Slowdown on Blogging

Recently I’ve been retyping probably 12 of my past blog posts that were on height and p-divisible groups for nLab articles. The original purpose of my blogging was to help myself learn things by explaining them to the internet. The things I was blogging about then were so basic that they existed in many, many forms all over the place, so it didn’t really matter where I was typing them up.

Here’s the state of things right now. My blog doesn’t get much traffic, and gets even less conversation. If my real purpose is to help myself learn things, then I can do exactly what I’m doing on this blog but over at the nLab. The difference is that at the nLab there will be more traffic, so it is a more useful place to publish. There is also the nForum where I can ask questions and get more points of view about the things I’m writing, i.e. more conversation. Ultimately, it seems like that will be much more beneficial for my learning than this isolated blogging activity.

Lastly, I’m probably going to want to import lots of what I blog into the nLab anyway, which is what I’ve been finding myself doing for the past few days. It feels really annoying and wasteful to be redoing all these posts I’ve already done, so I’d just rather start there.

Here is what I plan to use the blog for (math-wise) from now on. If I have interesting examples or computations I’ll probably put them here since that won’t fit as easily into the nLab. If I’m in really, really early stages of learning something I’ll probably start tinkering with the idea here until I can come up with better ways of writing it for the nLab. This means I’ll probably continue the crystalline stuff here since it is just a mess of information right now.

In summary, I’ll still be blogging, but much less frequently.


Crystalline Site 3: De Rham Spaces

I think my posts are becoming increasingly incomprehensible, and I have a partial remedy to that. At some point I want to take a few steps backwards and examine the “calculus of divided powers”. This will help us get a feel for how we are doing something that can be thought of as positive characteristic de Rham cohomology. I also want to look at some comparison theorems, so that we can see that this actually matches up with other cohomology groups that we know how to compute and it will give us useful information. But before doing any of that, I want to show the usefulness and motivation for all of this by tying it back to the height of varieties that we defined awhile ago.

Today I’ll just do a quick post on yet another way to think about this site when we are in characteristic {0}. For any space {X} (and by space, I mean that really loosely as just an object in {\mathrm{Psh}(\mathrm{Ring}^{op})}, but you can think scheme or variety if you want) we can form the associated “de Rham space”. This is the presheaf defined by {X_{dR}(R)=X(R/\mathcal{N}(R))} where {\mathcal{N}(R)} is the nilradical.

What this is doing in effect is identifying infinitely close points. A concrete way to visualize this is to take {X=\mathbb{A}^1_k}. Then {X(k[\epsilon])} are points plus a choice of tangent vector. But if you are used to thinking in terms of smooth manifolds or something, then you probably don’t think of {(0, v)} and {(0, -v)} where {v} is a tangent vector at {0} as different points because they are literally the same point with the only difference being some infinitesimal information. We call them being “infinitely close”. Since {k[\epsilon]/\mathcal{N}\simeq k}, we see that {X_{dR}(k[\epsilon])=X(k)}. So the infinitesimal information is killed off.

In the case of {X} being a smooth variety the natural map {X\rightarrow X_{dR}} is surjective and hence we can honestly think of {X_{dR}} as a quotient of {X} where our relation is to identify the infinitely close points. Now if we go back to a {X} being a scheme, we get the following: The big site {\mathrm{Ring}^{op}/X_{dR}} is just the crystalline site of {X}. When we are over characteristic {0} this is often called the infinitesimal site. We now have in some sense a categorical way to think of the construction.

Crystalline Site 2: The Topos

Last time we defined crystalline cohomology by defining the crystalline site. Our setup was to fix {(S, \mathcal{I}, \gamma)} a PD-scheme and look at an {S}-scheme {X} to which {\gamma} extends. Remember that the objects of the site {\mathrm{Crys}(X/S)} were triples {(U, T, \delta)} that we short-handed with {T}. Today we’ll start by re-examining these objects. We can give a little better description which will help us figure out what a sheaf on this site is. Recall that the full subcategory of the category {\mathrm{Psh}(Crys(X/S))} of presheaves (of sets) on the site is a topos which we’ll denote {(X/S)_{Crys}}.

