# Still alive

Just a brief note to my readers that I’m still alive. It turns out that summer is just as time consuming (if not more so…at the very least it is more stressful since I can’t fudge over details like I could during regular quarter since I’m presenting solutions in class and also there is always the burning in the back of my mind that it is my last chance before qualifying exams to actually make sure I can do it) as regular quarter. Hence, I’ve made essentially no progress on non-class work.

I did check out the Seattle gay pride parade yesterday. It was fun, but kind of disappointing. There were very few absurd floats. Quite tame compared to NYC or San Francisco.

# Consciousness Explained 1

Sorry this took so long, but I kept reading further in hopes of getting to something more meaty to talk about before my first post on this subject. I’m through 3 chapters and it has essentially only been basic definitions and a few thought experiments.

My first comment is that I always forget that I’m not actually interested in philosophy of consciousness. What I associate to phil. con. (my abbreviation from here on out) is not what the typical philosopher associates to it. I tend to view all of philosophy through the lens of philosophy of language. So I usually start form some sort of premise about how language is more primary and fundamental than consciousness. Now lets try to figure out how it is that language brings about consciousness. Of course, this is not at all how the book goes about it. In fact, I think the opposite assumption is implicit here. So needless to say, the book isn’t as interesting to me as I would have hoped.

The one argument that was presented right at the beginning that I had heard before but forgotten that was pretty interesting was about debunking the “brains in a vat” idea. Basically this goes back to Descartes who wanted to know if there was any way we could tell if we actually existed or if our brains were bodiless in a vat somewhere and scientists were just stimulating certain neurons to make us think we were people (well, so Descartes’ description was a little different, but this is the modern Matrix-esque interp). Essentially we don’t have to go through the trouble that Descartes went through to debunk this possibility. The standard sort of pragmatic argument is that it just isn’t possible for any reasonable interpretation of the term “possible”. The amount of computer power needed to do this would encounter a combinatorial explosion for even the simplest experience of the world. Thus, it is not possible we are being tricked (so before a torrent of arguments fill my replies, I watered it down, try to fill in the details yourself before arguing).

Other than that the only sort of important terms that might show up in later posts have to do with phenomenology. All of chapter 3 is essentially devoted to this. Essentially it is a method of philosophy of consciousness developed by Husserl that tried to remove subjectivity. I really don’t want to go into this much, since I feel like it won’t play much of a role later and it is giving me flashbacks of my 20th century philosophy class when we had long tedious arguments about the method of “bracketing”. Overall, what you should know is that it played a huge role in influencing major philosophers and schools of thought on philosophy, but in general is highly criticized and probably has been overtaken by neuroscience studies and interpreting them.

My one complaint so far is that the results of thought experiments (which play a major role in this book) are very skewed by leading questions. I don’t doubt that the visualization of X was harder than Y, but coming to that conclusion before asking, “Wasn’t visualization of X harder than Y?” would have been more convincing for your argument. I’m not sure if any were that bad, and I should look a specific one up, but I don’t really feel like it now.

My guess is that the next chapter is on a rejection of Husserl’s phenomenology, and then hopefully it will get into some of the crazy things our brain’s do from a neuroscience perspective. That could make things more interesting.

# Music 09 First Half

It is halfway through the year and thus time to present my thoughts on the music so far. As I did last year, I’m going to not give actually place rankings, just general categories. May as well do the COMAP ranking system.

Outstanding:
The Antlers – Hospice
Decemberists – The Hazards of Love
Doves – Kingdom of Rust

Meritorious:
Andrew Bird – Noble Beast
Animal Collective – Merriweather Post Pavilion
Loney, Dear – Dear John
Phoenix – Wolfgang Amadeus Phoenix

Honorable Mention:
Dan Deacon – Bromst
The Field – Yesterday and Today
Neko Case – Middle Cyclone

Successful Participant:
Bon Iver – Blood Bank EP
Duncan Sheik – Whisper House
M. Ward – Hold Time
Yeah Yeah Yeahs – It’s Blitz!

Some preliminary hilarious things to note are that despite Bon Iver being my number 1 best album of last year, it got the lowest ranking possible this year. Also, The Field is the only band I’ve ever liked an album so much that I actually wrote an Amazon review for it.

I attribute some of the uncharacteristic placements to the past couple of months in which my musical tastes having been doing an utterly complete shift. Also, what I consider to be great has a few extra components to it that it didn’t used to have which knocked Bird and Phoenix out of the top ranking.

Overall, I’ve had very few full albums I love from this year. Neko Case and M. Ward both have some of my favorite songs, but also have enough of songs that I just can’t stand that it was hard to rank as a full album.

