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, 43 chapters total:

1. Chapters 1-2: Abstract mathematical reasoning – a brisk introduction to the theory of proofs, naive set theory, functions, relations, induction, elementary number theory and combinatorics, basics of posets, groups, rings, first-order structures.

2. Chapters 3-5: 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 automation.

3. Chapters 6-18: 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, and complex analysis.

4. Chapters 19-26: General point-set topology, algebraic topology, and differential geometry – framed as "rubber-sheet and smooth geometry."

5. Chapters 27-33: Functional analysis, some abstract harmonic analysis, and 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. Chapters 34-37: 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. Chapters 38-43: Combinatorial alchemy – using tools from analysis and topology to study areas which I wrongly assumed were only done by counting. This includes some measure-theoretic probability theory, descriptive set theory, and topological combinatorics.

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

• Complete modulo typos: 3, 4, 5

• Mostly complete: 6, 7

• Actively in progress: parts 1, 2

More precisely, I can tell you exactly which chapters I'm still working on writing up:

• All of part 1 (Ch, 1-2): basics of rigorous math, posets, lightspeed theory of groups and rings, some very basic model theory,

• Ch. 5, finite-dimensional spectral theory,

• Ch. 34, a brisk review of ODE, 

• Ch. 37, interacting particle systems – specifically a section on mean-field limits for Coulomb/Riesz gases,

• Ch. 43, topological combinatorics. (currently in progress)

The good news is that it's summer and I type very vigorously... I'd like to have this project done before I need to start studying for quals.

The current public 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 well after the final draft, if at all. If you find factual errors or serious typos in the main text of the draft or any exercises, please do reach out to me.

Update (6/16/25): New public draft released! This contains Parts III-V as they will be in the final draft (modulo any typographical errors) and 83.33% of the Part VII content.

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. (errata)

• 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. (errata)

• 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 in the notes department:

• Fluids notes, assuming a decent rigorous background informed by two courses I'm taking at the CIMS this fall and a graduate fluids course I took back at CMU while I was still in the chemical engineering department.

• Generally: typed notes in similar style to Cookbook for my courses at the CIMS (probably not in a massive "all of undergrad" doc this time).

• Should I TA during my time at the CIMS (which I likely will – I enjoy teaching, plus it's a bit of extra cash), the notes I produce will be posted here.

Everything posted on this website is licensed under CC-BY-NC-SA 4.0.