Suppose {\mathcal{F}} is an object of {(X/S)_{Crys}}, i.e. a sheaf. This just means it is a functor {\mathcal{F}: \mathrm{Cyrs}(X/S)^{op}\rightarrow \mathrm{Set}} that also satisfies for any covering {\{T_i\rightarrow T\}} the following sequence being exact {\mathcal{F}(T)\rightarrow \prod \mathcal{F}(T_i)\stackrel{\rightarrow}{\rightarrow} \mathcal{F}(T_i\cap T_j)}.

Let’s fix some object {(U,T,\delta)} and some sheaf {\mathcal{F}}. If {T'\hookrightarrow T} is a Zariski open set then define {U'=U\cap T'} and {\delta'=\delta|_{T'}}. There is an inclusion {(U', T', \delta')\rightarrow (U, T, \delta)}, which by construction is a map in the category {\mathrm{Crys}(X/S)}, so by virtue of being a contravariant functor we get {\mathcal{F}(T)\rightarrow \mathcal{F}(T')}. The contravariant functor {T'\mapsto \mathcal{F}(T')} is a sheaf on the Zariski site of {T} which we denote {\mathcal{F}_{(U, T, \delta)}}.

We have a proposition that the data of a sheaf {\mathcal{F}} on {\mathrm{Crys}(X/S)} is equivalent to the data that for ever {S}-PD thickening {(U, T, \delta)} we assign a Zariski sheaf on {T} denoted conspicuously by {\mathcal{F}_T} and for every morphism {u:(U_1, T_1, \delta _1)\rightarrow (U, T, \delta)} we assign a map {\rho_u: u^{-1}(\mathcal{F}_T)\rightarrow \mathcal{F}_{T_1}} satisfying two compatibility conditions.

1) If {v:(U_2, T_2, \delta_2)\rightarrow (U_1, T_1, \delta_1)} is another map then we have a commutative diagram

{\begin{matrix} v^{-1}u^{-1}\mathcal{F}_T & \rightarrow & u^{-1}\mathcal{F}_{T_1} & \rightarrow & \mathcal{F}_{T_2} \\ \| & & & & \| \\ (u\circ v)^{-1}\mathcal{F}_T & - & - & \rightarrow & \mathcal{F}_{T_2} \end{matrix}}


2) If {u: T_1\rightarrow T} is an open immersion, the map {\rho_u^{-1}: u^{-1}(\mathcal{F}_T)\rightarrow \mathcal{F}_{T_1}} is an isomorphism.

Using this new characterization it is much easier to describe what the structure sheaf {\mathcal{O}_{X/S}} in this site should be, since to give a sheaf I need to give on every thickening {T} a Zariski sheaf and we have a natural Zariski structure sheaf {\mathcal{O}_T}. This assignment can be seen to satisfy the conditions and hence we have a crystalline sheaf. Note that this was not the only choice that works, though. We’ve already pointed out that {|U|\simeq |T|}, so we could also assign {T\mapsto \mathcal{O}_U} which is another crystalline sheaf that we’ll denote {\mathcal{O}_X}.

For our last example, let’s just create a new sheaf from our two above by taking {T\mapsto \mathrm{Ker}(\mathcal{O}_T\rightarrow \mathcal{O}_U)}. This is actually a sheaf of P.D. ideals in {\mathcal{O}_{X/S}} and we’ll call it {\mathcal{J}_{X/S}}. By construction it sits in an exact sequence {0\rightarrow \mathcal{J}_{X/S}\rightarrow \mathcal{O}_{X/S}\rightarrow \mathcal{O}_X \rightarrow 0}.

