Home again

After a quick week back in Ithaca for the start of the semester, I am back in Saugerties, in the house where I grew up, to welcome my parents back from their mid-Winter vacation. I still call this house, and this town, home. I am not quite so far removed that I call my apartment in Ithaca home instead, but I have progressed to the point where I refer to both Saugerties and Ithaca that way. In my head, the Saugerties home is a little heavier, but I don't think this is noticeable when I speak the words.

The drive partway across New York State last night was kind of fun. I was tired, but not tired in a prohibitive way. I was tired in the way that things start to seem a little more surreal than usual. I like that feeling, and last night I was only on the very cusp of it, so the crash afterwards wasn't too bad. I've had some extremely surreal mornings while in grad school, after having been up all night or most of the night, and when I run out of steam around 11 AM, the crash has been pretty awful. But last night was ok.

For most of the trip I was listening to the audio version of my favorite book, American Gods, by Neil Gaiman. It's read by George Guidall. He's narrated a couple of other audiobooks, and I'm pretty sure he's my favorite narrator. I've read this book once or twice, and listened to pieces or the full thing on audio countless times. It's wonderful, and I highly recommend it.

For the rest of the trip I listened to music. I put on a playlist on my iPod that I call Singalong, because it consists entirely of songs whose lyrics I've memorized and can hence sing along with (in theory). Even though I know most of these songs backwards and forwards, two of them struck me anew as I listened and sang. Here are their lyrics.

The Kid
(originally written and performed) by Buddy Mondlock

I'm the kid who ran away with the circus.
Now I'm watering elephants.
But I sometimes lie awake in the sawdust
Dreaming I'm in a suit of light.

Late at night in the empty big top,
I'm all alone on the high-wire.
Look he's working without a net this time.
He's a real death-defier.

I'm the kid who always looked out the window,
failing tests in geography.
But I've seen things far beyond just the school yard,
Distant shores of exotic lands.

There're the spires of the Turkish empire.
It's six months since we've made landfall.
Riding low with the spice of India,
Past Gibraltar, we're rich men all.

I'm the kid who thought we'd someday be lovers,
Always held out that time would tell.
Time was talking, guess I just wasn't listening.
No surprise if you no me well.

And as we're walking toward the train station,
there's a whispering rainfall.
Across the boulevard, you slip your hand in mine.
In the distance the train calls.

I'm the kid who has this habit of dreaming.
Sometimes gets me in trouble too.
But the truth is, I can no more stop dreaming
Than I can make them all come true.


I Wish I Could Go Back to College
from the musical Avenue Q

I wish I could go back to college.
Life was so simple back then.
What would I give to go back and live
In a dorm with a meal plan again?

I wish I could go back to college.
In college you know who you are.
You sit in the Quad and think,
"Oh my God, I am totally going to go far!"

How do I go back to college?
I don't know who I am anymore.
I want to go back to my room
And find a message in dry-erase pen on the door.

Oh, I wish I could just drop a class,
Or get into a play,
Or change my major,
Or %$&*$&*(& my TA.
I need an academic advisor to point the way!

We could be sitting in the computer lab,
Four AM before the final paper is due.
Cursing the world 'cause we didn't start sooner,
And seeing the rest of the class there too.

I wish I could go back to college.
How do I go back to college?
Oh-oh. I wish I had taken more pictures.

But, if I were to go back to college,
Think what a loser I'd be.
I'd walk through the Quad,
and think, "Oh my God.
These kids are so much younger than me."


These are both very nice songs. I've heard several different versions of the first one, including the original, but I think my favorite is by Peter, Paul & Mary.

I'm very happy for the day I first realized that a trip of several hours is the perfect time to play a song over and over and over to memorize the lyrics. My poor cat, who often goes on these trips with me, is, alas, not so happy. I think she finds it pretty upsetting, actually.

