Here are a collection of various mathematical notes that I have made for diverse reasons. They are at different stages of completeness, and I may at some point revise them further. I am posting them here in case they will be useful to people who come across this blog.

This is a paper containing material that I submitted to a science fair in 2007, on a version of Riemann integration in abstract spaces. I posted about it here.

I gave this as a talk, on the p-adic numbers, in spring 2009 at the conclusion of an independent study course on class field theory. Here are the notes I handed out to the people that attended.

I gave this as a talk on elliptic curves in fall 2008 based on the conclusion of an independent study course on the arithmetic of elliptic curves.

I wrote this expository paper on integral equations in 2007 as part of a seminar course in linear algebra.

Here are some notes on group cohomology I wrote in spring 2009 as part of an independent study course. The main source was Serre’s *Local Fields. *Eventually I will post some of the other notes I took as well.

This was intended as a crash course in complex analysis, intended for the reader familiar with Stokes’ theorem on manifolds. Some of it was posted here.

Here is a write-up of the Synge-Weinstein theorem in Riemannian geometry. Someday, when I am less lazy and/or busy, I shall add figures. I posted the material here.

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January 31, 2011 at 9:59 am

Hi Akhil,

I was wondering where I could find the template that you used for your presentations on p-adic numbers and elliptic curves? I have seen many presentations with that template, and it seems very appropriate for a math talk.

Thanks!

January 31, 2011 at 3:13 pm

Dear Vijay, I used the “beamer” class, which is standard for this sort of thing. I’ll email you the .tex source for the presentation.

April 25, 2011 at 10:24 pm

Dear Akhil, on your algebraic geometry notes, in the bibliography, you’ve referenced Matsumura’s book on commutative algebra. However, you’ve misspelled the name as “Hideki Matsumura”. I think it should be “Hideyuki Matsumura”. (Actually, I think these are the notes at your Harvard website.)

April 26, 2011 at 5:04 pm

Dear David, thanks for the correction! I’ll fix that once the semester is officially over and I get some time.

May 3, 2011 at 11:31 pm

I’ve just fixed it.

August 7, 2011 at 8:36 pm

Dear Akhil,

Despite a BS Physics, MS Mathematics, and PhD physics, I have done most of my learning on my own. Perusing your blog blows my mind. You exemplify rapidly growing mathematical maturity and deep self education.

Throughout my career, I have often associated with undergraduate and graduate students. They often pose questions similar to the ones you’ve received on how to best learn math. To this end, I have been putting together a guide over the years for the physicist, mathematician and engineer to better appreciate mathematics and its historical motivations, limits, and capabilities. Ditto for physics, the foundation of engineering. This work sits in a word document, it is mostly edited, and I would gladly share it with you. Your thoughts would be greatly appreciated.

Sincerly,

A. Alaniz

August 8, 2011 at 7:47 pm

I’m not sure that I’m the right person for this (you might try someone who has taught mathematics classes, for instance), but I’d be happy to look at it and offer comments, if you think I’d be useful.

August 9, 2011 at 12:51 am

Akhil,

You are very kind. What I’m after is a good, though not necessarily unique unified presentation of what math and physics are to the limit that this is possible AND useful, e.g., how the work of Sophus Lie provides a very general language for partial differential equations, topology and group theory, AND therefore to much of applied mathematical physics and theoretical physics, and allows the physicist the freedom to conjure up reasonable alternate universes in a general language. People, “students”, those who do far more than just the minimally required effort to obtain a degree, crave this kind of deeper understanding in general, and history is full of semi-failed attempts to provide such general frameworks, e.g., the work of Hilbert, and Bertrand-Russel, Zermelo-Fraenel, etc. Your posts cover a lot of mathematical variety (I’m looking at your Tags); you’re a “student”. Thus I’m interested in seeing how much unity or disparity you perceive in mathematics in the epic sense that mathematician Morris Kline protrays in “Mathematics, The Loss of Certainty”, or if such a question is even meaningful to you at this point in time. I strongly recommend this book to you. It is in essence the foundation of my thoughts on what are the minimal set of ideas in both math and physics that give us “reasonable” freedom to create “reasonable” mathematical and physical constructs, and on what limits we suffer, e.g., on the work by Cohen (1963) on whether the axiom of choice is independant of the Zermelo-Fraekel axioms, a deep, foundational math question, and how this math question might influence, or constrain recent notions that the universe is a holographic, information-theoretic processing machine if spacetime is discrete or not—physicists are wiley beasts very capable of creating “work-arounds” that lead to breakthroughs in both math and physics, e.g., how the Dirac Monopole + Algebraic Topology = deep understanding of Gauge Field Theories (Topology, Geometry, and Gauge Fields, Foundations) by Gregory L. Naber. I’ll send you the short document to your university email, and I’ll let you decide if it’s worth reading. Meanwhile, I plan to enjoy reading your posts.

Cheers,

Alex

February 3, 2012 at 4:07 am

Hi Akhil,

I’m very new to the field of mathematics, and have just now sort of sparked a self-motivated interest. Currently, the only math classes I’ve taken are up to single variable calculus. I’m interested in studying elementary real analysis, though, through Spivak’s classic. Upon diving in, I noticed a solid foundation in proofs would be helpful, so this is where I come to you: How did you get started in building a solid foundation in learning to prove things? Any text recommendations would be great.

Thanks,

Miles

February 4, 2012 at 10:00 am

Hi Miles,

I’m not really qualified to answer this question, but my impression is that courses in real analysis are usually used additionally as a way of learning how to write proofs better. However, it probably helps to have some prior exposure. At that point, all I knew about proofs came from popular math expositions (e.g. the kind I could find in a local public library). Maybe these would be helpful.

June 21, 2013 at 7:36 pm

Topologically, a torus is a closed surface defined as the product of two circles: S1 × S1. This can be viewed as lying in C2 and is a subset of the 3-sphere S3 of radius √2. This topological torus is also often called the Clifford torus. In fact, S3 is filled out by a family of nested tori in this manner (with two degenerate circles), a fact which is important in the study of S3 as a fiber bundle over S2 (the Hopf bundle). The surface described above, given the relative topology from R3, is homeomorphic to a topological torus as long as it does not intersect its own axis. A particular homeomorphism is given by stereographically projecting the topological torus into R3 from the north pole of S3. The torus can also be described as a quotient of the Cartesian plane under the identifications

(x, y) ~ (x+1, y) ~ (x, y+1).

Or, equivalently, as the quotient of the unit square by pasting the opposite edges together, described as a fundamental polygon ABA−1B−1. Turning a punctured torus inside-out The fundamental group of the torus is just the direct product of the fundamental group of the circle with itself:

Intuitively speaking, this means that a closed path that circles the torus’ “hole” (say, a circle that traces out a particular latitude) and then circles the torus’ “body” (say, a circle that traces out a particular longitude) can be deformed to a path that circles the body and then the hole. So, strictly ‘latitudinal’ and strictly ‘longitudinal’ paths commute. This might be imagined as two shoelaces passing through each other, then unwinding, then rewinding. If a torus is punctured and turned inside out then another torus results, with lines of latitude and longitude interchanged. The first homology group of the torus is isomorphic to the fundamental group (this follows from Hurewicz theorem since the fundamental group is abelian).