The last thing to point out for today is that this alternate way of thinking about crystalline sheaves is that it really gives us a way of thinking in Zariski terms which we’re already familiar with. In particular, if {v:\mathcal{F}\rightarrow \mathcal{G}} is a map of crystalline sheaves we can check whether it is an isomorphism (or surjective or injective) on stalks by checking for each {x\in X} and each {S}-P.D. thickening {T} of a neighborhood if the map {(\mathcal{F}_T)_x\rightarrow (\mathcal{G}_T)_x} is an isomorphism.

Crystalline Site 1

I’ve decided on pulling the motivation back into the picture. Recall way back when we were thinking about the shortcomings of trying to replicate a de Rham type cohomology theory in positive characteristic. One of our motivations is that we want a theory that has no problem being done in positive characteristic, but actually gives us the de Rham cohomology if there is some lift to characteristic {0}. We even tried to just define it this way. Take a lift, do de Rham, and then check that the result is independent of lift. The problem is that there are things that don’t lift to characteristic {0}, and the lifting process is definitely not an efficient process for computing.

This is where crystalline cohomology enters the picture. We’ll make this more precise later on, but if we have a smooth lifting {X\rightarrow S} to characteristic {0}, then we’d like to have a canonical isomorphism {H^*_{crys}(X_0/S)\rightarrow \mathbf{H}^*(X_{zar}, \Omega^\cdot)}. Since we’ve already talked about what it means to do cohomology of a sheaf on a site, we can actually state pretty easily what crystalline cohomology is. Suppose {X} is a variety over {S}. There is the crystalline site {\mathrm{Crys}(X/S)}, and {H^n_{crys}(X/S)=H^n(X_{cyrs}, \mathcal{O}_{X/S})}, so the crystalline cohomology is just sheaf cohomology on the crystalline site. The work is going to be in figuring out how to think about this new site.

First, we define a P.D. scheme. This is exactly what it sounds like. There is no problem in extending all the definitions done for rings so far into definitions on the sections of sheaves. For instance, if {X} is a space and {\mathcal{A}} and {\mathcal{I}} are sheaves of rings on {X}, then we say {(\mathcal{A}, \mathcal{I}, \gamma)} is a sheaf of P.D. rings if {(\mathcal{A}(U), \mathcal{I}(U), \gamma)} is a P.D. ring for all {U\subset X} open. A P.D. ringed space is just a ringed space with a sheaf of P.D. rings on it {(X, (\mathcal{A}, \mathcal{I}, \gamma))}. Inverse image and pushforwards of sheaves under maps {f:X\rightarrow Y} preserve the property of being a sheaf of P.D. rings.

Given a P.D. ring {(A, I, \gamma)} we can define {\mathrm{Spec}(A, I, \gamma)} to be the locally ringed space {(|\mathrm{Spec}(A)|, \mathcal{O})} where {\mathcal{O}} is the sheaf of P.D. rings obtained under the canonical extensions we get of {\gamma} since localization is flat and we checked previously that {\gamma} extends to any flat {A}-algebra. A P.D. scheme is a locally ringed space locally isomorphic to {\mathrm{Spec}(A, I, \gamma)}. Morphisms in this category are morphisms of locally ringed spaces that are P.D. morphisms on sections.

We’ll spend a little more time with these definitions and related issues next time. The goal of this post is to define the crystalline site. Now we’ll want to fix a base, so let {(S, I, \gamma)} be a P.D. scheme. If {X} is an {S}-scheme, then by looking locally it makes sense to ask whether or not {\gamma} extends to {X}. If it does, then we can define the crystalline site {\mathrm{Crys}(X/S)} as follows: The objects are pairs {(U\hookrightarrow T, \delta)} where {U\subset X} is Zariski open and {U\hookrightarrow T} is a closed {S}-immersion defined by the quasi-coherent sheaf of ideals {\mathcal{J}} where {\delta} is a P.D. structure on {\mathcal{J}} compatible with {\gamma}. We abuse notation and call {(U\hookrightarrow T, \delta)} just {T}.