---

I wanted to add to my list of Fantastic Moments in Fast Food Sandwiches. This one belongs to Burger King. It is their Rodeo Cheeseburger. (Although I only discovered it a couple of months ago, I just learned from Wikipedia that this burger was released nationally in 1998, and can now only be found in certain regional locations, including, apparently, several in Central New York.) It is a single burger with American cheese, several onion rings, and barbecue sauce, served on a sesame seed bun. It's not as large Wendy's' Classic Single with Cheese, but it has the advantage of currently selling for about $1.29, which is a pretty nice price. It's a startlingly good combination of flavors, and of course catsup can be added if desired. The barbecue sauce is a little strong, but not really spicy at all. Five stars.

Title

Proposed titles for future blog entries:
(1) Stark Raving Calm
(2) I know, put my earmuffs on the cookie.
(3) My dog is smarter than me.
(4) He's my brother, but he's still heavy.

---

My first week of classes is over. (I only have a seminar on Fridays, but it's just an organization meeting tomorrow, and I don't think I have anything to contribute. I may go anyway, as a prelude to T.G.I.F. at the B.R.B.. It was really good. I'm sitting in on a lot of classes, and so far all of them are very interesting. One of them is something is Teaching Secondary Mathematics: Theories and Practices, and I've been very pleasantly surprised. I figured, quite correctly, that a lot of it would be discussion, but I'm happy to report that we've been discussing some issues that I haven't really thought about before, and that definitely deserve some thought.

Here are some that come to mind. Consider the "fact" that:

6 = Sqrt[36] = Sqrt[(4)(9)] = Sqrt[(-4)(-9)] = Sqrt[-4] Sqrt[-9] = (2i)(3i) = -6.

Obviously something is wrong here, because otherwise math is broken. But what exactly is wrong, and how on earth would you explain it to a high school student?

Something we talked about today is the difference between 2/0 and 0/0. They're both mathematical bad eggs, but one of the instructors of the class, Dave Bock, suggested that the two are, in fact, rather different. He called the first undefined, and the second indeterminate. He cited some pretty decent reasons for this distinction, and he did so without resorting to limits or advanced algebra. He did it based on the idea that a fraction a/b is the number which solves the equation b x = a. Using this logic, 2/0 can't be a number (because 0 times anything is 0), but 0/0 could be any number (again because 0 times anything is 0). We also talked about 0 raised to the 0 power, which is sometimes considered to be 1, and is sometimes considered undefined.

What I'm realizing now is that although we talked a little about the pure mathematical, foundational type reasons for defining things one way or another, the discussion leaders always steered things back to questions about how teachers can and do present these things to students. How can teachers help students make sense of these seemingly (and sometimes actually) arbitrary rules? How can we help them learn this stuff in a way that they'll remember it because they absorbed it into their structure of understanding, and not just their structure of random memory?

---

Here's one more issue that I found extremely interesting. It started because we were talking about the FOIL method of multiplying two binomials, and how much it sucks in regards to student understanding and extension to more difficult situations. We talked about learning to use the distributive property, and some nice graphical representations of that.

This led to Dave Bock talking about how he's always found it absolutely amazing how algebra, specifically the distributive property, inexplicably captures the very geometric idea of multiplying complex numbers. I was very curious. I said something like, "What is this geometric idea of multiplication? I mean, I know the geometric interpretation of multiplying complex numbers, but to me, this was always something that comes after the algebraic definition."

Dave Bock replied that he thought this was very sad. I decided not to take offense right away, and hear him out. (Yes, I am actually getting somewhat smarter as I get older.)

He explained that when he looks at a complex number a+bi, he sees a single number. The plus sign is kind of artificial, or just algebraically convenient. It is the same sort of thing as the hidden plus sign in the single number "1 and 2/3" = 1 2/3 = 1 + 2/3. When we think of complex numbers sitting in the complex plane, there's no reason for the plus sign to be there. In a way, the plus sign is there exactly because the multiplication of complex numbers, defined in the standard geometric way, obeys these nice algebraic rules when we put the plus sign in there.

(Thinking about it now, I realize that a standard way of defining the complex numbers in certain circles is as the algebraic closure of the real numbers, obtained by adjoining the number i, and that this field extension is isomorphic to the set of all polynomials in i with real coefficients modulo the relation i^2 = -1. This is very much not the way I think of the complex numbers, of course. I think of them of sneaky little bastards that are often uncooperative, and sometimes far too rigid. I love them anyway.)

