I've spent almost four years now writing a massive compendium of notes intended to cover everything analysis-adjacent I've learned in my undergraduate education, called The Analyst's Cookbook. 

It's broken up into seven parts:

1. Abstract mathematical reasoning – a brisk introduction to the theory of proof, elementary number theory, basics of groups and rings.

2. Abstract and concrete theory of finite-dimensional vector spaces and linear maps between them. I try to emphasize the value of matrices as a tool for making computations in the abstract theory practical for both hand work and especially automation.

3. Mathematical analysis, starting from the real numbers and taking a journey through metric space topology, continuity, differentiation in normed spaces, abstract measure theory and the fundamental theorems of Lebesgue integral calculus, the area formula and submanifolds of Euclidean spaces, mollification, basic vector field theory, fixed-point theorems and applications (and completeness), function spaces such as the C^k, Lebesgue, and Sobolev spaces, complex analysis.

4. General point-set topology, algebraic topology, differential geometry, and algebraic geometry – framed as "rubber-sheet, smooth, and singular geometry."

5. Functional analysis and the calculus of variations, in particular Tonelli's direct method and conditions for sequential weak lower semicontinuity of energies of the form u ->  ∫ f(Du) dx, where u is in a Sobolev space W(Ω, R^d).

6. Ordinary and partial differential equations, including a chapter on the intersection of probability and PDE through the study of mean-field limits and interacting particle systems. 

7. Analysis-adjacent topics: probability theory and descriptive set theory.

Here's the current part-by-part status of the notes:

• Mostly complete: parts 3, 4, 5

• Actively in progress: parts 2, 6, 7

• Not written yet: part 1

The current draft of the book can be found here. Because I only thought to form an index after it was comically late to do so, I won't have one until the final draft. If you find factual errors or serious typos in the main text of the draft or any exercises, please do reach out to me.

I've also written some other miscellaneous notes on other cool topics. 

Stuff explicitly designed for teaching

• My recitation notes for the CMU undergraduate course 21-268, a proofy multivariable calculus course for mathematics majors. These notes contain a brisk review of linear algebra, basic open and closed sets of Euclidean and metric spaces,  elementary continuity and connectedness, differentiability in Euclidean spaces, optimization, Lagrange multipliers, the implicit and inverse function theorems, multivariate Riemann-Darboux integration, elementary surface integration. 

• Lecture notes I took during a minicourse / lecture series given by Felix Otto from 4/9-4/10 on De Giorgi's ideas in the theory of gradient flows on metric spaces, and applications to mean-curvature flow.

• A brief note on an interesting theorem of geometry whose proof uses, of all things, valuation rings and Sperner's lemma. A friend of mine was TAing for the CMU undergraduate honors algebra course 21-238, and I covered her first recitation since she was sick. She was told "do any algebra that isn't forcing," passed that information to me, and this was the result.

• A pocket introduction to model theory (for those with sufficiently large pockets) which posed a unique challenge to write. I was contacted by an acquaintance who was in his senior year of high school asking me the simple question "what's model theory," having minimal background beyond calculus. This was the result.

• Some slides for a rapid-fire talk on infinite-dimensional linear algebra which I gave as a bonus recitation session for the honors undergraduate linear algebra course 21-242 Matrix Theory at CMU.

Stuff less explicitly designed for teaching

• A short exposition on realcompactness, measurable cardinals, and the fundamental groups of compact Hausdorff spaces, written as a final project for the CMU graduate course 21-752 Algebraic Topology.

• Some slides from a final talk for the CMU graduate course 21-830 Advanced Topics in PDE: Mean-Field Limits, on quantitatively comparing notions of chaos for measures in the abstract and interacting particle systems in particular. 

• Some slides from my talk at the CMU Math Club Colloquium on January 17th, 2024, about a powerful theorem in model theory and its applications well outside it.

Here's some future work I have planned:

• During Spring 2022 and Fall 2022, I took an undergraduate course and a graduate course on fluid mechanics in CMU's Chemical Engineering department, as I had initially entered CMU as a chemical engineer to get an insight on applied fluid mechanics. I have some handwritten notes for these, but have never bothered to type them up, so I might give it a crack once the Cookbook project is complete.

• A complete exposition of the model theory course I took with Rami Grossberg at CMU during Fall 2023.