The morphisms {u:T\rightarrow T'} are commutative diagrams

{\begin{matrix} U & \hookrightarrow & T \\ \downarrow & & \downarrow \\ U' & \hookrightarrow & T' \end{matrix}}

where {U\rightarrow U'} is a Zariski inclusion of open sets of {X} and {T\rightarrow T'} is a P.D. map over {S}. A covering is just a collection of maps {\{u_i: T_i\rightarrow T\}} such that {T_i\rightarrow T} are open immersions and {T=\cup T_i}. We’ll just end this post by making several remarks and giving an example that we’re aiming at.

First, the term for an object {(U\hookrightarrow T, \delta)} is an “S-PD thickening of {U}“. One of the consequences of requiring {\gamma} to extend to {X} is that it makes {(U\rightarrow U, 0)} an object of our site for any Zariski open {U\subset X}. Another consequence of our definitions is that all our thickenings {U\rightarrow T} are topological homeomorphisms since they are defined by nilpotent ideals, {\mathcal{J}}. The last remark is that if {\{T_i\rightarrow T\}} is a covering, it comes with a collection of P.D. structures: {\delta_i}. By compatibility, the collection {\delta_i} completely determines {\delta} and conversely, given {\delta}, we can restrict and find out what the {\delta_i} must be.

The example that we want to think about is when we have some lifting of {X} over a postive characteristic field {k} to {W_n(k)}. In this situation {S=\mathrm{Spec}(W_n)} with {\mathcal{I}=(p)} with the canoncial P.D. structure inherited from {W}. We’ll look at this more closely when we are working with actual examples of lifted schemes.

Divided Powers 4

Today we’ll look at the P.D. envelope of an ideal. To do this properly would take many pages of gory calculations, so we’ll be a little sketchy in order to get the idea out there. Before we do that we need to look at a construction I’ve been avoiding on purpose. Suppose {M} is an {A}-module. Then there is a P.D. algebra {(\Gamma_A(M), \Gamma_A^+(M), \gamma)} and an {A}-linear map {\phi:M\rightarrow \Gamma_A^+(M)} satisfying the universal property that given any {(B, J, \delta)} an {A}-P.D. algeba and {\psi: M\rightarrow J} an {A}-linear map there is a unique P.D. map {\overline{\psi}:(\Gamma_A(M), \Gamma_A^+(M), \gamma)\rightarrow (B, J, \delta)} with the property {\overline{\psi}\circ \phi=\psi}.

Let’s be a little more explicit what this is now. First, {\Gamma_A(M)} is a graded algebra with {\Gamma_0(M)=A}, {\Gamma_1(M)=M} and {\Gamma^+(M)=\bigoplus_{i\geq 1} \Gamma_i(M)}. Let’s denote {x^{[1]}} for {\phi(x)} and {x^{[n]}} for {\gamma_n(\phi(x))}. In fact, by abusing notation we often just write {[ \ ]} in place of {\gamma} for the P.D. structure. This is because {\Gamma_n(M)} is generated as an {A}-module by {\{x^{[q]}=x_1^{[q_1]}\cdots x_k^{[q_k]} : \sum q_i=n, x_i\in M\}}. This should just be thought of as a “generalized P.D. polynomial algebra”. We’ll soon see its importance. Now back to the regularly planned post.

