Whoops!

I seem to have broken my MaBloWriMo streak. I hope you’ll believe me when I say it was impossible for me to get a post up yesterday. Unfortunately, the rest of this week looks just as hairy (for completely different reasons), so I’m going to have to take a break. Here’s where we’re headed once I have time again:

I’m going to skip a lot of background at some point and just introduce several equivalent definitions of the Schur functions. My hope is that stating some of the important results of the theory, even without proof, will be enough to get other people interested in symmetric function theory. I also get a lot of material to go back to and flesh out in subsequent posts, since I haven’t gone through most of the proofs of the basic results very thoroughly.

After that, I want to meander slowly through parts of basic algebraic number theory and algebraic geometry. My goal here is to thoroughly understand the classical analogy between rings of integers in number fields and nonsingular affine algebraic curves. Since several bloggers have covered much of this material in some form already, I’ll try to link to other posts I’m aware of, but I’ll have to repeat some things because I want to motivate every definition I need.

Since I don’t have anything else to say at the moment, let’s make this post an open thread and we’ll see if the Scott Aaronson style of blogging works for me. General comments, questions, suggestions, requests, etc. welcome!

I don’t trust uncountable sets

I have a mathematical confession: I don’t trust uncountable sets.

Some time ago on MathOverflow somebody asked what a reasonable definition of “infinite permutation” would be. The first answer that comes to mind is a bijection . The set of all such bijections does form a group, but not only is it uncountably generated, it contains, as Darsh observes, a copy of every countably generated group (acting on itself by left multiplication). In particular it contains a copy of the free group on countably many generators. It also doesn’t seem to carry any natural kind of topology.

On the other hand, a much nicer candidate is the set of “compactly supported” permutations, i.e. those which fix all but finitely many elements. This countable group is generated by transpositions and therefore has a neat presentation given by the usual relations. I believe it’s also the largest locally finite subgroup of the full group of bijections.

I find this group much more philosophically appealing than the full group of bijections, and the reason is simple: each element of the group is computable. On the other hand, only countably many elements of the full group of bijections are computable: the rest can’t be written down by a Turing machine. And I don’t trust anything that can’t be written down by a Turing machine.

Corollary: I don’t trust the real numbers.

Instead of explaining what I mean by this, which I don’t think I have time for today, I’ll just throw a question out to the audience: how do you feel about all this?

Halmos on writing and education

John Ewing wrote up a nice collection of quotes from Paul Halmos for the Notices of the AMS; let’s meditate on his words.

For example:

The best notation is no notation; whenever possible to avoid the use of a complicated alphabetic apparatus, avoid it. A good attitude to the preparation of written mathematical exposition is to pretend that it is spoken. Pretend that you are explaining the subject to a friend on a long walk in the woods, with no paper available; fall back on symbolism only when it is really necessary.

I’d have to agree. I see this as one of the strongest aspects of, for example, Terence Tao’s expository style. His latest post on relativization is a perfect example; Tao is a master at recognizing when technical details would obscure his exposition and when they are necessary. A related point:

IMO 2009 and proof systems

The problems from IMO 2009 are now available. I haven’t had much time to work on them, though.

There are two classical geometry problems, which I already know I won’t attempt. While I am well aware that classical geometry often requires a great deal of ingenuity, I am also aware of the existence of the field of automatic geometric theorem proving. On this subject I largely agree with Doron Zeilberger: it is more interesting to find an algorithm to prove classes of theorems than to prove individual theorems. The sooner we see areas like classical geometry as “low-level,” the sooner we can work on the really interesting “high-level” stuff. (Plus, I’m not very good at classical geometry.)

Zeilberger’s typical example of a type of theorem with a proof system is the addition or multiplication of very large numbers: it is more interesting to prove symbolically than it is to prove individual “theorems” such as . Zeilberger himself played a significant role in the creation of another proof system, but for the far less trivial case of hypergeometric identities (which includes binomial identities as a special case).

But so I can make my point concretely, I’d like to discuss some examples based on the types of problems most of us had to deal with in middle or high school.

Exceptional structures

Recently Isabel Lugo asked about problems that are hard for intermediate values of some parameter, and in discussing the question I got to thinking about exceptional structures in mathematics such as the sporadic groups. In 2006 David Corfield asked about how “natural” the sporadic simple groups are at the n-Category cafe. In that discussion and more generally there seem to be approximately two extremes in perspective:

• Exceptional structures represent a lack of room for asymptotic behavior to occur; thus they are distractions from the “generic” case. This seems to be the case for certain exceptional isomorphisms; there are only so many groups of a particular small order. It also seems to be a good way to think about objects that behave fine in characteristic zero or high characteristic but behave badly in low characteristic, characteristic usually being the worst offender.
• Exceptional structures represent the deepest part of a theory, and the exceptional structures in different fields are often related; thus understanding exceptional structures is crucial. This seems to be the case for the octonions, which can be thought of as an underlying cause of Bott periodicity. It also seems to be a good way to think about objects related to the number ; John Baez tells a great story about connections between the Leech lattice, the Dedekind eta function, string theory, and elliptic curves all centered around this mysterious number.

