- I led a group of eight participants at a recent AIM workshop on a project to write down higher genus analogues of the Legendre curve and prove results similar to those in my papers 2014a, 2014c, and 2014d. We've completed a long paper (roughly 120 pages!) explaining the results.
- In Arithmetic of Jacobians in Artin-Schreier extensions, Rachel Pries and I study Artin-Schreier analogues of results proven in the Kummer tower in my article 2007b, Lisa Berger's thesis, and 2013a. Our article has been accepted for publication in the Transactions of the AMS.
- In Rational curves on elliptic surfaces, we study the objects of the title over the complex numbers and show that a very general elliptic surface of Kodaira dimension 1 has no rational curves other than the zero section and the singular fibers. Equivalently, a sufficiently general elliptic curve over C(t) has no rational points over any extension of the form C(u) where t is a rational function of u. This is in marked contrast to elliptic curves over global function fields, cf. 2007b. The yoga that leads to this result suggest a natural conjecture generalizing a famous one of Lang. The article will appear in the Journal of Algebraic Geometry.
- These slides from a 2010 conference in honor of my teacher Dick Gross give an overview of what I have been doing for the last few years. Several of the problems mentioned in them have been written up in 2014a, 2013b, 2014c, 2014d, and the AIM project mentioned above.
I have projects for students at several levels. This undergraduate research project (which is not as scary as it might seem at first ...) was atttacked by several students, including Sam Kim, with excellent results. It is no longer relevant, but the description gives an idea of one area where I have other projects. I'd also be happy to talk with graduate students about projects in number theory and algebraic geometry ranging from a quick MS with good prospects for a paper to more substantial PhD projects.
My office hours for Spring 2016 are Mondays 2:00 - 3:00 and Thursdays 3:00 - 4:00 in Skiles 234. You can also set up an appointment by e-mail.
My long-form CV.
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