To anyone in the Cambridge, MA area: a bunch of us will be organizing a learning seminar on higher categories and derived algebraic geometry at Harvard. Our goal is to understand some of the topics in the book “Higher Topos Theory” and some of the DAG papers. We will be having an organizational meeting (where we figure out what our goals and format will be) next Tuesday at 4:30. Let me know (at amathew (at) college (dot) harvard (dot) edu) if you are interested and can make it!

At the REU I’m at, we listen to daily lectures. The current topic is “geometry of polynomials,” by Sergei Tabachnikov; it will continue for two weeks. I’ve been live-TeXing notes in class.

It’s not something I anticipated doing—after all, typing is slower, right? I find that’s not really the case. First, out of concerns of laziness efficiency, I always predefine macros (e.g. \e = \mathbb) in my source files that reduce the amount of typing.  Second, since this is a talk, there are pauses in the mathematical exposition that allow one to catch up. (I actually fall behind very rarely–even though I run pdflatex and scan the output every now and then.*)  The most serious problem is that this is a geometry course. I may try whipping out an image editor and trying to copy down the various diagrams (and insert them as figures into the document later).  But it’d be hard to keep up when there are so many figures, as seems to be the case in this course—and it’ll likely be even harder in the next course (“fractal geometry and dynamics”).

But, on balance, I think I’m pretty sold on live-TeXing.  Mostly because my handwriting is awful, and I’m really bad at keeping organized sheaves of papers.  By contrast, LaTeX output is pretty and computer files don’t (usually) vanish.  I recommend it to others, as well as this post of Chris Schommer-Pries.

So, without further pontification, here are my notes from the past two days.

*On the subject, I definitely recommend using evince as a PDF viewer–it has the nice property of being able to update the document automatically without your having to close and reopen it.

I graduated! Senior year was rather drawn-out, so it is good to be done.  My plans for the near future are also set.  During the summer, I will be attending the REU at Penn State (as I have already mentioned).  This actually starts in less than a week.  In the fall, I will be an undergraduate at Harvard.

The extent to which I keep posting will likely depend on how much time and energy I have at Penn State, along with the topic of my research project (about which I know next to nothing now).  However, I do know that I will resume blogging about more expository (and older) topics, since I have finished talking about my last project.  It would be criminal to leave off class field theory right before the Artin reciprocity law, so I—as a mildly self-respecting blogger—will post a few times more on the subject.

In addition, I’ve been trying to collect together the various posts I’ve done on algebraic number theory into some sort of sheaf of notes, but it has yet to attain even quasi-coherence.  I know I will regret this someday, but here is the current messy version.  I’ll try to flesh these out a bit over the next few weeks and clean it up.  So far, except for Chapter 0, it literally consists of my blog posts, one after another.

As an undergraduate, it is necessary for me to shore up the basics before pushing too far into fancier stuff like class field theory.  Nevertheless, I will try to keep the subject matter on this blog as advanced as I can, insofar as possible.  (Which is to say that I’m making no long-term promises, since  readers know full well that I tend to break them anyway.)

Remember the Sokal affair?  That was when an NYU physics professor submitted a parody article ostensibly about science, but using meaningless jargon to a journal of cultural studies, and it got accepted.  Oops.

Well, David Simmons-Duffin, a graduate student in theoretical physics at Harvard, has created a similar parody site called the snarXiv.  So far, the site uses context-free grammars to randomly generate meaningless abstracts involving fancy terminology.  For instance,

We verify an involved correspondence between decay constants in supergravity deformed by multi-fermion operators and path integrals in superconformal superconformal QFTs surrounded by (p,q) instantons. The determination of superconformal effects localizes to AdS_n x P^m. Therefore, some work was done among mathematicians on a model of bubbles. This result has long been understood in terms of the Wilsonian effective action. The Virosoro algebra is also bounded. After reviewing fragmentation functions, we derive that spinodal inflation at $\Lambda_{QCD}$ depends on the Seiberg-dual of the Landau-Ginzburg Model.

There is also a game where you can try to distinguish the fake abstracts from the real ones (on the arXiv, the actual site). I’m ashamed to say that I’m worse than a monkey at physics.

Now, someone who knows about programming should do the same for mathematics, and use the key words: “moduli spaces, etale cohomology, $latex  \infinity$-groupoidification, Deligne-Mumford stacks, perverse sheaves, Calabi-Yau manifolds, and homotopical category theory.”

Edit: Wait, there’s more!  Apparently, the creator has a theorem generator and even a program that can generate philosophy.

Tim Gowers asked a really great question on MathOverflow recently, on examples of mathematical “cognitive biases”: false widely held beliefs in (higher) mathematics.  Which naturally enough reminded me of the embarrassing experience yesterday when I realized that, after assuming the contrary for several months, the kernel of A \oplus B \to C is not the same as the direct sum of the kernels of A \to C and B \to C. Whoops. 

