Dr Horrible

This is one of the funniest things I have seen of the internet in a while. I await Act III.

http://www.drhorrible.com/

What would you save?

When I was an undergraduate my Dad drove me to university at the beginning of each year and so the amount of stuff I could take was not massively constricted. Today I got out my suitcase to evaluate how much space I would have to fill with things to take to Germany. So when I was mentally appraising everything I have I began to think of things that I couldn't live without (aside from the obvious food, liquid). If I could only save 5 physical items from my life, what would they be?

1. My bear (actually a lion).
2. My maths notebooks.
3. Some letters I have kept in a box.
4. The photos of my family from when I was young.
5. My USB stick.

For me, these are the irreplaceable. The lion, photos, letters: they are all things of a time that I can't get back. A perhaps silly toy that is older than me, yet I have barely spent a night without. I remember when he had to be washed in a pillowcase (so he wouldn't fall apart) and as a result there were the occasional nights when he was wet and I couldn't have him, so my Mum and Dad put him at the side of my bed to watch me. Physical photos from childhood; a time before digital, complacent, replaceable photos. Of long summers and happy christmases. Letters and cards documenting relationships that events have broken and time has eroded. The dissertation/maths notebooks and the USB stick. These contain my "original" work. Perhaps wrong and mainly a bunch of crap I still would want to save my own maths notes and doodles above anything I've ever made in lectures.

All these things are things I am fiercely proud of and hate the thought of losing. But some I wouldn't want to take away with me. I suppose it is the difference between remembering something, but not dwelling on it.

I asked my brother the same question. He replied with:

1. Phone
2. Laptop
3. Money
4. Trainers

and he didn't need a fifth.

It is interesting to me how we both interpreted the question. For me, I immediately thought of the irreplacable. For my brother, he seems to have picked more practical, less sentimental items. However this makes me a little sad actually.

So sad that I just broke off from writing this post and collected a pile of old photographs and put them into a big photograph frame for him. I've put it in his room. I hope that (even if he never admits it) this would now be his fifth.

The age old battle

Possibly the biggest battle since good vs evil.. perhaps this is even that same battle...

https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhrMaq62UMWbjHvveSd3Fp5qyqmMxKb1cg71uVGyhWgc7GSAzIszquWFK977CxVMaCrJIr0AI94jD56o3rDzWxc6ErWsskqf24gm88lZfYqt0minPYMVV3gCNzQ1pTv30od_XfDVClLs_IZ/s1600-h/sciencevsreligion.jpg

Go science!

Options, Options, Options.

So I have to find accomodation. I imagine. Up until now I have done nothing about this but an email to my inbox this morning has filled me with panic. Mainly because I don't understand it- given it is in German. I have translated one word "accomodation." The email is quite long and I feel it is probably quite important as at the bottom there is a space for me to sign and date it (I presumably then have to post it back or something). So I have several options:

1. Ignore it
2. Sign the form and send it off and hope I haven't sold a kidney
3. Get someone to translate it for me and find out what it actually says.

Obviously option 3 would be ideal, but that will take more effort from me. Plus I would probably have to ask supervisor #1, but last night supervisor #1 and supervisor #2 emailed me to ask me to do something and if I email back immediately, they may assume I have already done what they wanted. Option 2 could be a good thing to do as if I sign the form they might just sort everything out for me; but of course.. I could just be selling a kidney. Then there is good old option 1. I think I will linger on option 1 for a while. At least until I have to go there at the end of August.

Blowing things up isn't all bad!

In fact, in Algebraic Geometry it is rather useful. We always want to normalise algebraic curves, to get rid of those pesky singularities and letting C be an irreducible algebraic curve in P^2 and S denote the set of singular points of C we have the normalisation theorem to do this for us. Here we obtain a compact Riemann surface C' and a holomorphic map

f: C' ---> P^2

where f(C')=C, the number of points in the preimage of S under f (denote this S') is finite and

f: C' \ S' ---> C\S

is injective. But what are we doing to the singularities? We are blowing them up. This is, informally where each singularity is replaced by the space of tangent directions at the point.

