[18:32] <@hochs> alright, so i'll talk about the proof of implicit function theorem, then i'll give the chalk over to capso/breeden for the more interesting talk
[18:32] <@hochs> the proof of implicit function theorem is typed up at http://tiny.cc/f2gsew, and it is this that i will follow
[18:33] <@hochs> hopefully i will talk more about the salient points of the proofs than little technical details
[18:33] <@hochs> and anyone who has questions/comments/addons, feel free to just ask/comment :)
[18:33] <Capso> sorry, I wasn't sure when the second part would be; I may not be ready for that today.
[18:33] <@hochs> ah ok
[18:33] <Capso> so you should carry on, and we can come back to it next time.
[18:33] <@hochs> right, ok
[18:34] <Capso> I will cover the sheaf definition of differentiable manifolds and what it tells us about local/global properties.
[18:34] <Capso> rightly a separate topic.
[18:34] <@breeden> sounds fair to me
[18:34] <@hochs> so i'll state the theorem - Implicit Function Theorem (it's stated in the link above) - here for completeness sake
[18:35] <@hochs> the setting is this: you have a function g: R^n x R^m -> R^m, which has continuous partial derivatives, and points a,b are given in R^n, R^m respectively such that g(a,b) = 0
[18:36] <@hochs> and you would like to find some open neighborhoods A,B of a,b respectively such that there's a C^1 function h: A -> B s.t. 1) h(a) = b and g(x,h(x)) = 0 for all x in A
[18:36] <@hochs> insert 2) after "and"
[18:36] <@hochs> implicit function theorem says that such h: A -> B exists, and this is what we'll prove
[18:37] * hochs changes topic to 'Readings in Geometry and Topology seminar. | http://tiny.cc/f2gsew'
[18:37] <@hochs> so ok, here goes the proof:
[18:37] <@hochs> we can assume that a = b = 0 by translation (this is just a minor convenience)
[18:38] <@hochs> well i forgot a condition there. the jacobian of the map y -> h(a,y) has to be nonzero at y = b
[18:39] <@hochs> jacobian, as you may know, is just the determinant of the derivative of the function
[18:40] <@hochs> going back, we let L = the derivative of the map y -> g(a,y) at the point y = b, and let f(x,y) = y - L^{-1}g(x,y)
[18:41] <@hochs> just a little remark here: this looks suspiciously like newton's method, except L^{-1} here is a fixed linear transformation
[18:41] <@breeden> for finding roots of polynomials?
[18:41] <@hochs> the usual calc 1 newton's method thing: iterate the function x -> x - f(x)/f'(x)
[18:42] <@hochs> which can be used to approximate roots of polynomials
[18:43] <@hochs> ok, so we need some small neighborhoods A,B and function h:A -> B such that g(x,h(x)) = 0
[18:43] <@hochs> that is to say, a function h such that f(x,h(x)) = h(x)
[18:44] <@hochs> and this we can interpret as wanting to find a fixed point of the operator map h -> f( _, h(_)) from the space of functions A -> B to the space of functions A -> B itself
[18:45] <@hochs> which is where the banach contraction principle comes in - it's precisely used to find the fixed point of a contraction map
[18:45] <@hochs> more formally, for open neighborhoods A,B of 0,0 (we're assuming that our a,b in the theorem are translated to 0)
[18:45] <@hochs> let F be the set of all functions from A to B
[18:46] <@breeden> are we going to determine A and B later?