It looks like the winning ones so far are about little facts in linear algebra.  It would, indeed, make the proofs of many of the technical results on Lie algebras easier if \mathrm{Tr}(ABC) = \mathrm{Tr}(CBA).

Also, Arrow’s theorem is a scam.  Support range voting!

Finally, on an entirely unrelated note, this quote is ridiculously awesome:

“I’ve had the chance, in the world of mathematics, to meet quite a number of people, both among my elders and amoung young people in my general age group, who were much more brilliant, much more “gifted” than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle — while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things I had to learn (so I was assured), things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates, almost by sleight of hand, the most forbidding subjects.
In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still, from the perspective of thirty or thirty-five years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve all done things, often beautiful things, in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have had to rediscover in themselves that capability which was their birthright, as it was mine: the capacity to be alone.”  -Alexandre Grothendieck


I have a math blog? What is this?

The main excuse I had for ignoring Mount Bourbaki for the past month or so was the Intel science competition, which ended last week.  It was a lot of fun—I met many interesting people and enjoyed numerous pleasant conversations.

To my surprise, I ended up coming in third place.  I was quite stunned by this especially after hearing the finalists called before me–I have to say that I was genuinely amazed by every project that I saw.

Of course, I can’t resist a picture.   Here is one from the gala, of the top three:

I’m the guy on the left looking in the wrong direction.

I probably will do a technical post at some point about what my project was all about, but for now here is a non-technical video I made:

Other than this, I also know now what I’m doing this summer.  I’m going to do an REU at Penn State, where I’ll be working on a topic that I should probably find out about soon.   What I’m going to do next fall is still undetermined.

As of late, I’ve been reading up on some logic and model theory and a bit of ergodic theory (from Walters’ book, which I recommend).  I tried to study Spanier while I was at Intel though didn’t get very far.  And I sincerely will try to get some entries up soon.  I don’t know whether I will be able to keep my promise of Grothendieck topologies just yet.

I don’t really anticipate doing all that much serious blogging for the next few weeks, but I might do a few posts like this one.

First, I learned from Qiaochu in a comment that the commutativity of the endomorphism monoid of the unital object in a monoidal category can be proved using the Eckmann-Hilton argument. Let 1 be this object; then we can define two operations on End(1) as follows.  The first is the tensor product: given a,b, define a.b := \phi^{-1} \circ a \otimes b \circ \phi, where \phi: 1 \to 1 \otimes 1 is the isomorphism.  Next, define a \ast b := a \circ b.  It follows that (a \ast b) . (c \ast d) = (a . c) \ast (b. d) by the axioms for a monoidal category (in particular, the ones about the unital object), so the Eckmann-Hilton argument that these two operations are the same and commutative. (more…)

I’ve created two new pages: a bibliography, and a collection of old writings.

The bibliography is for me to list the sources I use (or plan to use shortly) in writing this blog.  It’s also a list of books that I’ve found helpful in various fields, so perhaps it will be useful to others learning about the same type of material.  Currently I’ve found the books by Folland, Introduction to Partial Differential Equations, and Taylor, Partial Differential Equations, to be excellently written and well motivated. Taylor’s book also ties in the differential geometry (more so than Folland).   The style is uniformly clean but leisurely, and both books are more-or-less self-contained.

Also, I collected together some old notes I’ve written for various reasons and posted them in case anyone might find them helpful.  For instance, I wrote an expository paper on integral equations a few years back for a seminar course, which I was pleased to stumble across, since I did not remember it.  That may become a blog post in the future.

My name is Akhil Mathew.  I’m currently a high school senior interested in mathematics.  This is my mathematics blog.

I first became exposed to the blathosphere as a sophomore after a mentor of mine pointed me to Terence Tao’s blog.  I soon discovered after clicking on the links that there were numerous blaths already on the web, many of which were accessible to me. 

I joined the blathosphere in the summer of 2009 at the group blog Delta Epsilons, started by the mathematics students of that year’s Research Science Institute.  Eventually, prodded by comments there, I found that my typical style of blogging, which often includes long series of posts and a textbookish style, was ultimately unsuitable for the motto of Delta Epsilons: mathematical research and problem solving.

So, while I’m going to remain a contributor to Delta Epsilons, I’ve started this to create an additional outlet where I can post to learn math better.  I plan to post entries more suitable to Delta Epsilons–by which I mean resembling a crisp expository article than a chapter in a book–in both locations.

The tentative topics I intend to talk about in the future are diverse.  Right now I’m in the middle of a MaBloWriMo sequence on differential geometry.  But in the future I’m considering discussing topological K-theory, algebraic geometry, and harmonic analysis.  In the long run, one of my ambitious hopes is to understand the Atiyah-Singer index theorem (and its proof), which may also become a topic.  However, my interests change too quickly for me to predict with any reliability.   I hope this will make it interesting.

Incidentally, you might be wondering why I have an introductory post after over sixty posts in the archives: those are the textbookish entries that I made at Delta Epsilons, copied here for completeness.