My current research is in rational homotopy theory. My work is on characterizing rational homotopy types of spaces. Specifically, using the fact that the homotopy groups of a topological space naturally form a graded Lie algebra (with bracket induced by the Whitehead product), I am forming a moduli space of all rational homotopy types of simply-connected topological spaces with the same homotopy groups and bracket. Since the structure I'm preserving is a graded Lie algebra, I'm following Quillen's footsteps and using differential graded Lie algebra models in my classification. A short, user-friendly version of my thesis will be posted on the Arxiv early this fall.
![A bigraded model with differential db=[a,a] and dc=[b,a].](https://static.wixstatic.com/media/2e962b_597e859b0e324fdeaf555484f67861f6.png/v1/fill/w_504,h_126,al_c,q_85,usm_0.66_1.00_0.01,enc_auto/2e962b_597e859b0e324fdeaf555484f67861f6.png)
I'm also working on using Twitter to engage preservice teachers in mathematics communication, outreach, and mathematics education literature. This is only in the experimental stages at the moment. The second iteration of the Twitter experiment will take place in an Arithmetic and Problem Solving for Elementary Teachers class I am teaching this Fall ('15).
The use of technology by teachers is a topic of great interest to me. One direction I would like to go with future research would be to study how different technology training affects pre-service and in-service teachers. My full research statement can be found to the right.