks up* This is joint work with Michael Magee. We combine concepts from random matrix theory and free probability together with ideas from the theory of commutator...

This talk will discuss the question: To what extent are the fundamental groups of compact 3-manifolds determined (amongst the fundamental groups of compact 3-manifolds) by their finite quotients. We will discuss work that provides a positive answer...

The problem of comparing the shapes of different surfaces turns up in different guises in numerous fields. I will discuss a way to put a metric on the space of smooth Riemannian 2-spheres (i.e. shapes) that allows for comparing their geometric...

After a brief introduction to the dynamics of the $\mathrm{GL}(2,\mathbb R)$ action on the Hodge bundle (the space of translations surfaces), we will give a construction of six new orbit closures and explain why they are interesting. Joint work with...

The Langlands program is an intricate network of conjectures, which are meant to connect different areas of mathematics, such as number theory, harmonic analysis and representation theory. One striking consequence of the Langlands program is the...

Following the revolutionary work of Thurston and Perelman, the topology of 3-manifolds is deeply intertwined with their geometry. In particular, hyperbolic geometry, the non-Euclidean geometry of constant negative curvature, plays a central role. In...

There is a natural action of the group $\mathrm{SL}(2,\mathbb R)$ of $2 \times 2$ matrices on the unit tangent bundle of the moduli space of compact Riemann surfaces. This action can be visualized using flat geometry models, which allows one to make...

A conjecture of Read predicts that the coefficients of the chromatic polynomial of a graph form a log-concave sequence for any graph. A related conjecture of Welsh predicts that the number of linearly independent subsets of varying sizes form a log...