well, at least it's not an $\mathbb{R}$-tree ..

the button has been clicked.

so i just sealed the fates of 140+ caΙculus iii students. it's done. course grades are now in "approved" status.

in previous semesters, students that had unexpectedly low final exam scores wrote me back right away, or showed up randomly in my office during the next semester.

i guess they just wanted to know why, which i can understand.

so this semester i pre-emptively wrote them about it. whether or not this is a good idea remains to be seen: the optimist would say that since the instructor explained the situation first, then the news is easier to take ..

.. but i myself tend to shoot messengers
(at least in my own mind)

sometimes, bad news is just bad news.

old friends keep you honest.

it's good to have people who know you when you were young. a friend from my undergraduate years, she's begun a postdoc in the same department as me. tonight she reminded me of two things:

1. at some point i once arrived early at a particular TA meeting. the main instructor was late and apparently in the meanwhile time i started ranting (happily) about gershgorιn's circle theοrem for .. 5 minutes?
   

   Let $A$ be a complex $n \times n$ matrix, with entries $a_{ij}$. For $i \in \{ 1, \ldots, n \}$ let $R_i = \sum_{j \neq i} \left|a_{ij}\right|$ be the sum of the absolute values of the non-diagonal entries in the $i$th row. Let $D(a_{ii}, R_i)$ be the closed disc centered at $a_{ii}$ with radius $R_i$. Such a disc is called a Gershgorin disc.
   
   Theorem: Every eigenvalue of $A$ lies within at least one of the Gershgorin discs $D(a_{ii}, R_i)$.

   
   it wasn't until she reminded me that i remembered my love for this theorem.
   
   isn't it cool, though?!? it states that in some cases, it suffices to draw a picture in order to determine the invertιbility of a matrix!


2. to my discredit, though, she told me that i was the first person to tell her about the joke regarding $e^x$ and the differential operator.
   
   that only adds to my infamy .. \-:

in which i am surfing the web again (also: me on video, mathematically!)

despite the end of classes, there are still many tasks to do this week. on the other hand, i'm no longer responsible for appearing twice on mondays, wednesdays, fridays at designated times.

so returning to my own inclinations, i've been surfing the web more .. even for mathematical reasons.

---

in particular, i discovered that the arkaηsas sprιng lecτures are now available in video format online, which is wonderful.

here's the link.

in particular, all five of bιll minιcozzi's lectures on mean curνature flοw are available, and these are incredible lectures. this speaker knows how to explain intuition, yet at the same time discuss why the work is nontrivial.

in particular, during these lectures -- i attended the conference -- he explained harηack's inequality in a way that i have never thought about.

in fact, this guy can weaponize the harηack inequality towards geometry: it will actually suggest monοtonicity of curνature flows and why one should expect that boundaries will shrink to points.

incredible, i tell you.

as it happens, if you've ever wondered what i sound like, then i have a 15-minute talk available on the same archive.

during that talk, i recall being quite nervous. i asked the organizers if it would be fine to change my talk to fit better the subjects of the conference, which were mean curνature flows and minιmal surfaces.

in the end, it was probably a good idea: venturing off into discussions of metrιc spaces on the last day of the conference would have exhausted everyone, i think.

so i talked about this schοenflies stuff .. again.

so for those of you who know me, this is the best talk you'll ever hear from me. most of my talks are like episodes of curb your enthusιasm: disasters waiting to happen.

idle bits (teaching).

odd. most of my students dislike spherical coordinates. it surprised me that everyone did one particular homework problem using them, though it was avoidable.

Let $\mathcal{S}$ be the part of the sphere $x^2+y^2+z^2=4$, $y \geq 0$, oriented in the direction of the positive $y$-axis. Compute $\iint_\mathcal{S} \vec{F} \cdot d\vec{S}$, where $\vec{F}$ is ..

[thinks]
[sighs]
[gives up]

.. some vector field that has nice derivatives. (i forget.)

when i wrote the solutions, i used a parametrization with coordinates $x$, $z$. out of 100+ homeworks, only 1-2 students did the problem the same way that i did it.

