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. The current draft is close enough to final that I am happy to declare it "version 0.1," which can be found here.  

It's broken up into seven parts, 45 chapters total:

1. Chapters 1-4: 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, and a detailed exposition of some topics in pure and applied model theory – the compactness theorem (proven with ultraproducts), applications to algebra and nonstandard analysis on the applied side and some infinitary combinatorics, saturated models, special models, prime, primary, atomic models, indiscernibles – all the theory we need to prove incredible results about the spectrum function such as Vaught's never-two theorem and Morley's categoricity theorem. 

2. Chapters 5-7: 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. Chapter 5 works purely at the level of vector spaces and linear maps, Chapter 6 deduces the theory of matrices from the theory of linear transformations and explores how both perspectives on linear algebra inform one another, and Chapter 7 is all finite-dimensional spectral theory.

3. Chapters 8-20: Mathematical analysis. With the basic structure of the real numbers given, we take 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 21-28: General point-set topology, algebraic topology, and differential geometry – framed as "rubber-sheet and smooth geometry."

5. Chapters 29-35: 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 35-39: Ordinary and partial differential equations, including a chapter on the intersection of probability and PDE through the study of mean-field limits of interacting particle systems. 

7. Chapters 39-45: Combinatorial alchemy – using tools from analysis and topology to unify the discrete and continuous worlds. This includes some measure-theoretic probability theory, descriptive set theory, and topological combinatorics.

Version 0.1 contains Parts III-VII. Part II is almost done, and part I is in progress, so expect a Version 0.2 somewhat soon-ish.
Note that the chapter numbers in Version 0.1 will not match the chapter numbers shown here.
This is for the simple reason that Chapters 1-7 are not present in Version 0.1, so all the numbers seem to be off by seven. 

Version 1.0 will be the version after extensive typo-checking and possibly compiling an honest index, depending on whether I have the patience to do so. As usual, if you find factual errors or serious typos in the main text of the draft or any exercises, please do reach out to me.

Cookbook Changelog:

• previous drafts --> version 0.1: 

  • put fully finished Parts VI and VII out to the world (so that Parts III-VII are now public),

  • added forewords on every part giving a concise description of the content, 

  • added list of book recommendations in overall foreword for those who seek further or supplemental reading.

I've also written some other notes which might interest you:

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)

  • The content here will be folded into Chapter 3 of Cookbook, but I will still keep this guide up as it exposits this material assuming less mathematical maturity.

• 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. (set-theoretic errata)

• 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 a course 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).

• When I TA during my time at the CIMS, the notes I produce will be posted here.

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