# About me

I am a husband, a father of two, and a math Ph.D. candidate at the University of Colorado under the supervision of Katherine Stange.

# Research interests

Algebraic and geometric number theory.

# About the cute people up there

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. As you can see, Maddie is a bit advanced, having read Lang cover to cover by 14 months. You can also see she
currently instructs at CU, with that particular picture taken from her algebraic geometry class. Our second daughter, Olive, was born almost two
years later in 2018. Olive is the calm and thoughtful one. Her favorite hobbies are sleeping and being a sweetie.

# Contact

Please email daniel.e.martin@colorado.edu.

# Research interests

I study algeraic and geometric number theory with emphases in Diophantine approximation, lattice-based cryptography, and Apollonian circle packing.
My research statement details the projects that are currently underway in each of these subfields.

# Papers

## Continued fractions in non-Euclidean imaginary quadratic fields

In the Euclidean imaginary quadratic fields, continued fractions have been used to give rational approximations to complex numbers since the late
19th century. A variety of algorithms have been proposed in the 130 years following their introduction, but none are applicable
outside of the same five fields. Here we overcome the non-Euclidean obstacle. We show how continued fractions can be produced in any imaginary
quadratic field, and we prove that they share many of the properties enjoyed by their classical forebear. The inspiration for the algorithm is a
fractal arrangement of circles arising from subsets of GL_{2}(**C**) acting on the Riemann sphere. The geometry of these arrangements reveals
an analog of the Euclidean algorithm that points us toward a more general continued fraction.

Here are slides presented at the March AMS joint sectional meeting in Hawaii and a
lecture video from the recent conference at Brown University's ICERM. Here is the
tool used to create the images in the paper.

# Philosophy

My main focus in teaching is to make students so curious about the inner workings of the subject matter, that they wouldn't accept the exam solutions
in advance even if I offered (I don't). In my experience the most effective way to approach this goal is with inquiry-based learning. Even when time constraints
make this difficult, I teach with as many thought provoking questions as I can create and find. Here is my full teaching statement.

# Current course

Fall 2019: I am exclusively researching in this semester.

# Past courses

Here are the CU Boulder course websites for Calculus 1,
Calculus 2, and
Calculus 3
.

Spring 2019: Math 2400-003, Calculus 3. Monday - Friday, 9:00 - 9:50 in KCEN N100

Fall 2018: Math 2400-003, Calculus 3. Monday - Friday, 8:00 - 8:50 in MUEN E432

Spring 2018: Math 2400, Calculus 3 (TA)

Fall 2017: Math 1300-010, Calculus 1. Monday - Friday, 4:00 - 4:50 in MUEN D144

Spring 2017: Math 1300-007, Calculus 1. Monday - Friday, 12:00 - 12:50 in MUEN E123

Fall 2016: Math 1300-004, Calculus 1. Monday - Friday, 9:00 - 9:50 in MUEN D144

Spring 2016: Math 1112-001, Mathematical Analysis in Business. Monday - Wednesday, Friday, 8:00 - 8:50 in FLMG 157

Fall 2015: Math 1300, Calculus 1 (TA)