Tai-Danae Bradley

tai [dot] danae [at] math3ma [dot] com

I am currently a research mathematician at Sandbox AQ and a visiting professor of mathematics at The Master's University where I help run the Math3ma Institute. I finished my PhD in mathematics in spring 2020 at the CUNY Graduate Center under the supervision of John Terilla and spent some time as a postdoctoral researcher at X, the Moonshot Factory (Google X). My research interests lie in the intersection of quantum physics, machine intelligence, and category theory.

This website was originally created in 2015 as tool to help me transition from undergraduate to graduate level mathematics. Quite often, I'd find that the ideas of math are hidden behind a dense fog of formalities and technical jargon. Much of my transition process was learning how to fight through this fog in order to clearly see the ideas, concepts, and notions which lie beneath. Throughout this process I learned that writing—and drawing—helps immensely. Eventually I decided to share my writings and drawings on the web in hopes that others may find them helpful as well.

Below is a list of my research, publications, and more.

PhD Thesis

At the Interface of Algebra and Statistics
My thesis uses basic tools from quantum physics to investigate mathematical structure that is both algebraic and statistical. arXiv: 2004.05631. Here is a 10-minute trailer video:


Topology: A Categorical Approach [online version]
A graduate-level textbook that presents basic topology from the perspective of category theory
with Tyler Bryson and John Terilla / published by MIT Press 2020


An Enriched Category Theory of Language: From Syntax to Semantics [video]
with John Terilla and Yiannis Vlassopoulos‍, La Matematica (2022) doi.org/10.1007/s44007-022-00021-2
A mathematical framework for passing from probability distributions on text to copresheaves on a category enriched over the unit interval, which contains semantic information.

Entropy as a Topological Operad Derivation [video]
Entropy 2021, 23(9), 1195. doi.org/10.3390/e23091195
A correspondence between Shannon entropy and derivations of the operad of topological simplices.

Probabilistic Graphical Models and Tensor Networks: A Hybrid Framework
with Jacob Miller and Geoffrey Roeder. arXiv pre-print: 2106.15666
A framework for passing between discrete undirected graphical models and Born machines using the physically-motivated concept of quantum decoherence.

Language Modeling with Reduced Densities [video]
with Yiannis Vlassopoulos, Compositionality 3, 4 (2021). doi.org/10.32408/compositionality-3-4
An assignment of reduced densities to words in a natural language, which is shown to be a functor between categories enriched over the unit interval.

Modeling Sequences with Quantum States: A Look Under the Hood
with Miles Stoudenmire and John Terilla, Machine Learning: Science and Technology (2020)
doi.org/10.1088/2632-2153/ab8731, code available at https://github.com/emstoudenmire/parity
An algorithm for training an MPS generative model inspired by the density matrix renormalization group

Translating and Evolving: Towards a Model of Language Change in DisCoCat
with Martha Lewis, Jade Master and Brad Theilman, Electronic Proceedings in Theoretical Computer Science 283 (2018) pp. 50-61. arXiv e-print: 1811.11041
A categorical approach to representing translations within the DisCoCat language model

On the Distribution of the Greatest Common Divisor of Gaussian Integers
with Yin Choi Cheng and Yan Fei Luo, Involve, A Journal of Mathematics, vol. 9, issue 1 (2016), pp. 27-40.
arXiv: 1502.02148
Asymptotics for the average norm of the greatest common divisor of Gaussian integers.


A New Perspective of Entropy [video]
A companion article to "Entropy as a Topological Operad Derivation" written for the ambitious lay person.
The Journal of The Math3ma Institute, vol. 1, issue 1, pp. 35–65. doi.org/10.55409/math3ma2022-112

What is Applied Category Theory?
A short expository booklet inspired by the 2018 Applied Category Theory conference
e-booklet, arXiv: 1809.05923

An Invitation to Category Theory
A non-technical introduction to category theory
Chalkdust Magazine, Issue 08 (2018) pp. 22-25.
online version here

Cognition, Convexity, and Category Theory
A short summary of the paper "Interacting conceptual spaces" by Coecke, et. al.
n-Category Café blog post (2018)


Linear Algebra for Machine Learning
A friendly, non-technical intro to basic ideas in linear algebra and some ways they appear in machine learning
TensorFlow YouTube channel (2021) - click here

PBS Infinite Series (Season 2)
A series of short videos produced by PBS Digital Studios about the wonderful world of mathematics
YouTube channel (2017 - 2018) - click here

In the Media

An article from The Brilliant (April 2023)

A video interview with Miles Stoudenmire at the Flatiron Institute (August 2022)

A podcast interview on Timothy Nguyen's The Cartesian Cafe (July 2022)

A podcast interview on Sean Carroll's Mindscape (November 2021)

An interview article in Math Horizons, volume 29, issue 2 (November 2021)

A video interview on Grant Sanderson's 3Blue1Brown Podcast (September 2021)

A video interview on MΣΣT a Mathematician (February 2021)

A podcast interview on My Favorite Theorem  (January 2021)

A podcast interview on The Local Maximum (November 2020)

An article from the CUNY Graduate Center (November 2020)