Growing up with C-C Spaces.

At some point I meant to write a blog post about this, but the damned preparations constantly get in the way. It might as well be now.

Anyways, I'm finally giving the talk that I've wanted to give for a while:

Tomorrow: an Introduction to C-C Spaces
[click here for an abstract at Student Analysis Seminar]

It's a lot of fun and a friendly mix of many a topic. Sometimes I believe that this area is what made me decide to be a mathematician, so many years ago.

Of course, it wouldn't be a very good conference talk or even a faculty-run seminar talk. There's the faux pas that I won't be talking about any of my own research, for one thing. Moreover at every conference I've attended, everyone seems to know this stuff already .. so like a little kid trying to run with the big kids, I learned about C-C spaces, too.

With that in mind, it feels like I'll be talking only about trivial things .. yet thinking about it further, it's taken me 4-5 years to sort out the scope of this topic. Each time I learn something new, something clicks in my head from C-C Spaces.

• some differential and Riemannian geometry,
  
• a little abstract algebra for Lie groups,
  
• metric topology,
  
• functional analysis and the calculus of variations,
  
• a little dynamics,
  
• measure theory and geometric analysis,

I wonder what that means. The perfectionist side of me is convinced that I could have learned these things in far shorter a time, but just this once I'll tell him (or me) to shut up. That's not the point, though.

Now I have to explain this stuff to someone else: a small audience, in fact, and composed of my own peers ..

[scratch that: they're all quite bright and far cleverer than me (;]

I feel like that little kid again, rushing back to the playground and telling all my friends about what I saw .. hoping to tell the story right and hoping they won't laugh at me if I tell it wrong.

How do I explain a theory to my friends, when it took me years to understand it? Will they understand? Will they think it's trivial or boring stuff?

I suppose everyone likes to be well-liked, even mathematically. Perhaps I've seen too many math grad students who grumble at the thought of analysis, or undergrads who groan at the sight of calculus. After a while it gets to you and leaves you uneasy and introspective.

Anyways, back to preparing notes for tomorrow's talk. Wish me luck!

A Call to Attention; Dealing with the Past and Inner Demons

As of two days ago I haven't had much time to read the Gromov article in my recently-celebrated acquisition, the Sub-Riemannian Geometry text. I still stand by what I've written last time, but I'm quickly reaching the impression that it will take a lifetime to fully understand all the nuances and connections that M. Gromov writes in those 100+ pages .. or rather, it will take me a lifetime, if I estimate rightly.

Realizing my mental faults and limitations doesn't bother me as much as it used to. In one sense it is relieving, just as you'd feel relieved at seeing the sheer number of books in a single Barnes & Noble Bookstore. The sheer amount of information unknown to you is limitless, and you can rest assured that there is far more knowledge in the world that what's between your two ears. In that way human civilization can progress merrily without my being responsible to push it forward.

Moreover there will never be a time in my life when I'd run out of things to do or to learn. From my perspective, that's quite a relief: there's no end to the fodder and fun of learning, and certainly that's one less thing to worry about.

Making sure that you learn what you want to learn and in a reasonable amount of time .. that's a separate issue. It was brought to my attention today when discussing my academic career with my funding director.

Right.

I'm supposed to be getting to know the faculty and finalizing a choice for a Ph.D. advisor. The standard suggestion is to ask a potential advisor for a reading course, and I believe that the suggestion is standard for reasons easily guessed. Yet I've done nil towards those ends.

I know these profs to a relative extent, but what do I really know? Am I really sure that my personality and tendencies of study will agree with theirs? Do I really have enough information?

No.

I don't know why it seems so difficult to do so, but it does. Why would either of them take me on? It seems that all I've done while at Michigan is blunder about, and when I've actually accomplished something I know that I should have done so a great deal sooner. This is the pessimist in me talking, but all I see from my past year and a half are:

• setbacks because of unpreparedness: passing my Qualifying exams took far more attempts than I deem necessary;
  
  
• some choices badly made and priorities misset; perhaps I could have organized my time more efficiently, as to keep my responsibilities in good standing and keep.
  
  
• an incomplete understanding of course material I should have learned properly, and shoddy coursework as a result;

I don't look like a very promising student from that account, and it's getting more and more difficult to determine how to troubleshoot my progress.

Am I doing something wrong? When learning, why do certain ideas seem to me so incomprehensible? There must be a clear approach that I'm missing, and it is increasingly frustration to figure out what that is, exactly.

Argh. No good.

I hate being non-clever, unresourceful, bad at asking questions and bad at seeking the weak points of an argument. I don't know how one acquires these rather useful habits, and rather necessary towards doing well in mathematics.

Strangely enough I remain hopeful that these matters will work out, but only if I act accordingly and promptly. It's never any dearth of love or interest for mathematics, but facing setback after setback, doubts eventually and inevitably catch up to you.

