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Michi’s blog » archive for February, 2006

 Anti-spam measures

  • February 21st, 2006
  • 8:24 pm

After having a visit of some 5+ spams (I’m known by the spam bots! Is that good?) I just installed a captcha-like plugin. It requires some basic arithmetic skills from y’all; but on the other hand, I -am- a math blog after all.

 Monads, algebraic topology in computation, and John Baez

  • February 21st, 2006
  • 11:06 am

Todays webbrowsing led me to John Baez finds in mathematical physics for week 226, which led me to snoop around John Baez homepage, which in turn led me to stumble across the Geometry of Computation school and conference in Marseilles right now.

This, in turn, leads to several different themes for me to discuss.

Cryptographic hashes

In the weeks finds, John Baez comes up to speed with the cryptographic community on the broken state of SHA-1 and MD-5. Now, this is a drama that has been developing with quite some speed during the last 1-1½ years. It all began heating up seriously early 2005 when Wang, Yin and Yu presented a paper detailing a serious attack against SHA-1. Since then, more and more tangible evidence for the inadvisability of MD-5 and upcoming problems with SHA-1 have arrived – such as several example objects with different contents and identical MD-5 hashes: postscript documents (Letter of Recommendation and Access right granting), X.509 certificates et.c.

 Reading Merkulov: Differential geometry for an algebraist (3 in a series)

  • February 13th, 2006
  • 1:55 pm

Since I cannot concentrate anyway, here comes the third installment of my reading Merkulov. We now get to the funky stuff – introducing sheaves and getting to what should be at the very start of basically any modern geometry course. At least if I am to believe the geometers I know.

So, let’s launch straight to it. A presheaf \mathcal F on the topological space \mathcal M is just a contravariant functor from Top(\mathcal M) to Ab, where Top(\mathcal M) is the category of open subsets of \mathcal M with morphisms being inclusion maps.

So that’s the one-line definition. But what does it mean?
Well, a functor is a map between categories that takes objects to objects and morphisms to morphisms. So we have that \mathcal F(U) is an abelian group for any open set U\subset\mathcal M. For such a map to really be a functor, it has to be sane in a rather precisely defined sense: namely morphism composition should still be associative and the identity endomorphism on a group shouldn’t actually, ya’know, change the morphisms before or after it.
For the functor to be contravariant means precisely that for f:U\to V we get \mathcal F(f):\mathcal F(V)\to\mathcal F(U) – all arrows reverse by application of the functor.

 This came faster than expected…

  • February 13th, 2006
  • 10:09 am

Breaking news! Just in from /.

According to this article, there is a Cincinnati-based company that just had two of its employees implant glass-encapsulated RFIDtags in their biceps as a part of the access control system to their datacenter.

And we’re one step closer to the artificial linking of identity verification to body parts.

I see two aspects to discuss here. One is of the inherent security problems with the solution, and the other is about the sci-fi feel and possible problems and antagonists.

So let’s start with the second aspect. I can remember a lesson in eight grade, discussing in our social sciences class, where I suggested use of passive radio transmitters to implant small chips in people that would work as a central for identification and verification. The implanted chip would be used as ID card, as credit card et.c. et.c. and you wouldn’t have to juggle cards at all any longer. I was quite taken by the vision I had – until my baptist pastor of a teacher started quoting relevations on me, claiming that such an implant would be a perfect example of how the Mark of the Beast would manifest.

 Reading John M. Lee – Introduction to Smooth Manifolds (1 of 1)

  • February 12th, 2006
  • 11:14 pm

If I’m going to take this renewed interest in trying to understand differential geometry more seriously, I might as well read more than one source on it. So, I’ll start a sequence of posts on this book as well.

Just as Merkulov, Lee starts with a short definition of what a topological manifold is, including definitions for the terms needed for the treatment.

Definition

An n-dimensional topological manifold is a second countable Haussdorff space of local Euclidean dimension n.

Next, Lee goes on to define coordinate charts. I won’t repeat the treatment, since he doesn’t really bring anything Merkulov hasn’t talked about in some manner or other. Atlases, smooth structures and equivalence of atlases also merits some treatment.

 Reading Merkulov: Differential geometry for an algebraist (2 in a series)

  • February 11th, 2006
  • 11:03 pm

So, in the last installment, we got to know smooth manifolds and charts, atlases and some nice topological tricks and tweaks. For this round, we follow Merkulov onward, and pretty soon stumble across category theory and sheaves. The notes I’m following here are from the link on Merkulov’s website. It starts, however, with a nice discussion of temperatures in archipelagos. Go read it – I imagine I’m almost comprehensible at that part of the text.

A map from a subset of a smooth manifold to \mathbb R is called a smooth function on the subset if for every x in the subset and a coordinate chart at x, the n-to-1 variable function f\circ\phi^{-1} is smooth at the point \phi(x).

 Reading Merkulov: Differential geometry for an algebraist (1 in a series)

  • February 11th, 2006
  • 2:23 pm

I’ll do this in posts and not pages on further thought…

Sergei Merkulov at Stockholm University gives during the spring 2006 a course in differential geometry, geared towards the algebra graduate students at the department. The course was planned while I was still there, and so I follow it from afar, reading the lecture notes he produces.

At this page, which will be updated as I progress, I will establish my own set of notes, sketching at the definitions and examples Merkulov brings, and working out the steps he omits.

Familiar parts in unfamiliar language

Merkulov begins the paper by introducing in swift terms the familiar definitions from topology of topology, continuity, homeomorphisms, homotopy, and then goes on to discuss homotopy groups, and thereby introducing new names for things I already knew. Thus, I give you, for a pointed topological space M

 New blog found!

  • February 10th, 2006
  • 3:08 pm

Stumbling across the blog Epsilon-Delta with a brilliant article series on mathematics as key to effective programming. This in itself merits a closer look at the blog; which in turn merits it a place in my blogroll. Go and read it you too!

 Borsuk-Ulam and West Wing

  • February 7th, 2006
  • 11:56 pm

In West Wing 4×20, CJ states that there are two antipodal points with identical temperature on the earth, as an argument why it should be possible to imagine that an egg could stand on its end at the spring equinox. This particular plotline also has her most emphatically claiming that this should not be possible at the autumn equinox. Why this particular physics is complete idiocy will be left as an exercise to the interested reader, and instead I will focus on the other claim.

This is, in fact, true. It’s a corollary to one of the prettiest theorem conglomerates I have ever seen: the Borsuk-Ulam theorem(s). Alas, I haven’t got my sources on it here at the moment, so I won’t give you the deep indepth survey I want to give; but I do want to give a bit of overview as to why the claim CJ supports her insane theory with is actually true.