---

on a related note, i have a soft heart. since it's finals week, i'm letting my undergrad graders off and grading the last homework and quiz myself.

no good deed goes unpunished. i thought that this week would be mathematically luxurious: spend all day with research and other matters. instead, even though classes are over, i am still working on teaching things ..!

my weekend, at a glance.

friday: i've an idea for a lemma.

saturday: the idea fails.

sunday: i realise that i'm proving too much; the lemma is "if and only if" and i just need the "only if" part (which is separate from friday's idea). proof done, cut and dried!

well, it's over, at least.

my last lecture for 2010 is done.
if i were less tired, then i might be happy.

(more on this, later)
anyways, back to work ..

---

a friend asked me last night how this semester went, particularly in terms of teaching. i didn't blink at all, and said:

"it was like an open wound for 2 months."

this, of course, scandalised him, so i had to explain:

the best mathematicians i know can switch modes very quickly:

two hours ago, they could be talking with their doctoral students, working out details. one hour ago, they would be teaching and talking about standard things. upon the hour, they would start brainstorming with me about ideas for a paper.

i don't have that ability, not yet anyway. between job applications, an NSF grant application, two conferences, and finding time for another paper, teaching often felt like an afterthought.

so i was surprised when one of my students chose to tell me how much he enjoyed my class. my own opinion was that it went sub-optimally, desolately.

at any rate, i feel wounds of the mental sort. i'm glad that the job applications are done, up to january.

it's been a while since i've "felt like a mathematician," struggling with new problems and looking for appropriate perspectives in which to think about certain problems. this week will be a good research week. maybe i'll accomplish something after all.

review sessions: an ambivalence.

philosophically, i am opposed to review classes for caΙculus classes. it's similar in view to why some believe that bicycle helmets do not save many cyclists' lives, but not as extreme.

perhaps i take an overly personal, extreme view on the subject, but:

if you have been listening to me all semester, then you know what's relevant and what's not.

i understand if, at a random moment, that you mightn't have a good picture of the course as a whole. then again, that's the point of studying outside of class. there is the old standby rule that for every 1 hour of lecture, one should spend 1 hour studying the material outside of class.

so if you haven't been listening, then why should i bother repeating myself? what evidence do i have that you will listen to me, this second time around?!?

realistically, these days one covers a lot of material in a calcuΙus course. it's hard to keep all the information in one's head, all at once, so it is good to remind the student of what has been discussed.

then again, isn't studying the responsibility of the student? looking over notes, working out more practice problems?

maybe i'm just become old-fashioned. it's harder to be a teacher, these days: one has to motivate students, all the time, while still show them nontrivial things. myself, i am not a natural teacher: i'm not patient enough.

some days i wonder if i should quit the business of mathematics!

on maths .. and grammar.

is "aforementioned" an appropriate word for maths papers?

this is an artificial sort of work, like "actuality" that is best avoided, i think, but modifiers like "as indicated above" or "previously discussed" become repetitive after a while.

on exposition.

since october i've been repeating this mantra: in december, i'll have my life back. by then i'll be a mathematician again.

(this was in regards to job applications, research travels, and other promises i meant to keep.)

to be fair, i was half-right. there is still one survey article to finish ..

.. but being more to do with research than, say, a statement of teachιng philosophy, i don't mind it at all. in fact, i'm enjoying it.

for one thing, writing makes me feel like i know something well.

it's been more than a year and 1/2 since i first worked out these details regarding de giοrgi's approach to reguΙarity theory (for certain ellιptic PDE) and only now does it feel .. natural.

that's the feeling, anyway: let's see if i can convince others of the same, through this article ..

the more i think about it, the more i like this idea of writing notes and expositions. in this last conference at οberwolfach, i was struck by the clarity of the research notes that colleagues of mine had written.

this is in regards to a generalised radεmacher theorem, in the setting of certain metrιc measure spaces.

the first proof was quite hard. there's a history of several versions of notes by several authors, in efforts to understand this result.

i liken it, actually, to how anyone in the open-source software community can contribute code for a particular program task. on the other hand, the best code gets passed around and used, for the greater good.

so i'm tempted to write my own notes on other subjects. specifically, there are a lot of folklore theorems out there, in this intersection between analysιs on metric spaces and geοmetric measure theory.

i could be deluding myself into thinking that my perspective is overly valuable .. but maybe by expositing, i can help as others have helped me.