[18:46] <@hochs> so for small enough A,B (we'll keep shrinking it in the proof as we go along), we want the map T -> f(_, T(_)) to be well-defined and a contraction
[18:46] <@hochs> yes, we'll keep cutting A,B down
[18:46] <@hochs> :)
[18:47] <@hochs> this is what analysts do it seems
[18:47] <@breeden> they like cutting things
[18:47] <@hochs> you want some functions/sets/space having some property, then you just keep cutting stuff out/down until you get what you want
[18:47] <@hochs> right
[18:48] <@hochs> i keep messing up the notations here, sorry. so we have the map F -> F, and we'll call this T: F -> F (so T is the operator, not an element of F)
[18:48] <@hochs> defined by Tw(x) = f(x,w(x)), for w in F
[18:49] <@hochs> so we want two things: 1) T to be well-defined, and 2) T to be a contraction
[18:49] <@hochs> if T satisfies these properties, then by the contraction principle you immediately get a map h:A -> B satisfying f(x,h(x)) = h(x), or g(x,h(x)) = 0
[18:49] <@hochs> so let's handle the well-definedness first:
[18:50] <@hochs> actually let F to be the set of all functions w: A -> B such that w(0) = 0 (added condition here)
[18:50] <@hochs> now take any w in F, and we want to cut A,B down enough so that Tw: A -> B and Tw(0) = 0
[18:50] <@hochs> well, Tw(0) = f(0,w(0)) = 0 is clear
[18:51] <@hochs> let's say the diameter of A = a and diameter of B = b (so we might as well use balls for A,B)
[18:52] <@hochs> actually just let A,B be balls of diameter a,b respectively, and let's cut down a,b
[18:52] <@hochs> so take any x in A, and we want to estimate |Tw(x)| (we want it to be <= b)
[18:53] <@hochs> Tw(0) =0, so |Tw(x)| = |f(x,w(x))| = |f(x,w(x)) - f(x,0) + f(x,0)| <= |f(x,w(x)) - f(x,0)| + |f(x,0)|
[18:53] <@hochs> well you don't need Tw(0) = 0 here
[18:54] <@hochs> anyway, you can estimate the first quantity |f(x,w(x)) - f(x,0)| by the mean value theorem
[18:54] <@hochs> use it on the map (y -> f(x,y) - f(x,0))
[18:55] <@hochs> so you get something like f(x,w(x)) - f(x,0) = sum f_j(x,y') * (y_j), where y' = (y_1, ..., y_m) is some point lying on the line joining 0 and w(x) in R^m
[18:56] <@hochs> and recall how f is defined: f(x,y) = y - L^{-1}g(x,y)
[18:56] <@hochs> so the the jacobian of the map y -> f(x,y) is just 0: (y - L^{-1}g(x,y))' = I - L^{-1}L = I - I = 0 at (0,0)
[18:57] <@hochs> i.e. cutting A,B, you get an estimation |f(x,y) - f(x,0)| < K*|y| for x,y in A,B respectively
[18:57] <@hochs> for some 0 < K < 1
[18:58] <@hochs> so you have estimation |Tw(x)| <= |f(x,w(x)) - f(x,0)| + |f(x,0)| <= K*|w(x)| + |f(x,0)|
[18:58] <@hochs> at least we know that |w(x)| <= b, so |Tw(x)| <= Kb + |f(x,0)|
[18:59] <@hochs> K < 1, so you can cut A down so that Kb + |f(x,0)| <= b for all x in A
[18:59] <@hochs> this is by continuity of f of course
[18:59] <@hochs> so ok, we have shown that you can cut down A,B enough so that T : F -> F is well-defined
[18:59] <@hochs> turns out that with these same A,B, T: F -> F is in fact a contraction
[19:00] <@hochs> (the metric on F, if i didn't mention it, is the uniform metric one)
[19:00] <@breeden> i think we need A,B to be closed to get a complete metric
[19:01] <@breeden> well, just A
[19:01] <@hochs> good point. so we'll add that condition also :
[19:01] <@hochs> :)
[19:02] <@hochs> and you can quickly convince that everything goes through with A closed neighborhood of 0
[19:02] <@hochs> the easy computation goes as follows: take any w,w' in F. then d(Tw, Tw') = sup_{x in A} |Tw(x) - Tw'(x)| = sup_{x in A} |f(x,w(x)) - f(x,w'(x))| <= sup_{x in A} K|w(x) - w'(x)| = K * d(w,w')
[19:02] <@hochs> so this is a contraction
[19:02] <@hochs> and so now simply apply the contraction principle on T:F -> F to get your sought-after map h !
[19:03] <@hochs> we aren't really done, since you want h to be C^1 and all that. but this is just the application of mean value theorem
[19:03] <@hochs> first the continuity of h: take any continuous function F -> F, then iterate T on it infinite number of times, then you get your fixed point
[19:04] <@hochs> and h = lim T^i(w), with w: A -> B any continuous function such that w(0) = 0
[19:04] <@hochs> sorry, not continuous function F -> F, but a continuous function A -> B that fixes 0
[19:05] <@hochs> but this is uniform limit of continuous function, so you get another continuous function
[19:05] <@hochs> and so h is continuous
[19:05] <@hochs> ok, what about differentiability
[19:05] <@hochs> take any x in A, then by translating, if necessary, we might as well assume that x = 0 again
[19:05] <@hochs> and we have g(x,h(x)) = 0
[19:05] <@hochs> just apply the mean value theorem on each of the image components of the function g
[19:06] <@hochs> i.e. if g = (g_1, g_2, ..., g_m), then apply the MVT on each g_i
[19:06] <@hochs> g_i(x,h(x)) - g_i(0,0) = 0
[19:07] <@hochs> and so you get sum dg_i/dy_k(p_i, q_i) * x_j + sum dg_i/dy_k (p_i,q_i)*h(x) = 0, for some point (p_i,q_i) in the line segment joining (0,0) and (x,h(x))
[19:07] <@hochs> dg_i/dx_j(p_i,q_i)*x_j for the first term i mean
[19:08] <@hochs> and just let i,j run through {1,2,3,...,m} while fixing i, and let x be a movement along an axis: x = (0,0,...,t,0,0,...)
