Welcome to my homepage! 

I am an RTG Postdoctoral Assistant Professor at the University of Michigan, where my mentor is Dick Canary.  I received my PhD in mathematics at Boston College under Martin Bridgeman and Ian Biringer (see my math genealogy).  

My research interests concentrate in and around mapping class groups, hyperbolic geometry, and Teichmüller theory. 

Several of my articles focus on identities on hyperbolic manifolds.  Here is an amazing identity on the moduli space of a punctured torus known as McShane's identity: Let T be a once-punctured torus with a complete finite-area hyperbolic structure, then

\sum_{\gamma}\frac1{1+e^{\ell(\gamma)}}=\frac12

where the sum is over all simple closed geodesics \gamma in T  with \ell(\gamma) denoting the length of \gamma. Amazing! Not amazed? Well, it's important to remember that there are lots of different complete finite-area hyperbolic structures on the once-punctured torus (in fact, the space of all such structures is known as Teichmüller space and is homeomorphic to \mathbb{R}^2 in this case).  Hopefully now you are amazed.

More recently, I have become smitten with infinite-type surfaces and their associated (big) mapping class groups.  Here are some fun surfaces (the only 4 infinite-type surfaces to regularly cover closed surfaces):


You will find more information under the Research and Teaching tabs. Thanks for stopping by!