short post first: seminar talk woes.

Apart from student-run seminars and analysis-related seminars, I have terrible luck with choosing talks to attend. I might look at the title and abstract, think

"Oh! That looks interesting. I'll certainly go to that!"

and as an afterthought, bring my folder of work with me .. just in case.

It is unfortunate that I have little background, and too often it is the case where the speaker has lost me for good, or that the jumps in reasoning are non-obvious (to me, at least; it is a relative term) and I cannot think fast enough to follow the argument.

It then becomes frustrating: I slip into self-driven ignorance, because I cannot summon the nerve to stop the speaker and ask for a reminder or two of what is going on. Then, eyeing my folder, I slip out my notes out or a copy of a research article and hope that the speaker doesn't mind too much.

One of my commentators from my LiveJournal blog has remarked about this so-called "courtesy" of ignoring the speaker when it is inconvenient to do so, and I don't have a good answer to support or defend this practice.

But should you pay the price of time and confusion because of a mistake in choosing the wrong talk to attend?

Maybe my expectations are too high. Every so often I attend a talk, hoping to learn what the words mean in the title.

I attended an Open Problem Seminar once to learn what a Kähler manifold was, but never found it out. I later asked a friend, who told me a casual definition and now I've forgotten what it is.

Another time I attended a Colloquium in order to learn what a Gromov-Witten invariant is, but to this day I still don't know what such invariants are. I suppose they are hard, if only because they sound hard. q:

But I'm going to try again. I think this is really going to work, this time. In 1 1/2 weeks' time there will be the following Colloquium for the UM Maths Dept:

What is K-Theory and What is it Good For?
Paul Frank Baum
Pennsylvania State University

This is a survey talk on K-theory and will consist of four points:

#1. The basic definition of K-theory
#2. A brief history of K-theory
#3. Algebraic versus topological K-theory
#4. The unity of K-theory

This sounds interesting!

I've heard the term tossed around by algebraic topologists, and it is always a fine thing to learn a new word, though I suppose when most people say that, they mean words like vitriolic or recondite.

At any rate, I remain hopeful this time. The speaker can't possibly avoid saying what K-theory is, right? (;

neuroses of a mathematician-in-training.

Earlier I think I over-exerted myself in terms of how long I could spend in the office while remaining sane. I eventually left for a little sushi, and lazybones that I was I took the bus home.

But this is what I wrote, before my escape.

---

Every so often I hesitate while doing perfectly normal activities in the office or chores at home. When other people are around, I think it looks like I 'zone out' or 'blank out' for a spell, and when I recover then all is well. Still, when it happens, it feels eerie for everyone involved.

For instance, I'm sitting in my office and staring at a white page of paper. I know what I'm going to write (or I think I do, at least) and the pen's already in my hand. But my hand is shaking slightly, you see, and other than that I can't seem to move it for a second or two. I can't write down the function or draw the diagram.

It sounds obsessive-compulsive or neurotic or some standard psychologically-motivated word and I can't help but wonder if something is wrong.

There's trouble with focusing, too, and trouble from distractions.

After teaching I can't return straight to work because the transition to silence and solitude feels too great and I must do something else, like grade quizzes or check my email .. and 15-20 minutes later my brain feels ready for tackling research or coursework or prelim reading or what-have-you.

After office hours I can't remain in my office. There's something about my desk now, which emits some residual state in which my mind rested. It's too computational and unfocused and if I start proving things in that mindset then I will argue myself in circles. I must leave it, leave to a coffeehouse or the second-floor Atrium/study area and work there. Let the experiential residue of teaching to dissipate, and then I can return to 1852 EH and work unhindered.

My god. This sounds like neurotic talk, and it feels like I'm less mentally capable than I used to be. It feels like something in me has changed -- maybe snapped or dissolved, say -- and as a result I've lost some peace of mind.

I don't know. Maybe it's all in my head, which could be relieving .. or troubling, depending on how you choose to see it. I really don't know.

after much objection ..

I will write down carefully the David-Jones Theorem, directly from the Mattila text. So here it is:

For any positive integers m and n and ε > 0, there exists an integer N(ε) such that if Q is the unit cube in Rⁿ, m ≥ n, and if f : Q → R^m with Lip(f) = 1, then there are B₁, ..., B_N in Q, N ≥ N(&epsilon), such that

Hⁿ_∞[f(Q \ (B₁ u ... u B_N)] < ε

and each f|B_i is bi-Lipschitz with Lip((f|B_i)^-1) ≤ N(ε).

Sorry for the confusion and errata, guys. Let me think once or twice before I comment on this theorem again!

plotting my own doom ..

Or as it's more commonly known here, writing a syllabus of topics for my Prelim Exam. A good bit of them I've seen before, but it still seems a scary task to remember it all .. or at least enough to keep my committee happy.

If you're curious, here is a tentative breakdown of the subjects:

1. Preliminaries
   
2. Lips¢hitz Functions and C0nvex Functions
   
3. Sob01ev Spaces and Fine Properties
   
4. Highlights from Ge0metric Mea$ure The0ry

Believe me, you can say a lot about each heading. I only hope that I can, too. Wish me luck!

