I just began working at Notre Dame’s Summer Math REU. For the uninitiated, the Research Experiences for Undergraduates program is funded by the National Science Foundation to pay for students to work at participating universities working on research in the sciences.

There are 8 students participating in the ND Math REU this year. Together, we are working on 4 separate math problems—2 students per problem. My partner and I are investigating Totally Positive (resp. Totally Nonnegative) Matrices, namely matrices whose minors are all positive (resp. nonnegative). (Note here that by minor, I really mean the determinant of any square submatrix). (See MathWorld for a description of a Totally Positive Matrix).

At the moment, we are looking at what inequalities between minors hold true for every totally positive matrix. This theory has been worked out for principle minors for up to 5×5 matrices, and generalized to some extent over non-principle minors. While there is a lot of work in this field, at present we are examining if there is a sort of ‘extremal’ or primitive set of inequalities that generate every other valid inequality.

Hopefully I’ll be able to write a more concrete description of what we are trying to accomplish at a later date.

Also, I must mention that the campus is beautiful in the summer (well, they are doing some construction, including regilding the golden dome). I’ll try to take some action shots of the REU students and the pretty campus tomorrow.

Update: Photos