At the time, I was only preliminarily convinced, but Dave Bock wasn't done messing with my mind. He pointed out the globally recognized difficulty of explaining to a high school student, or elementary student, or college student, or your average adult, why the product of two negative numbers is positive. He posited that this explanation is much like the geometric explanation of multiplication of complex numbers.

I immediately agreed, and was completely convinced. I also wasn't exactly sure why I was so convinced.

I think that the explanation of why (-2)(-2)=4 is that the best way to think of negation is as a reflection. In terms of the real number line, the number -x is the reflection of the number x about 0. When we do two such reflections, we don't change anything. Since negation is multiplication by -1, and not doing anything is multiplication by 1, this means (-1)(-1)=1. Since multiplication ought to be commutative, this means (-2)(-2)=4.

I believe this kind of thinking is exactly how one makes sense, geometrically, of multiplying complex numbers.

I smell like a monkey ...

... and I look like one too.

Yes, today is my birthday.



If you would like to know how old I am today, here are two clues. (The second clue is a little more fun, I think.)

(1) I was born one year to the day after the author of the Incompleteness Theorem passed away.

(2) My age is perfect, and I am old enough to vote, and I am not dead yet.

Because the weather forecast for today is bleak, (so far we only have nasty freezing rain), last night my brother and I took a drive and did a little shopping. I bought a couple of very nice toys, such as the one whose picture adorns the beginning of this entry. I also bought the DVD copy of the complete series Defenders of the Earth, which is, happily, just as awesome as I remember it. It has a fantastic theme song too.

We also looked at a lot of anime DVDs. I'm a little into them, and my brother is a lotta into them. I was trying to remember the name of one I remembered watching when I was very, very little. There was a train in it that traveled through space, and the title had the number 9 in it someplace. My brother fancies himself an expert on anime, but even he eventually gave up. He claimed I was making it up, since I lie to him all the time. But to be fair to me, I've only been lying to him for a little while, like since he was born.

He bet me a donut that I couldn't find out what it was. I took the bet. Unfortunately for him, he doesn't have the knowledge of computation theory that I do, because otherwise he would have known that he can't possibly win until I die, at which point I suppose he would have to get my estate to buy him a donut. Hehehe.

As it turns out, a quick use of the Google yielded the title. It is Galaxy Express 999. The sucker owes me a donut. Hehe.

On our drive back I drove around Wendy's to pick up my fast food favorite, the Classic Double with Cheese, as a pre-birthday treat. (I'm trying to cut down.) I doubt you are as carefully familiar with this pinnacle of sandwich-ness as I am. It is a double hamburger (square, of course) on a plain white bun with american cheese, lettuce, raw onions, mustard, mayonnaise, and catsup. (I always say "ketchup", but I like the spelling "catsup". At least, I do right this instant.) There are also tomatoes and pickles available, but I don't generally care for them.

Amounts of the various condiments may vary with franchise and with burger cook, but the results are usually nothing short of spectacular. In most cases, the balance of flavor is blissful.

Unfortunately, upon returning home I discovered that my burger bliss had been replaced with a chicken sandwich. Gasp! This was still very good, but not what I was hoping for. Sigh...

Well, today I will make a fantastic chicken salad, and all culinary matters will be well once again.

---

A couple of things to add to previous entries. I mentioned that for any of the super-heroes I mentioned, there is a ridiculously complete profile available on Wikipedia. I meant to add that, as an alternative, you can always ask me. In most cases I am very nearly as good as Wikipedia, if not better in some aspects. Knowing this may also help explain some of my behavior to you in the future. If I ever forget your name, or the date, or that we had plans, etc., please remember that the space where I ought to be storing that information may have inadvertently been used to store the fact that the original X-Men consisted of Professor X, Cyclops, Marvel Girl, Beast, Angel, and Iceman, and that their real names are Charles Francis Xavier, Scott Summers, Jean Grey, Hank McCoy, Warren Worthington III, and Bobby Drake, and that the occupations of their fathers, in order, were [I don't know], Air Force pilot, professor at Bard College, nuclear plant worker, CEO of a large corporation, and accountant.