So here is an individual favorite song list:

Andrew Bird – Anonanimal
Animal Collective – My Girls
Antlers – Wake
Dan Deacon – Get Older
Decemberists – Hazards of Love 4
Doves – Lifelines
M. Ward – Blake’s View
Neko Case – Red Tide
Phoenix – Lisztomania!

As usual. If you’re list looks at all similar to this, I’d love to hear about things you love that I may have missed/overlooked.

That’s all for now. I’ve made significant headway on Consciousness Explained so that will start up soon.

# Invisible Man

So I must admit that I’ve been having some sort of strange quarter-life crisis or something recently, and this book exactly hit the spot. It really resonated with lots of the things that I’ve been giving a lot of thought to. That being said, I should try to give an actual review now. As usual, I will not reveal any late plot points, so don’t worry, it will not be spoiled if you want to read it.

First, we really need to clear up the fact that Ralph Ellison’s Invisible Man is different from The Invisible Man by H.G. Wells. A quick historic note is that this was the only novel Ellison published in his lifetime. My first complaint is that even though I usually like my novels to be long, this clocked in at close to 600 pages and I felt that it could have been adequately done in 300.

Overall, the book is generally a fictional account of an African-American student that is kicked out of his university for a rather silly reason. He then moves to New York and becomes a leader of some sort of civil rights group.

Here is why I think the novel is too long. I think of it in three sections. The beginning and end are very personal introspective accounts by the narrator. He questions what it means to be stereotyped (in fact, this is what the title is referring to. He is not actually invisible. He just feels that way due to the society’s expectations). These two sections really show the emotional and subconscious effects that bigotry have on people. They were some of the most powerful, chilling works of prose I’ve read. These two sections also have some truly disturbing scenes about what was culturally acceptable behavior at some point (and, well, depending on what type of minority you are, possibly still today).

The middle third, although in some senses necessary, just seemed unimportant to me. You could cut it and be left with an utterly fantastic novel. This section was much more macroscopic in scale. It was the political activist section. It wasn’t as personal, and so it was hard for me to connect with. It was also filled with probably a good hundred pages of random political speeches. For the time period of the novel (1952), it was probably much more effective, but for me it was a major distraction and drag to what I considered to be the main point.

Overall, if you’re willing to put in the effort of a fairly long-winded novel, I’d say it is very much worth it (especially if you are going through a period of questioning your identity and how you fit into society and how to deal with stereotypes and being true to yourself). So I’ll leave you with some quotes that I found particularly profound.

“When one is invisible he finds such problems as good and evil, honesty and dishonesty, of such shifting shapes that he confuses one with the other, depending upon who happens to be looking through him at the time. Well, now I’ve been trying to look through myself, and there’s a risk in it. I was never more hated than when I tried to be honest. Or when, even as just now I’ve tried to articulate exactly what I felt to be the truth.”

“I looked at Ras on his horse and at their handful of guns and recognized the absurdity of the whole night and of the simple yet confoundingly complex arrangement of hope and desire, fear and hate, that had brought me here still running, and knowing now who I was and where I was and knowing too that I had no longer to run for or from the Jacks and the Emersons and the Bledsoes and Nortons, but only from their confusion, impatience, and refusal to recognize the beautiful absurdity of their American identity and mine. . . . And I knew that it was better to live out one’s own absurdity than to die for that of others.”

# Plans

I have settled on both short term and long(er) term plans for the blog. My immediate future will be a week in which I post no math. I will do my mid-point through the year music ranking soon, I am going to finish Ralph Ellison’s Invisible Man in a day or two, so I may talk about that.

For longer plans, I also found Dennett’s Consciousness Explained on clearance at the bookstore for like 4 dollars, and I’ve always wanted to read it. So that is a hefty read that will probably force me to catch up on some philosophy of consciousness, which I plan on doing through here. That is my main non-math project for the summer. Literary-wise, I haven’t decided whether to finish up some of those easier reads that just collect dust on my bookshelf, or to do the one main large work that is sitting there. Either way, that will posted about as well. In fact, the math posting will probably be fairly light during the summer, since my life will be reviewing stuff for prelims rather than learning new stuff. So it will probably just be when I make large revelations about things I probably should have already realized.

Thoughts? For the summer reading, my choices seem to be something along the lines of the large Sot-Weed Factor by Barth, or (after a brief glance at my shelf) it appears The Way the Crow Flies by Ann-Marie MacDonald, Paradise Lost by Milton, Sanctuary by Faulkner, Dune by Herbert, and Jude the Obscure by Hardy.