Let {(A, I, \gamma)} be a P.D. algebra and {J} an ideal of {B} which is an {A}-algebra. There exists a P.D. envelope of {J} which is a {B}-algebra denoted {\frak{D}_{B,\gamma}(J)} (or sometimes just {\frak{D}} when the rest is understood) with a P.D. ideal {(\overline{J}, [ \ ])} such that {J\frak{D}_{B,\gamma}(J)\subset \overline{J}} compatible with {\gamma} and satisfying a universal property:

For any {B}-algebra, {C}, containing an ideal {K} containing {JC} with a compatible (with {\gamma}) P.D. structure {\delta}, there is a unique P.D. morphism {(\frak{D}_{B, \gamma}(J), \overline{J}, [ \ ])\rightarrow (C, K, \delta)} which makes the diagram commute:

{\begin{matrix} (B, J) & \rightarrow & (\frak{D}, \overline{J}) \\ \uparrow & \searrow & \downarrow \\ (A, I) & \rightarrow & (C, K) \end{matrix}}

Where that vertical right arrow should be dotted. Like I said, the existence of this thing takes a lot of tedious calculations. You basically show by brute force that it exists in a few special cases and then reduce the general case to these. These calculations actually bear out a few interesting things that we’ll just list:

First, {\frak{D}_{B, \gamma}(J)=\frak{D}_{B, \gamma}(J+IB)}. Next, if our structure map actually factors {A\rightarrow A'\rightarrow B} and we have an extension to {\gamma'} on {A'}, then {\frak{D}_{B, \gamma}(J)=\frak{D}_{B, \gamma'}(J)}, which essentially tells us that it really is an “envelope” or some sort of minimal construction.

More importantly, we have a more concrete way to think about the construction. Namely, {\frak{D}} is generated as a {B}-algebra by {\{x^{[n]}: n\geq 0, x\in J\}} and any set of generators of {J} gives a set of P.D. generators for {\overline{J}}.

Let’s do two examples to get a feel for what this thing is. Let {B=A[x_1, \ldots, x_n]} and {J=(x_1, \ldots, x_n)}, then we’ll consider the trivial P.D. structure on {A} given by the {0} ideal. This gives us {\frak{D}(J)=A\langle x_1, \ldots, x_n\rangle} the P.D. polynomial algebra. Along the same lines but slightly more generally, suppose {M} is an {A}-module and {B=\mathrm{Sym}(M)}. Let {J} be the ideal of generated by the postive graded part {J=\mathrm{Sym}^+(M)} and do everything with respect to the trivial P.D. structure on {A} again. We get {\frak{D}=\Gamma_A(M)}.

The other example is that when we have {\gamma} extending to {B/J} with a section {B/J\rightarrow B}, then the compatibility condition is irrelevant. This just means that {\frak{D}_{B, \gamma}(J)\simeq \frak{D}_{B, 0}(J)}. This is just because the section gives us that {\frak{D}_{B, 0}(J)=B/J\bigoplus \overline{J}} and an application of the universal property.

I’m not sure what to do next. I’m sure I’ve alienated all my readers with all this. At this point I could shift over and at least define crystalline cohomology. We have enough to cover some of the basic definitions and actually show that all this has a purpose. On the other hand, we definitely have a ton more properties we should do if we want to get anywhere with crystalline cohomology.

Divided Powers 3

Today we’ll think about notions of compatibility for P.D. structures. The general setup is the following. We have {(A,I,\gamma)} a P.D. ring and {B} an {A}-algebra. We’ll say that {\gamma} extends to {B} if there is a P.D. structure {\overline{\gamma}} on {IB} such that {(A,I, \gamma)\rightarrow (B, IB, \overline{\gamma})} is a P.D. morphism. From what we talked about last time we immediately get that {\gamma} extends if and only if there is a P.D. ideal {(J, \delta)} of {B} such that {(A, I, \gamma)\rightarrow (B, J, \delta)} is a P.D. morphism since we’ve checked that {IB} is a sub P.D. ideal of {J}.

It also quickly follows from what we’ve done that if there is some extension, then that extension is unique. We should check that this definition isn’t useless by exhibiting an instance where it doesn’t extend. Just pick rings for which {B=A/J} and {I\cap J} is not a sub P.D. ideal of {I} (which necessarily means we picked {J} to have no P.D. structure).