[19:09] <@hochs> then you get a system of equations with variables the difference quotients of h_k, h = (h_1, ..., h_m)
[19:10] <@hochs> and if you paid attention in your linear algebra course you know you can solve for these difference quotients in terms of polynomials in the coefficients !
[19:10] <@hochs> bredon's says use cramer's, but you can do it with adjoint matrices ofc
[19:10] <@hochs> and the determinant of the coefficient matrix is precisely the jacobian of L
[19:11] <@hochs> so the system is invertible
[19:12] <@hochs> so well, the solution has a limit clearly, i.e. limit of the difference quotients of h_i's exist
[19:12] <@hochs> so h is differentiable
[19:13] <@breeden> done?
[19:13] <@hochs> i think so
[19:13] <@breeden> yay! :)
[19:13] <i_c-Y> *claps*
[19:13] <@hochs> yea i'm glad this is over too :)
[19:13] <@breeden> i love that proof btw
[19:13] <@hochs> me too
[19:14] <@hochs> it's quite clever i think. very resourceful use of the mean value theorem and contraction principle
[19:14] <@hochs> the trick was to put f(x,y) = y - L^{-1}g(x,y) and reinterpret it as wanting a function h such that f(x,h(x)) = h(x)
[19:14] <@breeden> the same technique is used in math other places, like proving the existence of solutions to some differential equations
[19:14] <@hochs> which is finding a fixed point
[19:14] <@hochs> right, i heard that
[19:14] <@breeden> the complex variant is even slicker
[19:14] <@hochs> i think the hardest part of the proof is discovering the definition f(x,y) = y - L^{-1}g(x,y)
[19:15] <@breeden> well for the inverse function theorem
[19:15] <@hochs> from there i can sort of see how one might come up with this proof
[19:15] <@breeden> i think proving that h is C^1 would've been my breaking point :P
[19:15] <@hochs> right, so the inverse function theorem is an easy consequence of the implicit function theorem if you just set g(x,y) = f(y) - x, where f is the function you want to invert
[19:15] <@hochs> heh
[19:16] <Capso> normally, I use the inverse function theorem to prove the implicit function theorem.
[19:16] <@breeden> when things are holomorphic, you have things like Rouche's theorem to find roots to by pass the contraction mapping thing
[19:16] <@hochs> same, i would normally do that as well. breeden/bredon just likes it this way
[19:16] <@breeden> (again for inverse function, heh)
[19:16] <@hochs> yea
[19:16] <@hochs> i love that proof too :)
[19:17] <@hochs> the argument principle
[19:17] <@hochs> counting principle, or whatever. you stick in that x in the usual log form to get an explicit formula for the inverse
[19:17] <@hochs> of an holomorphic function
[19:17] <@breeden> also, when things are holomorphic, it's much easier to find an estimate on the A and B
[19:18] <@hochs> i think this proof is pretty effective also. there was really only two estimations
[19:18] <Capso> Bredon's proof is nice :-)
[19:18] <@hochs> 1) |f(x,y) - f(x,0)| < K|y|
[19:18] <@hochs> 2) Kb + |f(x,0)| <= b
[19:18] <@hochs> both of which are pretty easy to estimate i think.. but i'm not a numerical analyst
[19:19] <@hochs> the first one you can just make all the partial derivatives of y -> f(x,y) < 1
[19:19] <Capso> By the way, since we are talking differential geometry here, an example use of the Implicit Function theorem is exactly to find implicit functions that define curves given by parametric equations in time.
[19:19] <@hochs> well alright that's it for implicit function theorem. we'll see a lot more of this when we get to morse theory and transversality :)
[19:20] <@hochs> yes, and immersion/coordinate change things
[19:20] <@breeden> indeed, i would like to say that the implicit function theorem motivated the definition of a manifold
[19:20] <@hochs> should we move on now?
