Mathematics | Astrophotography | Data Engineering

James Collier

Academic Writings

math Senior Thesis: Two-Colored Twist Formulae in Skein Theory
August 19, 2026

Personal Writings

Coming soon.

Recent

Coming soon
Knots as Hexagonal QR Codes

I reimplemented Bar-Natan and van der Veen's Θ invariant in Python and drew it as hexagonal QR codes.
Senior Thesis

My thesis on twist formulae for two differently colored strands, and what they give for 2-bridge links.
Astrophotography

Some of my photos of nebulae and galaxies.
Math Projects

Guided projects on hyperbolic knots, Mostow rigidity, and circle dynamics.

About

Portrait of James Collier.

Hi! My name is James Collier and I graduated from Yale in 2026 with a BS in Intensive Mathematics. I'm using this personal site as an opportunity to show off some of my passions and interests.

I study math, but I'm also passionate about astronomy, physics, statistics, linguistics, puzzles, and more. I've been an astrophotographer, an astronomy teacher, and currently do work with data engineering and creating LID pipelines for the Yale Library.

Contact

james.collier@aya.yale.edu

I plan to add more external websites in the future!

Posts

Senior Thesis

My thesis on twist formulae for two differently colored strands, and what they give for 2-bridge links.
Coming soon
Knots as Hexagonal QR Codes

I reimplemented Bar-Natan and van der Veen's Θ invariant in Python and drew it as hexagonal QR codes.

Projects

A toroidal knot (not hyperbolic) — A toroidal knot (which is not hyperbolic!)

Independent study - Ka Ho Wong

Hyperbolic Knots and the Volume Conjecture

Independent study which led toward my senior thesis with my advisor Ka Ho Wong. Studied the volume conjecture relating the Kashaev invariant to the volume of the hyperbolic knot complement, and proved it in the case of the figure-eight knot. Hyperbolic knots are of interest because the hyperbolic volume of their complement is an invariant when finite. This allows us to distinguish knots, since the hyperbolic volume is conserved under deformation. Draws on representation theory, complex analysis, hyperbolic geometry, and analysis.

Bounded diameter for a projection (Lücker 2010)

Fall 2024 - Dongryul Kim

Mostow's Rigidity Theorem

Guided project under Dongryul Kim. Built toward a proof of Mostow rigidity for hyperbolic 3-manifolds: given two compact hyperbolic 3-manifolds with isomorphic fundamental group, there exists a unique isometry between them. Leads to important consequences such as the invariance of hyperbolic volume. Explored topics such as ergodicity, hyperbolic geometry, geodesic flow, and quasi-geodesics.

The Denjoy Example (Wikimedia)

Spring 2025 - Ethan Cohen

Circle Dynamics

Guided project under Ethan Cohen. Explored classes of homeomorphisms and diffeomorphisms on S¹, Poincaré classification, rotation number, and bounded variation.

Coursework

Freshman: Multivariable Calculus, Fundamentals of Physics I & II, Linear Algebra, Discrete Math, ODEs
Sophomore: Analysis I, Group Theory, Classical Mechanics, Vector Analysis, Lebesgue & Fourier Analysis, Advanced Linear Algebra, Galois Theory
Junior: Complex Analysis, Graduate Algebra, Quantum Mechanics, Linguistic Phylogenetics, Introduction to Topology, Graduate Representation Theory, reading projects
Senior: Algebraic Topology, Bayesian Statistics, Stochastic Processes, Applied Machine Learning / Causal Inference, Knot Theory Thesis

Gallery

A wide-field image of the Orion Nebula (M42), showing its characteristic pink and blue emission structure. — Orion Nebula

A wide-field image of the Antares Nebula region, with red and gold dust clouds surrounding the star Antares. — Antares Nebula

A deep-sky image of the Andromeda Galaxy (M31) showing its spiral structure. — Andromeda Galaxy

Research

Six panels of light curves for V4641 Sagittarii in the B, V, I, J, H and K bands. Each shows black and green points over two cycles with a red periodic curve through the black points. — Light curves of V4641 Sgr in six bands over two cycles. Black is passive, green is active, and red is the periodic fit to the passive envelope.

STARS Summer Research Program, 2023 - Prof. Charles Bailyn

In the summer of 2023 I worked with Professor Charles Bailyn through the STARS program on V4641 Sagittarii, a binary star system where one of the two objects is a black hole that gives off X-rays. I had about twenty years of optical and infrared observations of the system to work with.

The system doesn't always behave the same way. Some of the time it sits in a passive state where it's fairly stable, and other times it's in an active state where it's much less so. In the plot above, the black points are the passive magnitudes and the green points are the active ones. I fit a periodic regression to the passive envelope (the red curve), and used that as a baseline to pick out the periods where the system was unstable. The bigger goal behind all of this was to find evidence of a stellar jet coming from the black hole.

For the analysis I used Python, Excel, Siril, and SAOImage, along with NumPy, AstroPy, and SciPy for building the timelines and preparing the optical data: alignment, dithering, and differential aperture photometry.