The editorial board looks great, including L. Caporaso, J. Ellenberg, D. Maulik and R. Pandharipande. They will definitely get my next algebraic geometry paper.

This is really good news. It’s seemed clear from the debates on journals of the last year that what is needed is for people and institutions of high reputation to commit to running open journals. Compositio, and the editors they have found, are top of the line. From a selfish perspective, what makes me really happy is that I didn’t wind up on the editorial board.

Good work, and good luck, to Algebraic Geometry.

Note some bizarre statements in the comment submission guidelines: They want submissions in Word or, if Word is not acceptable, another editable electronic format; they specifically state not PDF. (<rant> Why, oh why has the world forgotten RTF? Or plain ASCII? Or HTML? I understand why most people don’t want to use LaTeX, but the way that the world acts as if Word is the most convenient format drives me nuts. </rant>) They also say “[s]ubmisions become the property of the Committee.” That’s a bizarrely vague statement from a committee discussion copyright policy. I assume they mean that you are implicitly granting them the right to publish it, but if I were phrasing that I would say “you retain copyright to your words but grant us permission to retain your file/manuscript and to publish…”.

I don’t think that the points in the preceding paragraph represent bad intent on the part of the committee, but I do think they show an ignorance of how things are done in the academic world. So let’s help them out!

Hat tips: I learned about this from this blogpost, which I learned about from this comment. I am not familiar with either Glyn Moody or Alex.

But, also

We deduce that . And is clearly a positive real, so .

This immediately shows that the characteristic polynomial of has only real roots.

It is also easy to see that eigenvectors for distinct eigenvalues are orthogonal. If and , then is both and , so and .

The remaining point is to rule out the possibility of nontrivial Jordan blocks or, in the language of intro linear algebra classes, the possibility that geometric multiplicity is less than algebraic multiplicity. For a class where the Jordan canonical form theorem is proved, this is simple enough. If there is a nontrivial Jordan block, then there is some such that but . But then

so , contradicting that .

However, the Linear Algebra course that I am currently teaching doesn’t do Jordan canonical form. I haven’t figured out how to show that geometric mult. less than algebraic mult. implies that simply enough to fit in a lecture where I also talk about the rest of this stuff.

Nonetheless, I am amazed that every textbook I have seen uses the "optimize a quadratic form on the unit ball" argument rather than this algebraic once. Lots of students don't remember multivariable calculus well, and existence of maxima of continuous functions on multidimensional bounded domains is complicated. Plus, I find a lot of students have trouble with an inductive process like getting one eigenvector and splitting off an orthogonal complement.

This argument is just shuffling algebra around, combined with the fact that a sum of squares is nonzero. It seems clearly easier to me.

I'm just a little too short on time to present the whole argument this term. I'll do the two easy parts above and hand wave at the algebraic versus geometric issue. Next time I teach this course, I'll try to plan my discussion of algebraic multiplicity better so that I get the key lemma in.

I'm excited! I had thought that the spectral theorem would just be way too hard to prove in this class, but I think this will at least do a large part of it.

Theorem: On the category of (small) categories there is a unique model structure in which the weak equivalences are the equivalences of categories.

This unique model structure is of course the so-called “canonical” model structure of André Joyal and Myles Tierney. (The fact that it is the unique one with these weak equivalences lends credence, I think, to using the name “canonical”). It is proper, cartesian, simplicial, combinatorial, and every object is both fibrant and cofibrant. I first learned of this uniqueness result from Steve Lack’s comments on this MathOverflow question, though there were some details left to fill in. I hope to do that here.

Below I will give an elementary proof of the above theorem, partly so I have it written down somewhere for future reference. Charles Rezk has a nice write-up of this model structure, and I will start by describing it. It consists of:

• the canonical cofibrations, which are those functors of categories which are injective on objects,
• the canonical acyclic cofibrations, which are those equivalences of categories which are injective on objects (these are necessarily injective on morphisms too),
• the canonical acyclic fibrations, which are those equivalences which are surjective on objects, and
• the canonical fibrations, which are the “iso-fibrations“. These are the functors such that for any and any isomorphism in Y, then there exists an isomorphism in X which maps to under f.

Let’s fix some notation which will be useful later.

• Let E be the free walking isomorphism. This is the contractible category with two objects. (A category is contractible if it is equivalent to the terminal category pt).

There is an equivalence , which includes the terminal category as one of the objects. The canonical fibrations are precisely those maps which have the right lifting property with respect to this functor.

The Proof:

Now let us suppose that we have a model category structure on the category of categories in which the weak equivalences are the equivalences of categories. Such a structure consists of certain classes of fibrations and cofibrations. Our goal is to show that these must be the canonical fibrations and canonical cofibrations above.

The proof will use some basic properties about model categories:

1. The cofibrations and acyclic cofibrations are closed under retracts, compositions, and pushouts along arbitrary maps.
2. The acyclic cofibrations are precisely those cofibrations which are also (weak) equivalences. The acyclic fibrations are also weak equivalences.
3. The fibrations have the right lifting property with respect to the acyclic cofibrations and the acyclic fibrations have the right lifting property with respect to the cofibrations.

