The Work I Do
Some more details: my main area of research (toward my PhD) is currently time series analysis and machine learning (ok, ok, so that's two areas of research, but they're related). Time series analysis is about looking at data that evolves over time, such as stock market prices, the temperature in Central Park taken each day, or the fluctuations in a patient's heartbeat. I'm considering a specific type of algorithm to find a type of pattern among multiple streams of data. Machine learning is a cool combination of probability, combinatorics, and algorithms in which we develop algorithms which are guaranteed (in a sort-of vague way) to be good at learning certain types of problems. For example, many people in machine learning are interested in teaching computers how to understand human speech (this is a lot harder than you might think).
In previous years I also spent some time thinking about product integration. The product integral is a version of the regular integral (from calculus) which has been modified so that it multiplies the values of a function rather than adding them. The traditional use of the product integral has been for matrix-valued functions, which can assist in solving many types of differential equations. I am interested in both these classical uses and new ideas which may come from an abstract perspective.
During my third year as a PhD student, I studied DNA nanotechnology, but I decided to change my focus since I enjoy theoretical math much more than chemistry. Deoxyribonucleic acid is normally found as a long double helix structure, which is essentially linear from a topological perspective. I studied ways to utilize the self-assembling properties of DNA to create three dimensional structures. I was working with Dr. Dennis Shasha (computer science) and Dr. Ned Seeman (chemistry), whose lab has pioneered several techniques in this field.
I am also very interested in the general theory of computational complexity. So far I am not at the research level of knowledge; so I'm still teaching myself many of the principles and fundamental theorems of the field.
Of course I am not just a student but also a teacher. In fact, I really enjoy the opportunity to teach the capable students of NYU. Well, most of the students are pretty capable, anyway. I've had a lot of very bad teachers myself, and I see my work as a way to stop this tradition. Math is a truly beautiful discipline, and few courses give undergrads a chance to really see this; so I try to help out in that regard as well.
Finally, I love to program. Especially in C++ or Java. So if you have the time and inclination, please check out some of the programs I've written in the downloads page.