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EducationPh.D. (2007) Dartmouth College Research Area: mathematical logic, set theory, forcing, reverse mathematics and a little of everything elseI am interested in analyzing the relative complexity of various mathematical objects and methods. This broad goal has led me to branch into several areas of mathematical logic. Degrees of constructibility, or simply c-degrees, are used to measure the relative complexity of objects in the universe of set theory. I am investigating the possible initial segments of this structure. The word "possible" is essential here since the axioms of set theory are not sufficient to completely determine the structure of c-degrees. I use forcing, specifically forcing with Souslin trees, to obtain relative consistency results about the structure of c-degrees. In my thesis, I showed that all constructible dual algebraic lattices can be realized as an initial segment of the c-degrees in a forcing extension of the universe of constructible sets. Degrees of computability, or simply T-degrees (T stands for Turing), are another way to measure the relative complexity of mathematical objects. Instead of using the universe of set theory, as is the case for c-degrees, the basic universe for T-degrees is the natural numbers. Although this is a much smaller universe to work with, researchers in the field of reverse mathematics have shown that natural numbers and sets of natural numbers suffice to express many of the basic structures of analysis, algebra and combinatorics. Together with Richard Shore, I am currently investigating the relative strength of certain combinatorial statements along the lines of Ramsey's Theorem. A key feature of my main line of research is that it allows me to dabble all areas of mathematics. I am a passionate mathematician, interested in all areas of mathematics. In my spare time, I like to do some work in combinatorics, computer science and number theory. Recently, I have been involved in an exhaustive search for the very elusive third Wieferich prime. I wrote all of the code to efficiently search for such numbers and, together with Dominic Klyve, we spent over a year running it on research clusters at Dartmouth College. We are currently analyzing the data we gathered, sadly no new Wieferich prime was found... Selected PublicationsComing soon.Last modified: September 18, 2007 |