Also, I was talking about how so many of my current behaviors seem to have been predicted by the events surrounding my birth. I wanted to add that I was born at approximately 8 AM, and I am still something of a morning person. (Truthfully, I am also something of a nighttime person. I am mostly just not a middle-of-the-day person.)

---

I think that's it. Eat something tasty today, in my honor. Preferably something you normally wouldn't let yourself.

Faster than a speeding bullet

Most people who know me are very aware that I am a fanatic about super-heroes. I have many thoughts on the subject.

---

Some of my rabid fandom can be traced back to Saturday morning, and eventually weekday afternoon, and later still Sunday morning, cartoons. There were individual cartoons for Superman and Batman and Spider-Man. One of my favorite ones was Super Friends, which had many different incarnations. It featured Superman, Batman, Robin, Wonder Woman, and Aquaman, and various teen sidekicks. Later the show expanded to include many other characters, including the Flash, Green Lantern, and near the very end of the series, Firestorm, the Nuclear Man. (Firestorm become one of my favorite characters a little bit later, when I found a large collection of old comic books featuring him in a used book store. (To clarify, I found the comics in a used book store, and the comics featured Firestorm. The other interpretation of my sentence is much funnier, but not true.))


I should mention that you can read all about these characters on Wikipedia. There are entries for every single one of them, although I only provided links in this blog entry for some of them.


Another show I absolutely loved was Spider-Man and His Amazing Friends. This show featured Spider-Man, Iceman, and Firestar. Iceman is one of the founding members of the X-Men, and Firestar was at the time a very new and rather minor character from the pages of New Mutants, which was a close relation of the X-Men. It's not really clear why these three were put together. Firestar was apparently used because the more popular character of the Human Torch from the Fantastic Four wasn't available for some reason. But still, there wasn't really a precedent in the comics for these characters to work and live together.


I didn't know any of this at the time, though. It was a really fun series, and it had all sorts of other characters thrown in on an episode-by-episode basis.


One more show from that time period to mention is The Incredible Hulk. The big green fellow has had several cartoon series all to himself, but I remember this one especially well. It scared the stuffing out of me. I think there's a good reason for this. Angry adults are scary to children anyway, and if you throw in the fact that the angry adult will transform into a large green monster who likes to smash things, it is absolutely terrifying. In most stories, and certainly every cartoon story, the Hulk somehow never manages to hurt anybody, but I've had many dreams where this guideline was violently violated.


I remember a "friend" from nursery school who would occasionally double over, grab his head, and say, "Oh no, I'm ch-changing...!" I would run very very very fast to the nearest adult, every time. What a jerk!


In the time that I've grown up, there have been many more cartoons involving super-heroes, and most of them have been extremely cool. In fact, right now I am watching an episode on DVD of Justice League Unlimited, which is a SPECTACULAR show. This episode is especially great. The witch Circe transforms Wonder Woman into a pig, and at the end of the episode Batman is forced to sing a lounge-singer-type love song in front of a crowd in order to change her back. Somehow Wonder Woman finds out that he did this, and taunts him with the knowledge by humming the song as she walks away from him.

---

The other major influence that informed my love of all things super-heroic was my cousin Sam. He is two years older than me, and when I was little he was the epitome of all things cool to me. He collected comics, and drew super-heroes and other stuff all the time. So, I started collecting comics and drawing super-heroes. Today he is a professional graphic artist, and a very good one, but I don't think he reads comic books anymore. I still sketch a little bit, but by trade am a mathematician and teacher, and I still read comic books all the time. I'm trying to cut back on my collecting, because the darn things are increasingly expensive and I am increasingly mature, but I still read my old ones all the time.

---

Neither of these things were what I wanted to write about today, but that's ok. I'll write more next time.

Ithaca weekend

My dad and I went to Ithaca for a couple of days. I had a few chores to do, and he wanted to come along for the trip. We didn't actually go over a weekend, but we stayed for the length of a weekend, and since both of us are still on vacation there's currently not much difference between a weekday and a weekend day.


One of the things I did was have some friends over for homemade-pizza-from-scratch. I'd never made it before, but it seemed to turn out well. The biggest problem was that the pizza became atomically bonded to the tin foil it was resting on in the oven.


