I am a father of three and a Krener assistant professor in the UC Davis math department under the mentorship of
Elena Fuchs. I received my PhD in April, 2020 from the University of
Colorado under the supervision of Katherine Stange. My email address is daemartin@ucdavis.edu.

My wife, Kaitlyn, and I met teaching in Point Hope, Alaska in
2012. We taught in Alaska and California together for three years before moving to Colorado for graduate school. We married in 2014 and had our
first daughter, Madeleine, in 2017. She gave her first seminar at CU at around 18 months old (top-right). Olive was born almost
two years after Maddie. She is the rascal of the bunch. Our third daughter, Charlotte, was born in June (both bottom-right pictures).
Right now she just likes to spit up and laugh about it.

I study algebraic number theory with a focus on algorithmic and computational problems. My past research includes work in Diophantine approximation, lattice problems from cryptography, and computational and algebraic aspects of Bianchi groups.

## Fundamental polyhedra of projective elementary groups

For *O* an imaginary quadratic ring, we compute a fundamental polyhedron of PE_{2}(*O*), the projective elementary
subgroup of PSL_{2}(*O*). This allows for new, simplified proofs of theorems of Cohn, Nica, Fine, and Frohman.
Namely, we obtain a presentation for PE_{2}(*O*), show that it has infinite-index and is its own normalizer in
PSL_{2}(*O*), and split PSL_{2}(*O*) into a free product with amalgamation that has PE_{2}(*O*) as
one of its factors.

## A geometric study of circle packings and ideal class groups

A family of fractal arrangements of circles is introduced for each imaginary quadratic field *K*. Collectively,
these arrangements contain (up to an affine transformation) every set of circles in ℂ with integral
curvatures and Zariski dense symmetry group. When that set is a circle packing, we show how the ambient structure
of our arrangement gives a geometric criterion for satisfying the almost local-global principle. Connections to
the class group of *K* are also explored. Among them is a geometric property that guarantees certain ideal classes
are group generators.

Here is a lecture video from a conference at ICERM that
explains the connection between this paper and the continued fractions paper. Here is the
tool used to create the images in the paper. I mess with it
sometimes to create specific images, so it may or may not be functioning properly.

## Bounds on entries in Bianchi group generators

### To appear in *International Mathematics Research Notices*.

Upper and lower bounds are given for the maximum Euclidean curvature among faces in Bianchi's fundamental polyhedron for
PSL_{2}(*O*) in the upper-half space model of hyperbolic space, where *O* is an imaginary quadratic ring of
integers with discriminant Δ. We prove these bounds are asymptotically within (log Δ)^{8.54} of one another.
This improves on the previous best upper-bound, which is roughly off by a factor between Δ^{2} and |Δ|^{5/2}
depending on the smallest prime dividing Δ. The gap between our upper and lower bounds is determined by an analog of
Jacobsthal's function, introduced here for imaginary quadratic fields.

## Continued fractions over non-Euclidean imaginary quadratic rings

*Journal of Number Theory*, 243 (2022), pp. 688-714.

We propose and study a continued fraction algorithm that can be executed in an arbitrary imaginary quadratic field,
the novelty being a non-restriction to the five Euclidean cases. Many hallmark properties of classical continued
fractions are shown to be retained, including exponential convergence, best-of-the-second-kind approximation quality
(up to a constant), periodicity of quadratic irrational expansions, and polynomial time complexity.

Here is a C++ implementation of the continued
fraction algorithm. Here is a
C++ implementation of an algorithm that
precomputes admissible parameters for a given ring (used to make Table 2 in the paper).

## Reductions between short vector problems and simultaneous approximation

*ANTS XIV: Proceedings of the Fourteenth Algorithmic Number Theory Symposium*, 4.1 (2020), pp. 335-351.

In 1982, Lagarias showed that solving the approximate Shortest Vector Problem also solves the problem of finding good
simultaneous Diophantine approximations. Here we provide a deterministic, dimension-preserving reduction
in the reverse direction. It has polynomial time and space complexity, and it is gap-preserving under the appropriate
norms. We also give an alternative to the Lagarias algorithm by first reducing his version of simultaneous approximation
to one with no explicit range in which a solution is sought.

Here is a short, general-audience introduction to the problem
solved in the paper. It is the first half of the presentation at the 14th ANTS conference.

This quarter I am teaching Math 115A, Number Theory. Here is a list of past courses I have taught at
UC Davis with links to a list of topics covered in each:

Math 150C, Modern Algebra (field theory), Spring 2022

Math 21A, Differential Calculus, Spring 2022

Math 150A, Modern Algebra (group theory), Winter 2022

Math 148, Discrete Math (error-correcting codes), Winter 2022

Math 127B, Real Analysis, Spring 2021

Math 150B, Modern Algebra (ring theory), Winter 2021

Math 21A, Differential Calculus, Winter 2021

Math 21A, Differential Calculus, Fall 2020

I also taught Calculus 1 and 3 and Mathematical Analysis in Business at CU Boulder.