# Derived Functors II

First off, we need to get this pesky result out of the way that derived functors are independent of the choice of resolution. So we do this by proving a related result. Suppose $\cdots\rightarrow F_i \stackrel{\phi_i}{\rightarrow}F_{i-1}\rightarrow \cdots \rightarrow F_1\stackrel{\phi_1}{\rightarrow} F_0$ is a a complex of projective A-modules, and $\cdots G_1\stackrel{\psi_1}{\rightarrow} G_0$ is a complex of A-modules. Let $M=coker \ \phi_1$ and $N=coker \ \psi_1$. Suppose the homology of $G$ vanishes except $H_0(G)=N$. Then every map $\beta\in Hom_A(M, N)$ is a map induced on $H_0$ by a map of complexes $\alpha : F\to G$ and is determined up to homotopy by $\beta$.

Before proving this, note that as a corollary we get that any two projective resolutions are homotopy equivalent, and hence the derived functors have constructed on different resolutions have a natural isomorphism between them.

Proof: I knew I should never have tried to do homological algebra without a good way to do diagrams on wordpress. This is clearer if you draw it out…but the idea for existence is to inductively lift your maps. Lift $F_0\to M\to N$ to $\alpha_0: F_0\to G_0$, then $\alpha_0\phi_1: F_1\to ker(G_0\to N)=im(G_1\to G_0)$. Thus we lift this to $\alpha_1: F_1\to G_1$ and continue this process. This gives the map of the complexes.

We now want uniqueness up to homotopy. Suppose we have two lifts of $\beta: M\to N$ say $\alpha$ and $\gamma$. Then $\alpha - \gamma$ lifts the zero map. i.e. we need only show that any lifting of 0 is homotopic to 0. Suppose then that $\eta$ is a lifting of zero. We need that $\eta_i=h_{i-1}\phi_i + \psi_{i+1}h_i$ for some $h_i: F_i\to G_{i+1}$. Note that $\eta_0 : coker \ \phi_1\to coker \ \psi_1$ takes $F_0\to im\psi_1$. So we lift to $h_0: F_0\to G_1$ such that $\psi_1 h_0=\eta_0$. But now $\psi_1(h_0\phi_1-\eta_1)=\eta_0\phi_1-\psi_1\eta_1=0$. Thus $h_0\phi_1-\eta_1$ maps into $ker \ \psi_1=im \ \psi_2$. But $F_1$ is projective so we can lift to $h_1: F_1\to G_2$. Repeat this process.

I highly recommend just doing the diagram chasing yourself. This is sort of a mess to read, and so should only be used as a sort of guideline if you get stuck somewhere.

Hmm…what else did I say I would do? Oh right. If you’ve seen homological constructions, then you can probably guess that there is a connecting homomorphism type theorem. i.e. Something that is phrased, “a short exact sequence of complexes induces a long exact sequence in homology.” So this trick tends to be useful in actually calculations of your derived functor. I won’t go through it, since it is just your standard “Snake Lemma” construction.

When I said there was extra structure, I was thinking about going into putting a product on the whole thing to make it into a graded ring, but I’ve decided that that is getting a little far afield for now. This may be the end of my ramblings on derived things for awhile.

The other thing I thought I should mention was my confusion on what this blog has become. I’m at a sort of turning point. I’m not sure if I should eliminate the non-math/mathematical physics and turn it into a blog just on that stuff (it sort of accidentally has shifted to that unofficially), or if I should make a conscious effort to balance things more. There are positives and negatives to both in my mind. Actually changing to a more focused blog would help draw and keep readers that actually care about the stuff I’ve been doing recently. On the other hand, it sort of goes against everything I believe in. But how its been going now, I’ve probably alienated all readers that used to read for philosophy or random art things, and so randomly posting on those things seems sort of pointless if I’ve lost those people and it will just serve to confuse and possibly alienate people only interested in the math side.

No immediate decisions will be made, so there is some time.

# What is a derived functor?

First recall that functors do not have to preserve exactness. In fact, some of our favorite ones in really nice situations fail to be exact. For instance, tensoring (i.e. $-\otimes M$) is a covariant functor that does not preserve exactness on the left.

This is going to be our motivating example. Before working with the functor, we need some definitions. Let M be an A-module. Then we define a projective resolution for M as a collection of projective A-modules $\{M_i\}$ with maps $\{f_i\}$ such that $\cdots \stackrel{f_2}{\rightarrow} M_2 \stackrel{f_1}{\rightarrow} M_1 \stackrel{\epsilon}{\rightarrow} M \rightarrow 0$ is exact. Things to note. We definitely do not necessarily have M projective. If we did, then the resolution would be easy: $0\rightarrow M\stackrel{i}{\rightarrow} M\rightarrow 0$. Also, we don’t need to terminate at 0, we could end on M and just say that $\epsilon$ is surjective. Lastly, such a resolution always exists. I won’t write it out, but try it yourself (hint: you can form the free module with generators the elements of M…it is probably absurdly large, but there is nothing wrong with doing it).