The recipe was for kids, and so included lots of fun facts and extra activities. It talked about yeast and how it's alive, and how when mixed with water and sugar, the yeast releases gases from the mixture. It mentioned an activity where you mix these things together in a bottle and put a balloon over the top of the bottle and watch it slowly fill up with gas. It actually fills up very very very slowly. The photo at the beginning of this entry is the result of this experiment after the yeasty beasties have had a night to work.

---

On this trip, we were supposed to be looking around to see if we could find some string bags. These are bags that are very handy for groceries, because they are extremely lightweight, and if you leave them in the car then everyone can see that they are empty, so nobody is tempted to break in to the car to steal whatever is in them, (unless the thief is someone like us, who is looking for string bags). These bags are also extremely nice for the beach, because they won't retain sand or rocks. The mesh size is too large for that.


We didn't have much luck. We found some very expensive which were far too small to be useful. The closest we found were the sort of all purpose mesh bags that people often use for laundry or to store soccer balls or something. These were too large for our purposes, though. My dad made the brilliant observation that what we needed were mesh laundry bags for midgets.

---

Two of my friends who were there for pizza are going to have a baby in a couple of months, so several times the conversation turned to baby stories. I'd heard before that I was born exactly on my due date, but I hadn't heard (or maybe I'd forgotten) that my mom had to have a c-section because I was turned around the wrong way in the womb. When I heard this, it made perfect sense to me. I've always been a little confused, and been susceptible to being turned around easily, (physically and mentally).


Do you see? As a baby I was unbelievably punctual and easily confused. As an adult, I am still unbelievably punctual and easily confused. If only I had been studied more as a baby. Maybe we could have figured out what my dissertation topic is going to be, and where I should apply for a postdoc! Just imagine! The possibilities are limitless!! Exclamation points!!!!

It's been a while.

Wow! I haven't written in quite a while, have I?


The picture above is a sign on NY Route 206 that I pass almost every time I travel between Saugerties and Ithaca. Triangle is a very small village. If you drive a little fast, you can hold your breath all the way through it. I took a picture of it for my friend James, who has passed it himself many times, and has remarked what a funny picture it would make.


I've been spending a very relaxing winter break in Saugerties. It's good to see my family, and to spend enough time at home so that it feels like home again. It's also been great to see all of the animals on the great Goldberg Farm. I've posted some wonderful pictures here. My little Sofi-cat has really enjoyed being in Saugerties too. She loves visiting her cousins, and she gets to sit on a screen porch and watch birds and squirrels. It's like camp. I always feel a little bad about bringing her back to my stuffy little apartment in Ithaca.


I've even been doing some math, too! Some of it has been my own research, but a lot of it has been some more general textbook reading. Early on, I read through most of Humphreys text on Lie algbras. I realized that part of each of my meetings with my advisor often ends up being him reminding me about some Lie algebra theory that I should remember. (Even when I do remember it, it's still worth having him explain it, because he is a very, very good lecturer.) I still won't remember all of it, but each time I relearn the stuff a little more of it sticks.


The book I've been spending most of my time reading is Sharpe's text on differential geometry. I've started reading some of it many times before, but tended to get bogged down in all the structure and terminology. This time, I started from the very beginning and read through every chapter, which has been A-MAZ-ING. I highly recommend it, and I highly recommend going through it start to finish, (or at least start to wherever-you-stop-reading). The development of the Lie group-Lie algebra theory by focusing on the Maurer-Cartan form is especially interesting. And I am now ALL ABOUT principal bundles. They are AWESOME. Bundles ROCK.


I've been doing other reading also. One of the things I've been reading is Neil Gaiman's blog. He's my favorite human being that I don't actually know. His work is awe-inspiring. He is an amazing storyteller. I first read his work in the Sandman comic book, and since then have read all of his novels and many of his short stories. Read his stuff.


I'm going to have to stop for now, because one of the things I've done during this break is experiment, not always on purpose, with my sleep schedule. For instance, just last night I went to sleep at 3:30 AM and woke up at 5:30 AM. I took a nap earlier this evening, but I don't think I'm going to be able to make it much longer. I'll write more later.