So what do you think? Are exceptional structures accidents or miracles? (Or, as a third option: am I failing to distinguish carefully enough between interesting and uninteresting exceptional structures?)

Mathematical historical fiction

Bill Gasarch is right – writing technical posts is tiring! (I’ve been trying to finish the next GILA post for days.) So I’ll share some more thoughts instead. Today’s thought was triggered by David Corfield:

In the first of the above posts I mention Leo Corry’s idea that professional historians of mathematics now write a style of history very different from older styles, and those employed by mathematicians themselves. …

To my mind a key difference is the historians’ emphasis in their histories that things could have turned out very differently [emphasis mine], while the mathematicians tend to tell a story where we learn how the present has emerged out of the past, giving the impression that things were always going to turn out not very dissimilarly to the way they have, even if in retrospect the course was quite tortuous.

This in turn reminded me of something else Rota wrote about his Walter Mitty fantasies:

Going beyond your comfort zone

When someone linked me to Ravi Vakil’s advice for potential graduate students, I was struck by the following passage:

…[M]athematics is so rich and infinite that it is impossible to learn it systematically, and if you wait to master one topic before moving on to the next, you’ll never get anywhere. Instead, you’ll have tendrils of knowledge extending far from your comfort zone [emphasis mine]. Then you can later backfill from these tendrils, and extend your comfort zone; this is much easier to do than learning “forwards”. (Caution: this backfilling is necessary. There can be a temptation to learn lots of fancy words and to use them in fancy sentences without being able to say precisely what you mean. You should feel free to do that, but you should always feel a pang of guilt when you do.)

It’s great to hear this coming from an expert because this is exactly what I’ve been doing for the past year without realizing it. Without formally learning anything, I’ve begun extending tendrils into algebraic topology, category theory, and all sorts of subjects about which I still can’t say anything particularly intelligent. However, from my experience so far I have a tentative list of the benefits of this strategy:

1. It becomes easier to recognize related concepts or constructions across different subjects, hence to tie them together.
2. If you have a concept you don’t fully understand sitting in the back of your head, it may come to pass that once you learn the necessary tools to understand it you may re-derive the concept partially based on your memory. As Richard Feynman said, “what I cannot create, I do not understand.”
3. Certain things become better motivated if you can say to yourself something like, “oh, I know why we’re learning about Theorem X; it’s an instance of Phenomenon Y which has lots of other nontrivial instances.” Here I’ll give an example: Pontryagin duality.
4. You are naturally led to ask lots of questions, and questions are great. “This looks a lot like Theory Z,” you might ask your professor. “What’s the connection?”

The idea that constantly working outside your comfort zone is key to progress appears to me to be a general phenomenon; in two-player games and sports, for example, playing opponents who are better than you is a great way to improve.

What I’m curious about, though, is whether the undergraduate math curriculum explicitly encourages “tendril” behavior. Perhaps it’s just something every math major should be motivated to do independently, but I can’t help but think that Ravi’s advice, which I’ve never seen written down anywhere else, should be more widely acknowledged.

I hate axioms

(A more appropriate title for this post would probably be “I hate Bourbaki,” but I like it as is.)

I spend a lot of my free time reading research papers, usually in combinatorics; those tend to require the least background. Today I decided to read everything I could find written by one of the great champions of combinatorics, Gian-Carlo Rota, and in his philosophical writings I found the explicit declaration of an opinion I’ve held for some time now.

Consider the following passage from Syntax, Semantics, and the Problem of the Identity of Mathematical Objects:

The real line has been axiomatized in at least six different ways. Mathematicians are still looking for further axiomatizations of the real line, too many to support the justification of axiomatization by the claim that we axiomatize only in order to secure the validity of the theory.

Whatever the reasons, the variety of axiomatizations confirms beyond a doubt that the mathematician thinks of one real line, that is, the identity of the object is presupposed and in fact undoubted.

The mathematician’s search for further axiomatizations presupposes the certainty of the identity of the object, but recognizes that the properties of the object can never be completely revealed. The mathematician wants to find out what else the real line can be. He wants ever more perspectives on one and the same object, and the perspectives of mathematics are precisely the various axiomatizations, which lead to a variety of syntactic systems always interpreted as presenting the same object, that is, as having the same models.

Or the following passage, from Combinatorics, representation theory, and invariant theory: The story of a ménage à trois:

Some context

It occurs to me that I haven’t really explained what I’m doing here. My reasons for writing this blog aren’t the same as they were when I first started it, so I’ll discuss both the old and the new ones.

When I started this blog in the summer of 2006, I was a sophomore in high school fresh out of the Program in Research for Young Scientists, an experience that convinced me that I wanted to be a mathematician. The name and subtitle of the blog are a reference to Professor Glenn Stevens, the director of the program. At the time, I found a lot of truth in what he said. One of the biggest lessons of the program is a strong emphasis on rigor; that is, on annoying precision.