We consider blowing up a points in complex space. Let z1,...,zn be the coordinates of n-dimensional complex space C^{n} and let w1,...,wn be the homogeneous coordinates of (n-1)-dimensional complex projective space P^{n-1} . We consider embedding

g: D={zi wj=wi zj} ---> C^{n} x P^{n-1}.

Composing with projection we obtain the holomorphic map

g: D ---> C^{n}.

Formally D is the blowup and g the blow up map. Defining E as the inverse image of the blow up locus Z under g we obtain an isomorphism,

g: D\ E ---> C^{n}\Z.

This is seen in the normalisation theorem. E is defined to be the exceptional divisor.

When have I used this? When calculating monodromy of certain multiple polylogarithms I considered the picture














and it was necessary to blow up to obtain














This told me around where monodromy has to be calculated. So blowing things up is useful!

The Decline of the Hat

The more I watch Poirot, the more convinced I become that the decay of society began when the English gentleman took off his suit with turned up collar and hat and donned a track suit instead. When "gay" and "ho(e)" and "queer" all meant very different things and murders were confined to manor houses motivated by that long forgotten "last will and testiment." Perhaps not... but I have limited resources to base my knowledge on.

The currency of society has always been suspicion. The "society pages" have always dominated the minds of the individual- speculating about the famous and infamous figures of the day. But today suspicion seems much more dark. I was shocked to read an article the other day speculating on some long dead actor being a paedophile. Stripping the article of all rhetoric and spite the only evidence to this was that he met his wife for the first time when she was 12. But let us look into popular culture. Monica (from long-running sitcom Friends) met Richard (boyfriend who was best friend of her fathers) when she was very young. Was he branded a pervert?

This actor may well have been a paedophile, but why are some articles so quick to be so spiteful in their suspicions? Well, I fear I know the answer. I read the Daily Mail.

Necessary Complex analysis...

Let U be an open set in C^{n} and f: U--> C be a C^{1}-map. Let u be in U and let T_{U,u} denote the tangent space to U at u. (Note: there is a canonical isomorphism between T_{U,u} and C^{n}.)

Def: A function f is holomorphic if for all u in U the differential df_{u} in Hom(T_{U,u},C)=Hom(C^{n},C) is C-linear. Equivalently f is killed by differentiation w.r.t the complex conjugate to each coordinate z_{i} of C^{n}.

We note that the set of all holomorphic functions forms a ring: if f is a holomorphic function that does not vanish on an open set U then 1/f is holomorphic and if f and g are both holomorphic functions then equally f+g and fg are.

It is also worth noting that composition of holomorphic functions also yields a holomorphic function.

More often than not the term "holomorphic" and "complex-analytic" are used interchangably and this is a result of the, not exactly trivial, theorem:

Theorem: Holomorphic functions in complex varaible z1, z2 ,.., zn admit expansions as power series in variables zi.

There are two particularly important complex analytic results which are studied very early on in an undergraduate course: Stoke's Theorem and Cauchy's theorem. Stoke's theorem (as I hope to get on to at some point) is important as it is used in Algebraic Geometry to pair de Rham cohomology and singular homology and Cauchy's theorem is important as it is used to obtain some very useful analytic continuation results.

Theorem: Let U be an open connected set of C^{n} and let f be a holomorphic function on U. If f vanishes on an open set of U then f is identically zero.

Riemann Extension Theorem: Let f be a bounded holomorphic function defined on the complement of a set {z : z1=0} in U, where U is an open in C^{n}. Then f extends to a holomorphic map defined on U.

Hartog's Extension Theorem: Let U be an open set of C^{n} and f a holomorphic function on the complement of a set S={z : z1=z2=0} in U. Then f extends to a holomorphic map defined on U.

We note that Hartog's Extension theorem also holds if S is of codimension 2 in U.

So what can we get from these extension theorems? They tell us that possible singularities of a holomorphic function can not exist unless the function is not bounded and are not defined on the complement of some analytic subset of codimension 2.