Maybe I was wrong after all, in gauging how well I'm realizing my faults and limitations .. and how to reconcile them with ends and results. It wouldn't be the first time I've erred for the worse.

A Book Acquired: Synthesis and an Optimistic Viewpoint.

At last.

I've finally snagged a copy of Be11aiche's $ub-Riεmannian Ge0metry¹, a Birkhauser text and an all-around recommended and cited text for those who study CC (Carn0t-Carathe0dory) spaces. After months of harassing the library staff at UM and steadfastly refusing to pay the $100+ for my own copy of the text, it eventually occurred to me that I could just as easily take advantage of some other library's archive through Inter-Library Loan .. and here I have it: the earthy green hardcover and all.

So wonderful .. I might even be getting a little misty-eyed.

Half of the book is composed of an article by M. Gr0mov, and browsing through the first section (it seems only mathematicians and computer scientists are willing to name the first section: 'Section Zero'), I'll repeat the description which I had written to a friend of mine, just a few hours before:

It's like sitting down for a lecture without knowing how short or long it will last. The speaker begins by turning on several projectors to multiple, polygonal-shaped transparency screens, one devoted to a branch of mathematics.

During the introduction he narrates, pointing to this screen, then that one, often to two or three at the same time. Provided you recognise the content of most of the screens, it becomes a kaleidoscope where all the screens form prism-faces, and you find yourself looking through the ultimate looking-glass. The object in sight is the same, and it is only your perception which changes.

Well, that was more of a paraphrase than a repetition .. but you get the idea. (;

---

At moments like these it is easy to revert from thinking in mathematics to thinking about mathematics, or more precisely, thinking about mathematicians and mathematical thought.

What is it that we are doing exactly, when we study and make mathematics, and good mathematics, at that? What is 'good' mathematics, anyway?

I'd be foolish to give an answer, but I'll do it anyway - that is, I'll give a partial and personal answer to what sort of mathematics I like. One of the great beauties in pure maths is the sheer variety and at the same time, the surprising condensation of ideas. There seem countless ways to describe and discuss all sorts of concept and phenomenon, real or theoretic .. yet on many occasions, there is a deeper fact or idea which governs it all, which manifests itself in that multitude of ways.

I find it sufficiently difficult to understand even the fundamental ideas in a single field of mathematics. There seems no end to the possible intricacy and subtlety in a single definition or concept.

That is why it's so fascinating to me, when I learn a relation or correspondence between seemingly separate and disparate ideas. These persons who explain these things to me, they really know something of substance: they understand this and they undeerstand that, and now they tell me that the two ideas are really the same! No, no, you must simply look at it from this perspective, or perhaps through this funny pair of glasses, and when I do, the resolution is amazing. I see what they see, so very clearly in that single instant ..

It's a wonderful reduction when it occurs, an efficiency of memory and brainpower, a convenient structure one may fall back upon and it will not collapse. On a more philosophical level, it assures us, we bearers of human intellect, that perhaps the world is knowable, if only this world of ideas. We may know much more after having learned old lessons and how they fit into new context. Perhaps we need only understand a few things, and the rest will follow.

Now I'm rambling. Better to spend time reading and learning, rather than ranting and supposing. I've waited for this volume of Be11aiche long enough, and I might as well enjoy it before the late fees arrive.

¹More accurately, A. Be11aiche is one of the editors to $ub-Riem@nnian Ge0metry, which is a collection of several articles on the topic. He is, however, the author of one of the articles therein.

Topics Old and New ..

Odd how topics you've heard about have a tendency to come back. This is an abstract from the UM Math Dept Seminar Bulletin.

Analysis Study
Thursday, January 13, 3:00-5:00, 3866 EH

Juha Heinonen

Optimal transport and synthetic treatment of Ricci curvature bounds (after Lott-Villani and Sturm)

We hope to spend a few lectures in the study seminar on recent works by J. Lott and C. Villani, and K-T. Sturm, who have studied the problem how to define analogs of lower Ricci curvature bounds for metric metric measure spaces. The key idea is to use recent advances in optimal transport and so called Wasserstein distance in the space of probability measures on a given metric space.

This sounds great stuff! The last I heard of Wasserstein spaces was from attending a series of lectures ([1], [2]) by N. Ghoussoub, this past May at CNS Summer School '04. The topic concerned a general inequality which models the free energy interactions of gases, but oddly enough a special case allows a new derivation of Sobolev and Gagliardo-Niremberg inequalities. If I remember rightly, his last lecture advocated the further study of the Wasserstein space, because surprisingly it has some natural notion of a tangent bundle!

Amazing stuff, this! I wouldn't have expected this line of thought - to mix arguments of curvature and bounded geometry with notions of optimal transport and the space of probability measures. Maybe this is a very natural and obvious phenomenon, who knows? But this is one of those moments which I'll be happy to be a buffoon, and learn as much as I can!