It also rests on a

Key Fact: every equivalence class of objects contains a fibrant representative and a cofibrant representative.

(Recall that an object is cofibrant if the unique map from the initial object (the empty category, in this case) is a cofibration. Dually, and object is fibrant if the unique map to the terminal object is a fibration.)

What do these facts tell us? 

Trivial Lemma: The inclusion is a cofibration.

Proof: Since each equivalence class of categories contains a cofibrant representative, we know that there exists some cofibration   for a non-empty category A. The desired map is a retract of this, hence also a cofibration. ◊

The acyclic fibrations must have the right lifting property with resect to all cofibrations, hence with respect to this map  . This means the acyclic fibrations must be surjective on objects. Since they are equivalences too, this implies the following consequences.

1. the acyclic fibrations are a subset of the canonical acyclic fibrations, hence
2. the cofibrations contain the canonical cofibrations, hence
3. the acyclic cofibrations contain the canonical acyclic cofibrations, and hence
4. the fibrations are a subset of the canonical fibrations.

This means we are half-way there. We must rule out the possibility that there could be more cofibrations. (This includes the case there are more acyclic cofibrations).

A Somewhat Less Trivial Lemma: If the cofibrations contain a map which is not a canonical cofibration (i.e. fails to be injective on objects), then the following map is also a cofibration (hence an acyclic cofibration as it is an equivalence):

.

Proof: Suppose that we have a functor which is a cofibration but not injective on objects. Then there exists at least one pair of objects in the source category which map to the same object in the target category. Call these objects x and y, and their image p.

The cofibrations are closed under pushouts along arbitrary maps and this allows us to alter this map to make a new cofibration. First note that a functor from a category to E is the same as a partition of its objects into two disjoint sets. Thus we may choose a functor which separates x and y. We may form the pushout along this map to get a new cofibration:

.

At this point we would like to form a retract onto the desired morphism. The problem is that this might not be possible as the image in X of the non-trivial isomorphism in E might fail to be an identity. If that is the case we will not be able to retract onto the desired map.

However cofibrations are also closed under composition. Let be the contractible category with the same objects as X. There is a unique functor

which is the identity on objects. Since it is injective on objects it is a canonical cofibration, hence this map must also be a cofibration. Composing gives us a new cofibration:

and now this retracts onto the desired map. ◊

Whew! That was the hardest part of the proof. Glad that’s over.

So we have learned that if the cofibrations contain more than just the canonical cofibrations, then they also contain , which is then necessarily an acyclic cofibration. This leads us to define the following:

Definition: A category is gaunt if every isomorphism is an identity.

Gaunt categories are what you get when you take category theory and strip away the fleshy meat of topology (in this case 1-types or groupoids).  We also have this:

Another Trivial Lemma: If the acyclic cofibrations contain the map , then the fibrant objects are necessarily gaunt.

The proof is just unraveling definitions.  We also have a

Trivial Observation: Not every category is equivalent to a gaunt category (e.g. non-trivial groupoids).

But now we see a contradiction emerge. For a model structure, every equivalence class of objects must contain a fibrant representative. If the cofibrations contain more that the canonical cofibrations, then is a cofibration and hence  the fibrant objects are gaunt. The equivalence class of, say, a non-trivial groupoid cannot be thus represented. We are thus led to conclude:

Theorem: There is precisely one model structure on the category of categories in which the weak equivalences are the equivalences of categories. It is the canonical model structure.

]]> http://sbseminar.wordpress.com/2012/11/16/the-canonical-model-structure-on-cat/feed/ 6 Chris http://sbseminar.wordpress.com/2012/11/14/shameless-conference-promotion/ http://sbseminar.wordpress.com/2012/11/14/shameless-conference-promotion/#comments Wed, 14 Nov 2012 15:57:41 +0000 Ben Webster http://sbseminar.wordpress.com/2012/11/14/shameless-conference-promotion/ ]]> So, obviously posting on this blog has ebbed a little (I keep hoping to reverse this trend, but I think we’ve all found that demands on our time that come before blog posting tend to ratchet upward, not downward).  I assume there are still a few people reading, though, and I wanted to do a little promotion.  

One of the thing that’s been demanding my time lately has been conference organization.  We’re planning a conference at Northeastern next spring in recognition of the 60th birthday of Andrei Zelevinsky, with the exteremely original name of “Algebra, Combinatorics and Representation Theory”. I think it’s going to be great, and I encourage any of you who are able and interested to attend; we have a really great line-up of speakers.  

For more information, see our website.  I want to particularly encourage young people to attend; we’re really hoping (cross your fingers) to have funding for grad students and postdocs.  (Furthermore, it will help us to obtain said funding if young people express an interest in coming.  So, if you would like to come, please register).