We can guarantee that {\gamma} extends in certain nice situations. For instance, let’s prove that {\gamma} extends to any {B} when our ideal {I} is principal. Suppose {I=(t)}, then we need to define {\overline{\gamma}} on {IB} or in other words we need to understand how to define {\overline{\gamma}_n(tb)} for any {b\in B}. First, let’s check that the natural guess is well-defined. Define {\overline{\gamma}_n(tb)=b^n\gamma_n(t)}. Suppose {tb=tb'}, then {b^n\gamma_n(t)-b'^n\gamma_n(t)=\sum_{i=1}^{n-1} b^ib'^{n-i-1}(b-b')\gamma_n(t)}. Now since {\gamma_n(t)\in I} we get that it is just another multiple of {t} which shows that {(b-b')\gamma_n(t)=0} making the whole sum {0} and hence {\overline{\gamma}_n(tb)=\overline{\gamma}_n(tb')}. Now that we have well-defined, we can just use the fact that {\gamma} is a P.D. structure to check the properties for {\overline{\gamma}}.

Here we have the main definition of today. Suppose {(A, I, \gamma)} is a P.D. ring and {B} and {A}-algebra with {(J, \delta)} a P.D. ideal of {B}. We say that {\gamma} and {\delta} are compatible if any of the following equivalent conditions are met.

1. {\gamma} extends to {B} and {\overline{\gamma}=\delta} on {IB\cap J}.

2. The ideal {K=IB+J} has a P.D. structure {\overline{\delta}} such that {(A, I, \gamma)\rightarrow (B, K, \overline{\delta})} and {(B, J, \delta)\rightarrow (B, K, \overline{\delta})} are P.D. morphisms.

3. There is an ideal {K\subset IB+J} with a P.D. structure {\delta ' } such that {(A, I, \gamma)\rightarrow (B, K', \delta')} and {(B, J, \delta)\rightarrow (B, K, \delta')} are P.D. morphisms.

The equivalence is straightforward to check using the if and only if condition listed in the first part of the post for extending a P.D. structure. Let’s go back to thinking geometrically for a second. Then what’s going on is we have some affine variety, and possibly some thickening of it. The original variety has a P.D. structure, and so does the thickening. Compatibility is just saying that the original P.D. structure can be extended over the whole thickening (meaning there is a P.D structure on the thickening that restricts to the original one) and it can be done in such a way that restricting to where the {\delta} P.D. structure is defined also agrees. So as opposed to just an extension, we have what could be thought of as a simultaneous extension of both P.D. structures.

Next time we’ll start talking about some more complicated things by introducing what could be thought of as formal completions of P.D. structures. This shouldn’t be hard to guess what it is if you are comfortable with formal completions of affine varieties.

Matt Nathanson’s Modern Love Reviewed

I haven’t spent that much time with this album, but I felt an overwhelming need to do a review. I’m predicting that this will be one of the most underrated albums of the year. The reason I say that is that you could easily listen to this album five times and think to yourself that it is just another standard pop album to be lost and forgotten among the hundreds that come out every year. The purpose of this review is to explain some of the subtleties that make it much more than that.

First off, let’s place the album in context. It really is necessary. Matt Nathanson on every album he has every released has been the most optimistic, hopelessly romantic songwriter you will ever find. I could be having the worst day of my entire life, and there is one thing I can always rely on: popping in Some Mad Hope will instantly make me feel exceedingly happy to be alive.

The song “Lucky Boy” contains the lyrics “They tell me it’s a cruel world that I’ve found/It’s a cruel world and I’m a lucky boy.” The song “Come on Get Higher” has the chorus “Come on get higher loosen my lips faith and desire and the swing of your hips just pull me down hard and drown me in love.” In “Heartbreak World” he says “In this heartbreak world where nothing matters C’mon lets make this dream thats barely half awake come true.”