---

As I was browsing through books for syllabus items, I found this theorem in a book of P. Mattila. The result is credited to P. Jones, which generalises a previous result of G. David.

Maybe I am rather dull and ignorant, but it seems a little unbelievable.

Let Q be the unit cube in Rⁿ and let m ≤ n. Given ε > 0, there is a number N_ε where for all 1-Lipschitz functions f mapping Q into R^m, there are balls {B_i : i = 1 to N} with N < N_ε, where Hⁿ_∞(f(Q \ U_i B_i)) < ε and f|B_i is N_ε-biLipschitz!

Essentially, this means that off a small set set of small image, 1-Lipschitz functions from the unit cube to a higher-dimensional Euclidean space are locally bi-Lipschitz (and probably with very large biLipschitz constant). It's not even clear to me that a 1-Lipschitz function should be locally invertible, much less locally biLipschitz!

Edit: perhaps thinking of $ard's Lemm@ gives plenty of motivation. Now that I think about it, I'm not as impressed. Since n ≥ m, then the Hausdorff content given by Hⁿ_∞ doesn't detect the image that well, since in some ways, the image is an "m-dimensional set." I'd be much more impressed if the theorem used H^m_∞ instead.

Anyways, back to work.

Another Friday night at the office, and why.

The day winds down and I'm more awake now than I was sixteen hours ago, and that would likely be true even if I hadn't bought that cup of coffee an hour or so ago from Espresso Royale Caffe. It's the curse of years of late nights; some I spent doing things of importance and some I spent doing nothing in particular.

Regardless of event and memory, my nocturnal inclinations remain. It's why I'm still in the office, trying half-heartedly to finish a little more work. I don't think this is futile for a few reasons:

1. The weekend flies, and it flies quickly. It's unwise to fit all that you couldn't accomplish during the 5-day work-week and stuff it into a 2-day weekend.
   
   But even weekends have schedules: I'd have to fit some time for grading team homeworks and time for my own studies. Saturday afternoons are basketball and Sunday evenings are BSG nights at a friend's house, and this Saturday morning is a meeting I should attend and I probably will.
   
   Just to make sure there is time for fun, sometimes it's best to do a little work early at unorthodox times.
   
   
2. I've been social enough. I mean it for this particular week. A few of us UM grad students met with the GFT speaker, a grad student from Finland (visiting by way of Cinti), and thought to make her welcome and do a few fun things.
   
   It was nice: I can't remember the last time I went out for coffee with friends .. at least when it wasn't at a conference .. and had long conversations not about mathematics and not about complaining about math grad life. I've forgotten how pleasant such occasions can be.
   
   But that's what they are: occasions. Work beckons loudly and often to me, so it comes first before other matters; whether or not that is wise and correct, I'll debate it when I have less on my plate.
   
   
3. It's quiet, for once. As much as I like my office-mates and my friends and colleagues in the department, their number is great and during the workday it can be a cacophony of movement, conversation, and katzenjammer.
   
   In the late evening it's nice to have the solitary peace and quiet to think.
   
   
4. I'm relaxed. I may be working, but I'm doing so in a leisurely way. I can read without rushing and enjoy those pauses to reflect over a few paragraphs and what this or that really means. It almost feels like leisure reading, but instead of reading Haruki Murakami novels, I'm reading about geometric measure theory.
   
   The best circumstance might be the freedom of not working, but the second-best circumstance is the freedom of working without a deadline and at your own pace.

So that's why I'm working, and it seems perfectly reasonable to me. I just wish people would stop bothering me about it.

Telepathy! (that is, they read my mind)

I received this forward from my department. Talking with others, one suspects that this is a common feeling .. but this common? I didn't realize it had a proper terminology! This better not be in D$M-IV .. \:

Either the world is crazy, or that a sub-world is crazy and we are slowly accepting it .. q:

---

From: [NAME/EMAIL ADDRESS OMITTED]
Sent: Tuesday, November 01, 2005 9:53 AM
To: [NAME/EMAIL ADDRESS OMITTED]
Subject: First Year Grad Students: THE IMP0STER $YNDR0ME

First Year Graduate Students, have you ever had these thoughts?

What am I doing here? I am not smart enough to be in graduate school. Admissions must have made a mistake!

If so, you are definitely not alone. Your feelings are part of a common phenomenon that many graduate students experience over the course of their careers.....

THE IMP0STER $YNDR0ME

The Imp0ster $yndr0me may be undermining your success by making you feel as if you don't belong in graduate school or don't have the capability to do well. DON'T LET THIS BE THE CASE! Develop strategies to deal with your feelings of insecurity, fear and uncertainty.

Join us for ... CONFRONTING THE IMP0STER $YNDR0ME

with [NAME OMITTED], a C09nitive Behavi0ral Therapist from the U of M Psych01ogical Clinic on Thursday, November 10th from 12-2pm in Rackham's Assembly Hall ..

teaching today, and on the nature of "mathgradness"

So I finally got around to it, today: I decided to shut up.

Instead of a tiresome lecture about the finer points of sequences and series, I made a worksheet of problems, some from the book and some I wrote myself, and suggested that the students work together and discuss them at their tables.