[19:21] <@breeden> well, sure, but I have to be honest, I didn't prepare very well at all, hehe :)
[19:21] <Capso> hmm
[19:21] <@hochs> as we mentioned in the first talk, we'll try to keep this more interactive
[19:21] <@hochs> so the spotlight doesn't have to be on just one person only
[19:21] <Capso> can we have the basic definition of a differentiable manifold, and then see some example applications of the implicit function theorem?
[19:22] <Capso> I'd like it if the topics interconnected. If we must wait for things to be useful, then we may as well postpone those topics until they'll appear again.
[19:22] <Capso> (that is, we don't have to follow the format of the book; it is perhaps advisable that we don't.)
[19:23] <@breeden> right, so we can also say (by the inverse funciton theorem) that h is a diffeomorphism
[19:23] <@hochs> the easiest one i can think of now without developing further stuff is just that if you have change of coordinates x_i' = h_i(x_1, ..., x_n) from R^n -> R^n which has nonzero jacobian then it's invertible in neighborhoods
[19:23] <@hochs> i.e. if it's injective on the tangent space
[19:23] <@hochs> or surjective
[19:23] <@hochs> whichever
[19:25] <@breeden> Ok, let's review the classical definition of a manifold.
[19:26] <@hochs> ok
[19:26] <@breeden> Intuitively, we are trying to give a characterization of surfaces/spaces in an abstract setting
[19:26] <@breeden> and there are many good reasons for doing this
[19:27] <@breeden> when we learned group theory, we learned that groups can be embedded into a symmetric group of some cardinality
[19:27] <@hochs> like cayley's theorem
[19:27] <@hochs> ?
[19:27] <AreEssay2> yeah
[19:27] <@breeden> however, we decided to maintain the abstract definition of a group (and leaving the symmetric group setting, as Galois did), for a few reasons
[19:28] <@breeden> hochs, yes, Cayley's theorem, or some sharper variants
[19:28] <@hochs> yea
[19:29] <@breeden> Why, well, we often talked about quotient groups, and then if you wanted to translate this to the symmetric group setting, you would have to ask your self? Well where does this quotient group live?
[19:29] <@breeden> or if we took semi-products of a group, etc...
[19:30] <@breeden> and we will find ourselves asking the same questions if we always insisted that our geometrical figures lived in R^n for instances
[19:30] <Capso> is this in relation to embedded manifolds, or glued up diffeomorphic pieces of R^n?
[19:30] <@breeden> it's if we insisted that our manifolds lived in R^n in a classical sense
[19:31] <@hochs> i think breeden just wants to say that embedded manifolds aren't useful ways of looking at things
[19:31] <@hochs> and we should look at it in terms of glued up things
[19:31] <Capso> ah, well, beyond that: you won't have a function space manifold that will easily be embedded into R^n (if at all) :-)
[19:31] <@breeden> right, the embedded is usually not what is important
[19:31] <@breeden> you mean banach manifolds?
[19:32] <Capso> so I think this goes beyond that groups as symmetric groups analogy: somethings you just can't push into R^n.
[19:32] <zeno> Well, you can for any connected paracompact manifold, right?
[19:32] <@breeden> well, i agree, but we are going to start with smooth manifolds, which can be embedded into R^n if we wanted
[19:32] <zeno> any such n-dimensional M embeds in R^(2n+1)
[19:32] <@breeden> yes, and that was proved by Whitney I believe
[19:32] <@hochs> yes zeno
[19:33] <zeno> (or into R^(2n) in the compact case)
[19:33] <@hochs> for compact n-manifolds is the result i know
[19:33] <@hochs> paracompact, R^{2n+1} i guess
[19:33] <@hochs> btw, zeno & capso can give the next talk. i'll add zeno and capso in the website
[19:33] <@hochs> zeno be ready
[19:34] <zeno> Aren't most manifolds people seriously talk about paracompact anyway? (= separable = metrizable)
[19:34] <@breeden> ok, i dont have Bredon on me, but we start with a topological space that is Hausdorff and 2nd countable (ie has a countable basis)?
[19:34] <@hochs> zeno: yep
[19:34] <@hochs> breeden: yes
[19:34] <@hochs> and bredon has an example in the baire category section where if you lose the 2nd countability then you get a really weird space you don't want to touch
[19:34] <zeno> Well, by far the most fun non-paracompact manifold is the long line.
[19:35] <@hochs> a connected 2-manifold which has a countable dense set, has uncountable discrete subset, and not a normal space
[19:35] <zeno> it even admits smooth structures that are not parallelizable.