My project for the next few weekends is to skim through as many algebra texts as I can and pick one to use. So I thought I’d put up a request for your opinions. Below the fold, some of my criteria:

The text should cover finite group theory, rep theory, Galois theory and, ideally, some Dedekind domains. Abstract linear algebra, including tensor products, would also be a strong plus, although in theory they’ve all had that already.

I’m a dynamic lecturer who is good at generating excitement and drawing connections. (Or so I like to tell myself.) By comparison, I am not as good at presenting technical arguments and definitions. I believe that teachers should choose textbooks which complement their style, so I would prefer a book which is careful and precise at the expense of being duller.

I’d like a good reference book. These are grad students, or undergrads who are very likely to go to grad school. The textbook should be useful to them beyond the class.

Ideally, I’d like a book which shows off connections of algebra to the rest of mathematics. All of our grad students take this course (except for those who already know the material), including lots of analysts and geometers. Let’s convince them algebra is useful and beautiful.

In recent years, the course has been taught from Dummit and Foote, from Artin, and from Lang. I definitely plan to look at these. My favorite algebra text is Jacobson, but I think I have to reject it on the grounds that he doesn’t do rep theory until the middle of volume 2, after a lot of other intimidating stuff. (I love this book, though, so feel free to talk me into using it.) Please let me know other great options I’m missing, or what you think of these.

Integrable systems, toric geometry and Okounkov bodies The answer to a question which many people have asked: What’s the relationship between integrable systems and torus actions? Allen Knutson has been telling people roughly what the picture should be for a while, but the details seemed very hairy; now they are all resolved. That gets us half way to the question I want to know the answer to: “What is the relation between these integrable systems and cluster algebras?”

A closed formula for the decomposition of tensor products of Specht modules for the symmetric group A positive formula for stable Kronecker coefficients! (Corollary 4.08) And a proof which relies on the sort of planar diagram philosophy that people on this blog love. Congratulations to Bowman, de Visscher and Orellana.

Causal diagrams and causal models Not new, but new to me. I had always learned that, if and are correlated, then the only way to tell which one causes which (or whether they are both caused by something else) was by a randomized trial. Not true! You can make this determination in a purely observational manner, by seeing how and both correlate with . Apparently, this was known since the late 80′s, but my stats course never covered it. Makes me want to go back and work in algebraic statistics.

And I’ll take the opportunity to plug a paper of my own which has been a long time coming. Schubert problems with respect to osculating flags of stable rational curves The Shapiro-Shapiro conjecture, now proved by Mukhin, Tarasov and Varchenko shows that any distinct points on give Schubert problems whose solutions are, astonishingly all real. What happens when the points collide? And what does this have to do with the work of Henriques and Kamnitzer?

The particular point I want to address is posters who write that the fault is with contest organizers, for not designing their contests with internet age security in mind. A typical exemplar of this viewpoint writes

[Cheating] is a problem with folks using antiquated methods for tests, contests, etc. – methods that are a poor fit to the current information age. Any problems they encounter should be fixed at the source – not kludged here.

I strongly disagree.

How many times in your mathematical career have you heard “math contests test nothing but speed and trickery, and have nothing to do with deep mathematical ability?” “Math contests reward those with strong test taking skills, and discourage those who dislike high pressure gimmicky environments.” “Math contests allow no time for contemplation or discussion?”

I believe that these are all true. This isn’t a case of sour grapes — I test very well, think very fast, and enjoyed math competitions greatly. But I have met many colleagues who think more slowly than I do and nonetheless are superb mathematicians. So the problem emerges, how can we design contests which show students that mathematics has room for contemplation, cooperation and insight, not simply rapid tricks?

In answer to this question, contests like the USAMTS and OMO were born. These contests ask challenging and intriguing problems, require detailed written answers and give weeks to solve them. I loved the USAMTS when I was in high school; every question was well written and interesting, and I pushed myself to write the best answers I could. The OMO is after my time, but it looks like it was designed with similar love.

Such contests are impossible to secure. You can’t ask high school students to live under proctored conditions for a week.

Cheating in contests is damaging to everyone who competes. In theory, of course, participants could simply focus on achieving their own personal best, and not think about their own placement. In practice, seeing cheaters prosper produces disillusionment, disgust and a feeling of wasted effort. This true in any field of competition, but it is worse in competitions where there is no absolute gauge of accomplishment, such as these contests. A runner who finishes the SF marathon in 2:44:05 knows she has a superb race no matter what others may have done; a Math Olympian who sees that someone got 74/75 on the USAMTS has no way of knowing whether his own 72/75 was still a superb performance or not.

If we want there to be contests which reward prolonged thought, we run competitions which can be cheated on. If we want competitors to value their achievements in these contests, we who answer questions on math fora need to be vigilant about shutting cheaters down.

PS A much harder question is how to deal with university courses. Many of the same arguments against high pressure timed exams apply, but part of what universities provide to their students, and to society, is certification of mathematical ability. I am still struggling with these issues.