For meticulous exposition, and documentation, of the results is almost as important to me as the proofs of results themselves…
Here, you’ll find the following kinds of docs:
These are LaTeX pdf's that contain logs of the results I prove during the courses I take, while reading math books, or when inspiration strikes. I obsess over maintaining linearity and being precise. My earlier works used to tolerate zero abuse of language or notation resulting in razor-precise, but that resulted in excruciatingly long statements for even simple results. However, over time, I believe I have learned how to strike a balance so that more can be said in less. Nevertheless, I still try to be extremely explicit about any abuses, via remarks, that occur throughout the pdf's.
I am also quite nitpicky in mentioning whenever any choice (CC, DC or AC) gets used.
These are Google Drive links to my notes that will sometimes accompany the "Organized results" pdf's. These are where I write all the proofs—really recording the journey whose end products are the "Organized results". Expect these to be messy. But sometimes, I do re-write things nicely.
Primarily, they are pdf's of handwritten notes. Only lately have I started writing these in $\mathrm{\LaTeX}$.
Courses
Though the “Organized results” here often contain all the results proved in the lectures (including tutorials), these also occasionally contain the results that I prove independently, some of which I have highlighted in comments.
At IIT Gandhinagar
| Course | Links | Comments |
|---|---|---|
| (MA 504) Introduction to Linear Algebra | o.r., notes | – |
| (MA 501) Basic Algebra | o.r., notes | §4 of Chapter 1: An original formulation to derive the basic properties of permutations. |
| (MA 628) Algebra II | o.r., notes | Very incomplete; content of only the first half of the semester. Appendix is my favorite bit, culminating in the relation between ED’s, GCD domains, atomic domains, UFD’s and Bézout domains. |
| (MA 605) Commutative Algebra | o.r., notes | – |
| (MA 627) Algebraic Topology | o.r. | Until about 2/3-rd of the course. |
| (MA 509) Topics in Real Analysis | o.r., notes | §3 of Chapter 1: Some results on base representations of reals. Theorem 3.8 of Chapter 3 (“Extending a continuous function at limiting values”). Chapter 4: Generalized definition of differentiability to be able to talk of at limit points; the results that follow are also of such generality. |
| (MA 502) Complex Analysis | o.r., notes | §1 of Chapter 3: Tried laying the foundations for complex line integrals on a firm footing. |
| (MA 626) Functional Analysis | o.r. | Until about 2/3-rd of the course. |
| (MA 629) Introduction to Differential Geometry | o.r., notes | Very incomplete; just multi-variable calculus is included. However, what’s proven is quite generalized, in the same spirit as in Cartan’s amazing book Differential Calculus. |
o.r.: Organized results
On NPTEL
| Course | Links | Comments |
|---|---|---|
| Introduction to Rings and Fields | o.r. | Doesn’t include last week’s content. |
| An Introduction to Point-Set Topology | o.r. | Only contains the first half of the course. |
By topic
| Topic | Links | Comments |
|---|---|---|
| Topology, metrics and linear spaces | o.r., notes | – |
| Category theory | o.r. | These are notes of CatCafé, a lecture series delivered by Prof Sanjay. |
| The Moore-Aronszajn theorem | o.r. | A component of my Master’s project with Prof Gadadhar Misra |
Books
I do hope to finish a book someday. :’)
| Book | Links | Comments |
|---|---|---|
| Aluffi’s Algebra: Chapter 0 | o.r. and e&s | Until some of Chapter 2. |
| Tao’s Analysis I (3e) | o.r.v1, o.r.v2, e&s, notes | v1: Until Chapter 6. v2: Until Chapter 4. The Appendix contains some nice general results. |
| Artin’s Algebra (2e) | o.r., e&s | Until some of Chapter 4. |
| Halmos’ Naïve Set Theory | e&s, notes | Until Section 18. |
e&s: Errors and suggestions