Mathematics (or something like that)

This blog on PF is a copy of the posts on my blogspot.com blog. There are pages and content not compatible with PF blogs, so please check out the original! The URL is http://letepsilonbegreaterthanzero.blogspot.com

I will very quickly summarize the main points of the blog here

For the summer of 2012 I am reading through two books, Fourier Analysis by Stein and Shakarchi, and The Road to Reality by Penrose. I am blogging about thoughts I have from these books. I think it would be pretty cool to have feedback here =)

In fall 2012, I will be starting a PhD program with plans of continuing the blog over my course work. If I pass quals in two years (fingers crossed) this blog may change focus.

Anyways, every time I have a new post I will post a link here. I won't have any of the pages or other info on this blog, but at least the bulk of the post will be the same.
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# road to reality: chapters 6-7

Posted Jun3-12 at 10:44 PM by theorem4.5.9

View full entry at my blogspot.com blog: road to reality: chapters 6-7

Levels of smoothness:

I really enjoyed how Penrose spent so much time talking about what a mathematician means by smooth. The hierarchy he has is

one continuous derivative $\rightarrow$ derivatives of all order $\rightarrow$ analytic (has a power series expansion)

I like mentioning this because before I had not considered analyticy as the next level of "smoothness", though I think it's a pretty simple and natural extension. These levels of smoothness are different enough in real variables.

Indeed, one of the most useful functions in analysis, the bump function, is not analytic. For those unfamiliar, a bump function is a smooth function (in the usual sense of having derivatives of all order) with compact support. Compact support means (the closure of) the set of points where the function is nonzero is bounded. A typical bump function is one whose graph is shown below.

This $C^\infty$ function is zero for $x\notin (-2,2)$ and one for $x\in [-1,1]$
A bump function on $\mathbb{R}$ cannot be analytic; if you expand outside of the support (where the function is zero) you end up with the zero function.

If a function is analytic, then once you know the function at a single point, its power series dictates what all the others values are within the radius of convergence. We can extend this further by analytic continuation (I'm really more concerned with functions that are analytic everywhere, i.e. entire). Hence you aren't able to define an analytic function by prescribing values in different regions and then filling in the rest analytically. This is what we did with the bump function, we chose where we wanted the function to be one, and where it should be zero, and smoothly continued the function (non-uniquely) for the rest of the domain. Side note: this is what Penrose was driving at for why Euler only thought of analytic function, though I thought his prose distracted from this point.

A marvelous result of complex analysis is that the three levels of smoothness are actually all the same. That is one derivative implies analyticy (take a minute to think what this means!).

You can try to define a complex bump function by the same ideas above, but the previous result tells us something remarkable; there are no differentiable complex bump functions.

Multi-valued complex functions and algebraic topology:

I was delighted to see the connection Penrose made of algebraic topology and contour integration. Let me remark that I have never studied algebraic topology and I know extremely little of the subject (but that should change next year!). The fact that the complex logarithm is multi-valued (see the previous post in this project) means we can "wind" around the origin and arrive at a different value. This is a beautiful example of homotopy. The value of a continuous multi-valued complex function after you have traveled in a "loop" is determined by the homotopy of the path itself. In the case of the logarithm, the homotopy class is determined by how many times you wind around the origin.

I hope I have this right, this is something that I will definitely look into again after I learn some algebraic topology.
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