How to Read Mathematics and ML Papers

This course draws from two distinct literatures: pure mathematics (category theory, algebraic topology) and machine learning research. Each has its own conventions, and reading them effectively requires different strategies. This guide covers both.

Reading Mathematics Papers and Textbooks

Mathematical writing is dense by design. A single page can contain more information than ten pages of prose. The standard approach of reading linearly from start to finish will fail. Use the following strategy instead.

First Pass: Identify the Main Theorem (15–20 minutes)

Read the abstract and introduction to identify the main result.
Find the statement of the main theorem — usually in bold or a numbered environment.
Read the statement carefully without trying to understand the proof.
Look at any key diagrams or commutative squares; these often contain the entire argument visually.
Read the conclusion or summary section.

Second Pass: Follow the Proof Skeleton (1–2 hours)

Read the proof of the main theorem, accepting lemmas as black boxes.
Identify the key steps: what is constructed, what is verified, what is the punchline.
Sketch the logical flow in your own notation.
Note every place you do not understand — do not stop to resolve them yet.

Third Pass: Fill the Gaps (as needed)

Go back and work through the lemmas you accepted on faith.
Verify every claimed isomorphism or equality with a small example.
Try to construct a counterexample to each hypothesis — this tests whether you understand why each condition is necessary.

Reading ML Research Papers

First Pass: Survey (5–10 minutes)

Read the title, abstract, and introduction.
Read all section headings and look at every figure and table.
Read the conclusion.
After this pass: What problem is solved? What is claimed? Is this relevant?

Second Pass: Comprehension (30–60 minutes)

Read more carefully, skipping complex derivations on first read.
Understand every figure — if you cannot explain a figure, you have not understood the paper.
Summarize the method in one paragraph in your own words.

Third Pass: Critical Reading (1–2 hours)

Work through the math and derivations.
Question every assumption. Ask: what breaks if this assumption fails?
Compare claimed results with the theoretical guarantees (or lack thereof).
For this course: ask whether the method is an approximation to a categorical ideal, and if so, what the morphism gap is.

What to Extract from Every Paper

Section	What to Extract
Abstract	Problem, method summary, key result — one sentence each
Introduction	Motivation; why existing solutions are insufficient
Related Work	Context and positioning; often skimmable on first read
Method	The core contribution; focus the majority of your time here
Experiments	Datasets, baselines, metrics; are the comparisons fair?
Results	Key numbers; look at failure cases, not just successes
Conclusion	Summary and future directions; often reveals open problems

The Course-Specific Question

For every paper read in this course, ask one additional question beyond the standard ones:

The Categorical Lens Question

What is the categorical ideal that this method is approximating?
What are the morphisms being preserved, ignored, or approximated?
Where does the method break outside the training distribution, and what morphism structure would prevent that?