Alright, so now we can form an exact sequence of “nice” modules from a given module. We want to pull in the tensor now. So if we hit the projective resolution (note it does not include M itself) by the functor $-\otimes N$ where N is an A-module, then we get another sequence: $\cdots \stackrel{f_{2*}}{\rightarrow} M_2\otimes N \stackrel{f_{1*}}{\rightarrow} M_1\otimes N\rightarrow 0$. But we’ve lost exactness. We have not however lost $im f_i\subset ker f_{i+1}$. i.e. This is a chain complex and we can measure how far off it is from being exact by taking the homology $H_i M_. = \frac{ker f_i}{im f_{i-1}}$. This is the derived functor $Tor_i^A(N, M)$.

So a quick check will show you that $Tor_0^A(N, M)=N\otimes M$ and that $Tor_1^A(A/(x), M)=_xM$ the torsion subgroup, so it is a good name.

There was nothing special about this construction. This is called a left derived functor. You can do exactly the same thing for any right exact functor. $Tor^A(-, M)$ is the left derived functor of $-\otimes M$. I won’t go through the left case in generality since the process is the same as the specific case, but you will get the idea from below.

We can also form a right derived functor of a left exact functor. Let $\mathcal{F}$ be a left exact functor (out of the category of A-modules for now). Then take the injective resolution $0\rightarrow M \rightarrow I_1 \rightarrow I_2 \rightarrow \cdots$ (an exact sequence with each $I_i$ injective). Apply the functor to the resolution (don’t forget M is not a part of it). We get a cochain complex $0\rightarrow \mathcal{F}(I_1)\rightarrow \mathcal{F}(I_2)\rightarrow$ This is no longer exact, so to check how far off it is we take the cohomology $H^i(I_.)=\frac{ker f_i^*}{im f_{i-1}^*}$. This is the ith right derived functor of $\mathcal{F}$.

In a specific case, we get that $Ext_i^A(M, -)$ is the ith derived functor of $Hom_A(M, -)$.

The astute reader may have noticed that in forming derived functors, there is an arbitrary choice, namely which resolution do you take (they are not unique!). It turns out that the choice is irrelevant. Any will yield the same construction.

Maybe I’ll talk a little more about this, how these are actually useful, and some more structure we actually have on them next time.

# Schemes

There was talk about schemes in the comments of my last post, so after reviewing what I’ve already posted about, I decided I may as well package it all up nicely in a brief post so that I’m allowed to use the term freely from now on.

First, recall the sheaf structure we already have. For any ring, R, we have the associated topological space $Spec(R)$ and the sheaf of rings $\mathcal{O}$. Then the stalk for $p\in Spec(R)$ is $\mathcal{O}_p\cong R_p$. Also, $\mathcal{O}(D(f))\cong R_f$ for any $f\in R$.

Let’s extrapolate what was the important structure here. We really have a topological space and a sheaf of rings on it. We call this a ringed space. Morphism in this category are a pair $(f, g): (X, \mathcal{O}_X)\to (Y, \mathcal{O}_Y)$, where $f:X\to Y$ is continuous, and the sheaf structure is preserved, i.e. $g: \mathcal{O}_Y\to f_*\mathcal{O}_X$ is a map of sheaves of rings on Y.

A ringed space is called a locally ringed space if each stalk is a local ring. I’m not sure how technical I should be about the definition of a local homomorphism. Essentially, we want to preserve localness on the homomorphisms induced on the stalks by the sheaf homomorphism. So a homomorphism is local if the preimage of the maximal ideal in one go to the maximal ideal in the other.

So without proof I’ll just state that a homomorphism of rings $\phi : A\to B$ induces a natural morphism of locally ringed spaces (contravariantly), and conversely, given A and B, any morphism of locally ringed spaces $Spec(B)\to Spec(A)$ is induced by a ring hom $A\to B$. The first statement essentially follows from laying down definitions, but it is not trivial. The second one requires some more thought.

Now we define a scheme. An affine scheme is a locally ringed space that is isomorphic to the spectrum of some ring. A scheme is a locally ringed space in which every point has an open neighborhood $U$ such that $(U, \mathcal{O}_X\Big|_U)$ is an affine scheme. Morphisms are in the locally ringed sense.

The easiest example would be a field, where the topological space is a point and the structure sheaf is the field back again. If we step the dimension up by one (and require the field to be algebraically closed for sake of example), then $Spec (k[x])\cong \mathbb{A}_k^1$

I may or may not return to elaborate. I sort of want to consolidate the algebra I’ve learned this quarter through a series of posts before doing anything else along the algebraic geometry side of things.