Research

I am primarily interested in algebraic number theory and arithmetic statistics. In particular, I am interested in the shapes/geometry of lattices arising in number theory, studying lattice distributions, and in counting questions/asymptotics of number fields. If you are interested in hearing more about these projects, or talking about other projects, send me an email!

I have also had the opportunity to work on a few fun projects in arithmetic dynamics, algebraic geometry, and a biologically inspired project involving permutation groups and game theory.

Brief introduction to shapes (...in progress)

• You might think of the shape of a lattice as the shape of it's fundamental domain. More precisely, given a lattice $\Lambda$ we can define its shape, $\text{sh}(\Lambda)$ to be the lattice up to homothety (scaling, rotation, and reflection). Often people view the shape as an element of a double coset space, $\text{GL}_n(\mathbb{Z})\backslash \text{GL}_n(\mathbb{R})/\text{GO}_n(\mathbb{R})$, which is mentioned more in the shape papers below.
  • For an example you can take a rank 2 lattice in the plane: rotate it so that one vector is on the $x$-axis and the other vector lies above the $x$-axis, then we can scale the lattice so that the real vector has length 1. So, given a rank 2 lattice we obtain a point in the upper half plane. The only thing left to do is act on the lattice by change of basis, i.e. the general linear group, $\text{GL}_2(\mathbb{Z})$. Note that if we act instead by the special linear group, $\text{SL}_2(\mathbb{Z})$ we get the (possibly familiar) tesselation of the upper half plane, where each point is equivalent (under the group action) to one in the gray shaded region. Extending this action to the general linear group splits each region in half so the space of shapes of rank 2 lattices, $S_2$, can be represented by the hyperbolic triangle that connects the three point: $\infty, i$, and $\frac{1+\sqrt{-3}}{2}$.
  • Given a family of rank 2 lattices we can ask many questions about the shapes in $S_2$ and how they change as we vary over elements of the family: do they lie on particular subspaces? Do they seem to be somewhat random in the space/on the subspaces? Are there any limit points? What is the measure of the space and how the shapes distributed with respect to this measure?
• Given a number field, $K$, there are different lattices associated to it that we can talk about the shape of. For example, we can consider the lattice coming from the maximal order (the analogue of the integers in these larger fields), or the lattice coming from the units (invertible elements in the field), or lattices generated by ideals. My work, so far, has focused on the latter two cases where we examine the shapes in various families of fields with prescribed Galois conditions (symmetries), try to classify their shape, the subspaces they lie on, and answer questions about their distribution.
  • In the case of unit lattices forthcoming work of Rob Harron, Sameera Vemulapalli and myself consider Dihedral number fields of degree $p$ with signature $[1,(p-1)/2]$ which, by Dirichlet's unit theorm, yield a lattice of rank $(p-1)/2$. We came upon this family, which we were already intersted in from the lens of arithmetic statistics, while computing shapes of rank 2 unit lattices for various families. We prove that the shapes of unit lattices in $D_5$ fields lie on a fixed hypercycle in $\mathcal{S}_2$. Generalizing this result to all $p$ we show that the unit shapes of $D_p$ fields, under certain conditions on $p$, lie on a finite union of hypersurfaces (in a dynamical sense, these spaces are presented as torus orbits). The canonical shape is determined by the trace form of the totally real field $\mathbb{Q}(\zeta_p + \zeta_p^{-1})$. We provide a visualization of this obit in $\mathcal{S}_2$ HERE which desribes the space on which $D_5$ unit shapes lie. (This is all made more precise in our draft but feel free to reach out it you have questions or are intersted in this!). The entire hypercycle is given HERE.