Really. You have to believe me. If there is a person that is endlessly optimistic about the possibility of true love, it is him. Let’s start with the title of this new album. It is called Modern Love. One might guess that it will be more of the same. Very quickly you learn it isn’t. This is basically a subtley constructed concept album. It sounds like a bunch of disjoint songs, but really it is basically like a bunch of connected diary entries.

The songs all have to do with a single topic. He has given up on love. Instead he has replaced it with this new idea of “modern love”. This is basically the idea that all that exists are quick hookups and nothing that lasts. Love is a thing of the past. The only thing that exists nowadays is modern love. This shift is really shocking in context. It makes this often upbeat album quite depressing when you really listen to what’s going on.

The greatness and subtelty to this album is the fact that he is trying to put on a happy face and make his songs sound happy and uplifting just like his old ones. Most people that listen to this will undoubtedly come to the conclusion that it is just the same old stuff. But the lyrics are these profound struggles where he starts to fall for someone and he has to remind himself that he can’t let that happen. Love is not what is wanted in today’s world and he has to stamp it out in place of modern love before he lets his feelings take over too much. He has to be satisfied just with superficial quick relationships. This may sound like some cheap emo type stuff, but really it is spectacularly done.

Let’s take a few of the songs one by one and see how it manifests itself. “Love Come Tumbling Down” I think the title says it all, but here is part of it “Cities made of stone sinking in the flood All the people wait around to rust And all the stars above our heads Faster they came, faster they went … Love comes tumbling down.” I think it is pretty clear that cities sinking and people rusting is a reference for how much he believes love being a thing of the past is damaging civilization and people.

In “Room @ the End of the World” he says “I let go of love once it finally had me figured out … Sad can’t catch me Call me baby now When it’s all I used to believe.” He is admitting here that it he has let go of love, but it was all he used to believe. In “Modern Love” he explains the concept “And so we stumble and disconnect
Over and over again This modern love is not enough She said, “watch your back I’m nobody’s girlfriend.” He is connecting and unconnecting with girls, but none of them want to be called girlfriend. He still believes that modern love is not enough, and that is the struggle of the album. Despite believing this he has come to believe also that modern love is all there is.

“Kiss Quick” is probably my favorite song on the album. It is the most moving to me. “Kiss quick, I got a line out the door who all think they can save me. One by one they lay the world at my feet, one by one they drive me crazy. Shut your mouth, pull me out before this all goes grey. One by one they lay the world at my feet, one by one they go away.” He apparently has people in his life trying to pull him out of this slump that he’s in. They think they can save him. But he sees that it isn’t real. They come and then they go away not really willing to do the work to pull him out of it and teach him that love is real.

I’ll just do one more song. In “Kept” he says “I should have kept my head I should have kept my arms inside I believe it now I should have kept my head I should have kept my heart, my heart.” This song is absolutely tragic. He was warned not to fall for someone. It happened anyway. He couldn’t help it. The song is about how he should have kept everything to himself. He was warned. Why would he think this person was different and wanted something more than modern love?

I could keep going on, but I think you see my interpretation of the album. Once you come to the realization, you won’t be able to listen to a single song without hearing that this is what it is really about. The album is so intimate and personal. It is like reading a diary of someone’s transformation from eternal optimist about love to someone who doesn’t even believe love exists anymore. It is utterly fantastic. I have to end by reminding you that if this sounds like too depressing of music to you, he is keeping the same feel of his optimist music from the past.

Most people listening to this may miss the lyrics and think it is just happy pop music, so it is quite upbeat stuff. There are huge climaxes with lines like “I’m so amazing when I’m around you.” These are moments in his life where he has glimpses that love might still be possible, of course the song still ends with it being false. Maybe I’m completely insane and this isn’t at all what the album is about, but I’m still naming it most underrated of the year anyway. It has caused me great amounts of emotion and contemplation. I’m fascinated to see if his next album is back to his usual self showing this really was a temporary slump, or if he just has a new permanent attitude towards love and life.