ChapterZero

It’s unbelievable how hard it is to find a good working definition of a branch point– I have looked through more than 15 books of varying sophistication, and am still at a loss.

The simpler complex analysis books substitute handwaving for a rigorous definition, and implicitly give the message that you should be able to look at an expression for a function and quickly find the branch points. But notice how all the examples are contrived: or . And notice also the way they tip-toe around the issue of branch points at infinity: the standard approach here seems to be to define infinity to be a branch point of a function if is a branch point of . Besides begging the question of what makes 0 a branch point, why is this a reasonable/useful definition? Why do I care about branch points at infinity? I read in one book that the identifying character of a branch point at infinity is, if you travel from a point back to itself in a really big circle around the origin, you get a different value starting than the one ending. What the hell? Isn’t that the ‘definition’ given for a branch point at 0?

The more advanced complex analysis books invariably give a reasonable sounding definition, closer to the end of the book than the beginning or even the middle. Said definition relies on some heavy concepts like local biholomorphicity or holomorphic mappings between Riemann surfaces, or analytic continuation. Hence they are practically useless to me.

What’s the problem here? Is there no simple exposition of the basics of branch points: what they are, how to spot them, how to make cuts, etc.?

The closest I’ve found to a reasonable explanation of branch points was given in Visual Complex Analysis, where the author gives a neat diagram illustrating why (sort of) 0 is a branch point of :

In this image, is the starting point, are the different values of , and are closed paths that are traveled from back to . His point here is that traveling these paths in the domain means the image in the domain travels a corresponding path at a radius as large, and with the angular speed. Notice how paths start off at in the image domain, but after transversal wind up at different values of , corresponding to the number of circuits they take around 0.

I don’t see why 0 has this special property, or how to identify it as a potential branch point from just looking at the formula , but at least this author isn’t expecting me to mine through crap looking for gold.

Not that the two are related…

I rediscovered an excellent book yesterday, Janich’s Vector Analysis, which despite the title, is not about curl, div, grad, and all that Instead, it covers manifolds and differential forms in what seems to be a straightforward and readable manner. I had picked it up a couple of years earlier, but found the first chapter was above my head. On this second reading, it seems just at my level. Definitely a book I’d recommend to anyone trying to learn manifolds and differential forms. Now I need to stop collecting books.

On to the fast food. I just finished reading Chew On This today; the thought occurred to me as I was closing the book, that fast food eating habits are prime candidates for inclusion in the high school health curriculum. When I was in high school, the incursion of fast food companies into our high school cafeteria was just beginning, and the options– e.g. chicken baskets, which I ate just about every day– offered by the school itself weren’t very healthy. It’s disturbing to compare this to some of the assertions the book made: that becoming obese at a young age trains your body to remain obese (as a specific statistic, 90% of teenagers obese at the age of 13 while be obese in their midthirties), that type II diabetes, which used to be called adult onset diabetes because it was rarely seen in children, is now common place among teenagers, etc. Definitely, facts like these should have been made available to my friends and me in high school, when we were in the process of taking more control over the foods we eat.

Just wanted to share the good news: manifolds and the associated concepts– tangent spaces, local coordinates, diffeomorphisms between manifolds, etc.– are starting to make a lot of sense to me. Ironically, most of this is due to the fact that I’ve been reading several books concurrently, all of which define (differentiable) manifolds differently: this makes me have to identify the quiddities of the involved objects in order to correlate the information meaningfully.

As a prime example of the benefits of this strategy, just yesterday one of my persistent questions was answered: why is it stipulated that if and are two overlapping charts on a manifold , they must satisfy is a mapping? What I had expected as a natural condition was that and would be identical on their intersection, but that is too restrictive a condition because then we could just define a new chart as the ‘join’ of and … following this kind of definition, you would end up saying that each (connected) manifold has a global chart. Instead, we acknowledge that in general, some points will have different coordinates under different charts, but stipulate that the corresponding change of coordinates must be nice in some sense. Exactly how nice a manifold is depends on the smoothness of the change of coordinates between its charts. This justification for the standard definition is simple and satisfying, but is not explicitly stated in any of the pure math books I’m reading, only in this analytical mechanics book.

Primarily I’m reading Tensor Analysis on Manifolds, which is good enough to stand alone, and the reading would go much faster, but I’m going to stick with the skipping around. If all goes as planned I’ll be done with the first chapter of that book tonight, review it tomorrow, and then proceed to the next two chapters on tensors.

Today was the first day of the graduate orientation, which will last all week. In the morning the Dean of Graduate Studies gave us a welcome speech, with pictures showing us where Caltech is in relation to: 1) the universe, 2) the world, 3) the state, 4) LA, using satellite photographs, radar pics, and Google Earth– in fact, that part of the speech was pretty much the same he gave when I came down here the first time for the Grad preview program. Then we had a free lunch, and in the afternoon I attended the convocation.

The convocation was… unique. The first speaker, in addition to being a Nobel laureate, has a ridiculous amount of honorary degrees (30, I believe), is on boards of several organizations and universities, directs an interdiscplinary research center at CalTech, is a political activist, and still does research. The second speaker has slightly less credentials, but still very impressive, and the final speaker was Michael Browne, the guy who discovered Xena/Eris, the object that led to the creation of the distinction between planets and ‘dwarf’ planets. It was obvious that the arrangers were trotting out the big names to impress us. And it worked

According to “Chew on this“, Pasadena is the home of everyone’s favorite fast food place:

Eager to cash in on the new craze, Richard and Mac McDonald opened their own drive-in restaurant in 1937. Located in Pasadena, California, it had three carhops and sold mostly hot dogs.