If I hadn't mentioned it before, the classroom is arranged so that there are square tables. Four students sit at one table, and each takes one side. As a result, two students per table almost always have their backs to me, during class.

You have no idea how much that bothers me .. \:

But it worked out: the students were still rather quiet, but they discussed the problems, and I wandered around and lent a hand when a team had trouble with a particular concept or computation.

Working this way almost makes me a disbeliever of large-scale teaching. I know it bothers me when I have to lecture in front of 30+ students. It's not because of nervousness, but because I have only a small sense of whether they actually understand what I say.

It's easy for me to feel that the lecture is boring and pointless and nobody's getting anything out of my careful method of explaining this or that. If that happens then it's a waste of everyone's time.

What's the point of that?

This also raises another question: how do diligent, thoughtful students become that way? I like to think that I'm not that lazy and confused student in the world ..

(and at the least, people do say that I take good notes)

.. and how do students become maths students? I've seen enough of my friends and familiars to realize there is a rich variety and diversity in personality and manner when it comes to one's approach to academics ..

.. yet there seems an intangible invariant: "mathgradness," as Jo so clearly put. What is this special quality, and what drives us to do maths and keep others from experiencing this coolness?

I'm not sure if that was a rhetorical question. It needn't be. Any takers?

Let me not go into nature and nurture arguments, and I'm happy to know why I like mathematics. Some days I can actually explain why, too. q:

Thoughts before the Brown Jug ..

(.. and after the weekly meeting with my advisor)

It feels good to be back "in the game." Granted, I'm still doing more reading than anything else. I was suggested a new paper to read, but on a wholly different topic this time: last week it was Hajlasz, Sobolev spaces, and approximations on manifolds and metric spaces, and this week it's Milnor, smooth functions, and manifold theory.

I've learned a few lessons already:

• Fibre bundles are pretty cool, once you hear about them from someone who understands them.
  
  
• I will never again underestimate how hard the smooth category is. Before I was mistaken: I thought once you had a smooth function, then life was relatively easy.
  
  Suffice to say, that is far from the truth!

It fascinates me, how long it takes to read mathematics and to understand our lessons well. On the other hand, it makes me wonder how long, on average, it takes for people to write a good article!

So long a time spent, writing something, and so equally long a time spent reading the same something! I could make an analogy to Achilles and the tortoise, but just mentioning that probably gives my meaning already.

One last thought about reading and work: the list piles up and up. Besides Milnor, there is material to read about:

• the Ahlfors Measure Conjecture, for a student talk for my Hyperbolic manifolds class;
  
  
• Gromov-Hausdorff convergence of metric spaces, for a talk I promised the folks at Student Geometry-Topology Seminar;
  
  
• and certainly not least, my prelim reading!
  
  The books by Evans-Gariepy and by Mattila are sitting in my bookbag and if I don't start now, I won't know the results and details well enough to talk competently about them!

Never a dull moment. But it is Thursday, and close to Brown Jug time. The reading can wait an hour or two. q:

"wait .. where am i?"

I think I misrepresented myself, or gave an inaccurate depiction of what I was actually doing (see last post).

It is true that I am reading a paper of J. Milnor's from 1956 or so, but only part of it [1]; more accurately, I'm reading enough of it to determine whether a certain property of spheres [2] is false in the C¹-smooth setting; it is known to be false in the C^∞-smooth setting.

On that note, it's actually been a bit of fun. I'm reviewing some differential topology that I hadn't thought about in ages, and am finally applying some facts that I thought would be useful only for passing my Topology qual.

I've struck upon a regular bit of confusion, though: perhaps I'm bad with the notation, but it doesn't seem easy to tell where a particular function is 'living' (i.e. its domain, and whether I am looking in the chart or on the manifold ..). Hence the title.

Of course, the fun might end soon. There's a sizable section or two which requires some knowledge of homology and cohomology. As you can imagine, algebraic topology isn't my strong point .. \:

Oh well. Que sera, sera, and it's back to work for me!

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[1] In particular, it's "On Manifolds Homeomorphic to the 7-Sphere" from the Sept 1956 Annals of Mathematics.

[2] and yes, I am deliberately being vague. q:

Disparate bits ..

I finally finished a write-up of my work from the last month or so. I think it took a year or so off my life. Maybe from now on I should LaTeX the results as I prove them, and save myself those nights of brain-disintegrating toil.

I wonder if my advisor will read it, at some point. Maybe I should have done something else, these last few weekends.

---

Today was the first day in which nobody came to my Office Hours. It was quiet and pleasant and I got a bit of work done, but strangely enough, I actually missed the student questions a little.

I must be getting soft.

---

There are some results in mathematics that everyone's heard of and which you never expect to worry about. Take J. Milnor's result on differentiable structures on the sphere; it serves as a moral to those who study manifolds that smooth structures can be very, very strange and eerie algebraic topological invariants may arise out of the ether.

But does anyone ever look it up and read the paper? I never thought I would. But guess what: I'm reading it for Thursday, as a suggestion of my advisor.

I'm not sure how I feel about that. Milnor, eh?

---

Well, once more into the breach, my friends. Until later.