Research Opportunities

Undergraduate and Masters Research Projects

If you are interested in doing research, here are a few projects you can work on with me.  If you're interested in these or other projects, stop by to chat or send me an email.

Mahler Measure of small degree polynomials: The Mahler measure of a polynomial is the product over all the roots (real and complex) outside the unit circle of the distance from the root to the origin. The Mahler measure is equal to 1 only when all the roots lie on the unit circle (or are zero). Mahler measure is related to Weil height, which generalizes the notion of height of a rational number as the maximum (of the absolute values of) the numerator and denominator. There are many aspects of Mahler measure to explore, including computational projects about Mahler measure of small degree polynomials.  

Knot Invariants -  Informally, you can make a knot by taking a circular rope, breaking it, making loops and reattaching it. In a precise way, knots (and links) are the building blocks of 3-dimensional "manifolds" and play an important role in geometry and topology.  Knot groups, the fundamental group of the complement of a knot, carry a wealth of information and are connected to many invariants. There are also several ways to represent knots (for example, the petal presentation) which lead to interesting quesitons about asymptotic behavior.  Other questions surround the computational complexity of related decision problems.

Undergrad Research Programs

If you are interested in research at UMD, check out Undergraduate Research Opportunities Program (UROP).   

Many colleges and universities have summer REU (Research Experiences for Undergraduates) programs in which you get a stipend to participate in a research project with other students under the direction of a faculty member. Applications for these programs are often due in January and February. The NSF funded programs are listed here, and the AMS has a list here. Some other pages can be found by searching for 'REU math'.

In addition, there are several math specific study abroad programs, where you can spend a semester or a year in in-depth study with other interested students from across the globe. Math in Moscow and the Budapest Semesters in Mathematics are two such well-known programs.

If you're interested in applying to graduate school, many programs require the general and subject GRE tests. You will also need to write a statement of purpose and obtain recommendation letters from 3-4 professors. I recommend talking to a faculty member to compose a short list of schools to apply to.