Why don’t Pasadeans proudly shout this from the mountaintops: is it because the nearest mountains aren’t in Pasadena, or is this a shameful secret?

While in Houston, I discovered that I like the taste of homecooked Indian food, so I said I would learn how to cook it in California. My mom donated one Indian cook book she got from a friend at work, I bought another one, and at the same time got a book on curries which was on sale (and yes, I do know that not all curries are Indian). Recently I also found another excellent resource, in the form of a blog on indian cooking. That should be enough to get me started.

Today, I made my first trip to the ‘local’ Indian grocery. I say ‘local’ for two reasons. First because it is only 3.46 miles away, but I had to walk… that was an informative three hour roundtrip: now I know I walk on average about 2 mph. And second, because it is actually in San Gabriel, not Pasadena. Since I couldn’t settle on a particular recipe to get ingredients for, and I realized how far I’d have to tow my purchases, I got what exotic ingredients I remembered are used in a range of recipes: ghee, asafatida, garam masala, paneer, citric acid, fresh mint, and ginger (ok, so the last three aren’t exotic). I was disappointed that there weren’t any fresh breads available– this wasn’t surprising, given that the entire grocery is (efficiently) packed into a single storefront in a strip mall– but I got frozen stuff naan and parathas.While there, I also drank a glass of lychee juice out of curiousity: a friend had once taken me to a Vietnamese restaurant in Houston, and I remembered declining a lychee flavored ice-cream– this seemed a less risky way to try the fruit out for the first time. The taste was somewhere between lemonade and soursop. Is lychee common in India?

Now I’m trying to decide what recipe(s) I should attempt first. The more modern book, although it claims to be practical, is annoying in that it assumes you have fresh peppercorns and fresh cardamon pods and a spice grinder, etc. in the interesting recipes. The directions in the older book are amusingly vague, e.g. ‘cook’ instead of ‘fry’ or ’sautee’ or even ‘bake’.

I’m tempted to look for one of those simple 5 ingredient recipes in my slow cooker book, add garam masala, asafetida, and paneer indiscriminately and call the result Indian food.

As an interesting aside, I discovered that Jains and Hindu Brahmans use asafetida in place of garlic and onion because those are forbidden to them. Why is that?

I feel like a loser. I rarely leave the house– but then, that’s what I did when I was in Houston; it’s just that there I actually had the roam of the house, so it didn’t feel so restrictive. Here, I feel most comfortable when I’m in my bedroom behind a closed door, mostly because the other guys are giving off a I-want-my-personal-space vibe.

What can I do/where can I go to kill time that doesn’t cost money, and is interesting?

The implicit and inverse function theorems are two of the most important theorems in analysis, and are foundational in differential geometry, yet I’ve found that the way they are presented is more often confusing than not. Here’s my attempt to provide a simple intuitive approach to looking at the two theorems:

The gist of the implicit function theorem is that a set which satisfies a continuously differentiable implicit relation also locally satisfies explicit continuous differentiable equations.

The gist of the inverse function theorem is that a continuously differentiable mapping is locally a diffeomorphism.

Here are formal statements of the theorems (taken from “Differential Geometry: Curves, Surfaces, Manifolds” by Kuhnel):

Implicit Function Theorem
Let and be open sets and let be a continuously differentiable mapping. Let be a point for which and for which the square matrix

is invertible. Then there are open neighborhoods of and of and a continuously differentiable mapping such that that for all , if and only if .

and

Inverse Function Theorem
Let be an open set in and let be a continuously differentiable mapping with the property that the Jacobian at a fixed point is invertible. Then there is a neighborhood with on which the mapping is also invertible, i.e., is a diffeomorphism.

The crucial hypothesis of both theorems is that the derivative of the map has to be full rank; this is often condensed to ‘the map is full ranked’. The term immersion, which refers to an everywhere (within its domain) differentiable and everywhere full ranked map, is suggestive of the conclusion of the implicit function theorem: an immersion locally immerses a lower dimensional hypersurface in a larger ambient space in a recoverable manner, due to the injectivity of (an appropriate ‘partition’ of) the derivative.

Hmm… what I just wrote makes sense to me, but I doubt it will be clear to anyone else.

I just saw TransAmerica, a film about a transsexual who discovers he has a 17-year old son, and the son of course. Generally speaking, I avoid films about transsexuals: they are either superficial or depressing (cf. Soldier’s Girl, the first one I ever saw), however, it is undeniable that there is something emphatically unsettling about gender dysphoria (previous post) which makes it a prime subject for films. In fact, this one wouldn’t have been high on my list if it weren’t for the irony of a woman (Felicity Huffman) playing a man in the process of becoming a woman. I wonder what made them cast her: even though she did a superb job, throughout the movie the fact that the actress was indeed a woman kept drawing my attention. The movie would have had a very different impact if Bree had been played by a man; I suspect it would have been more depressing (because it would have been closer to the reality of gender dysphoria).