Starting in Fall 2019 I will be a tenure track professor at Bucknell University. I am currently in my last year as a RTG Lovett Instructor (postdoc) at Rice University. I received my PhD in 2016 at the University of Texas at Austin and my B.S. at UIUC in 2010. I grew up in Chicago, IL. These days, my main interests outside of math include cooking/baking, painting, attending concerts, reading, playing board games, various additional arts & crafts projects, and bouldering. |
My research focuses on problems in arithmetic geometry and algebraic number theory. In particular, I often work on questions related to failures of the Hasse principle. Many of my current and recent projects concern obstructions to integral and rational points on surfaces (e.g. K3 surfaces) and higher dimensional varieties. My thesis advisor was Felipe Voloch. My postdoc mentor is Tony Várilly-Alvarado.
I am the organizer of the Algebraic Geometry and Number Theory (AGNT) seminar at Rice for the 2018-19 academic year.
Abstract: We show that odd order transcendental elements of the Brauer group of a K3 surface can obstruct the Hasse principle. We exhibit a general K3 surface Y of degree 2 over the rationals together with a three torsion Brauer class A that is unramified at all primes except for 3, but ramifies at all 3-adic points of Y . Motivated by Hodge theory, the pair (Y, A) is constructed from a cubic fourfold X of discriminant 18 birational to a fibration into sextic del Pezzo surfaces over the projective plane. Notably, our construction does not rely on the presence of a central simple algebra representative for A. Instead, we prove that a sufficient condition for such a Brauer class to obstruct the Hasse principle is 3-adic insolubility of the fourfold X (and hence the fibers) and local solubility at all other primes.
Abstract: We consider the Brauer-Manin obstruction to the existence of integral points on affine surfaces defined by x^2−ay^2=P(t) over a number field. We enumerate the possibilities for the Brauer groups of certain families of such surfaces, and show that in contrast to their smooth compactifications, the Brauer groups of these affine varieties need not be generated by cyclic (e.g. quaternion) algebras. Concrete examples are given of such surfaces over the rationals which have solutions in the p-adic integers for all p and solutions in the rationals, but for which the failure of the integral Hasse principle cannot be explained by a Brauer-Manin obstruction. The methods of this paper build on the ideas of several recent papers in the literature: we study the Brauer groups (modulo constant algebras) of affine Chatelet surfaces with a focus on explicitly representing, by central simple algebras over the function field of X, a finite set of classes of the Brauer group which generate the quotient. Notably, we develop techniques for constructing explicit representatives of non-cyclic Brauer classes on affine surfaces, as well as provide an effective algorithm for the computation of local invariants of these Brauer classes via effective lifting of cocycles in Galois cohomology.
In the summer of 2018 I co-led a 2 week intensive program for middle school students entitled Patterns, Math, and You at Rice University.
In the summer of 2012 I worked as a teaching assistant at the Summer Program for Women in Math (SPWM) at George Washington University.
Below are some photos from my two main hobbies: baking and art.