[19:36] <@hochs> so yes, let's assume 2nd countability breeden :)
[19:37] <@breeden> right, most of the time people consider manifolds that are actually metrizable, but sometimes it's nice to drop this restriction so we don't have to worry about proving that a space is metrizable (while keeping most of the properties that we usually need)
[19:37] <@breeden> at least that's how i felt when i took a first course in differentiable geometry
[19:38] <@hochs> funny thing we don't assume 2nd countability for schemes
[19:38] <@hochs> but the zariski topology is too weird (it's not even T_2) and such
[19:38] <@breeden> idk, does anyone want to hear the definition of a manifold?
[19:38] <@hochs> for example, the usual fundamental group of all connected "schemes" are trivial
[19:39] <@hochs> which is why people consider etale fundamental groups
[19:39] <@hochs> so continuing, n-manifold M is just a T_2, 2nd countable space that chart/coordinate system U -> R^n with all its transition maps smooth.
[19:39] <@hochs> i think we can leave it at that?
[19:39] <@breeden> yeah
[19:39] <@hochs> and you can define a sheaf of smooth functions on M, and you can glue these to get a sheaf on the entire set of open subsets of M
[19:39] <Capso> well, you can often also lose the whitney embedding if you quotient by a group action
[19:40] <@breeden> Capso, what do you mean?
[19:40] <@hochs> exercise by next week's seminar perhaps: glue the sheaves on certain open subsets of M to get a contravariant functor from *all* open subsets of M to R-algebra of smooth functions
[19:41] <@breeden> is that like taking a maximal atlas?
[19:41] <@hochs> yea
[19:41] <@hochs> it's sort of a pedantic process of gluing these sheaves. it's in hartshorne's 2.1 exercise also
[19:41] <@hochs> i remember having to write this in detail for HW in the AG course that followed hartshorne long ago, so i remember how pedantic this is
[19:41] <@hochs> :P
[19:42] <Capso> breeden: if you have a lie group acting on a manifold, you can apply the quotient topology to the space given by gluing things in the same orbit
[19:43] <@breeden> ok, so how do you lose the embedding?
[19:43] <@breeden> do you mean, the embedding might change, or it's no longer a smooth manifold?
[19:43] <Capso> hochs: ah, maybe you should talk about the functional structure and making a differentiable manifold.
[19:45] <@hochs> there's an old abandoned section of a project of mine (abandoned after stack project came out) when i was an undergrad: http://dl.dropbox.com/u/3799589/Math%20Notes/Algebraic%20Geometry/Hartshorne/Scheme%20Theory.pdf has the glueing problem and its solution written down
[19:45] <@hochs> hmm ok, so you have the classical definition, and you have the definition by sheaves:
[19:45] <@hochs> you just keep track of all the smooth functions on a T_2, 2nd countable space M
[19:45] <Capso> breeden: I believe if the group action is not nice (some string of conditions) then you can lose smoothness
[19:46] <@breeden> i see
[19:46] <@hochs> with sheaves (functional structure according to bredon)
[19:47] <@hochs> it's just a contravariant functor from the set of all open subsets of M (with inclusion map as the only arrow/morphism) to R-algebra of real-valued functions
[19:47] <@hochs> i.e. if X is a T_2, countable basis space (let's just call it a space from now on )
[19:48] <@hochs> then package this X with a data of real-valued algebra of functions, F_X(U), for each open subset of X
[19:49] <@hochs> and you have the natural maps F_X(U) -> F_X(V) for each open subset V of U
[19:49] <@hochs> concretely, F_X(U) = set of functions s: U -> R^n that is closed under +,*, and has all constants
[19:50] <@hochs> so so far it's a "pre"-sheaf
[19:50] <Capso> yes
[19:50] <@hochs> the sheaf part requires you to have uniqueness and the glueing part
[19:51] <@hochs> specifically, if you have some open covering U = union U_i of an open set U, then whenever you have data of sections s_i in F_X(U_i) that are compatible with each other (i.e. s_i restricted to U_i \cap U_j is the same as s_j restricted to U_i \cap U_j) then you can glue these to get a section s in F_X(U) whose restrictions on U_i is s_i
[19:53] <@hochs> hmm i think that's it for the "functional structure" on X
[19:53] <@breeden> so how does smoothness play a role here?
[19:53] <@hochs> so now we have a more fancy definition of differentiable manifold:
[19:53] <@hochs> we haven't gotten to the smoothness yet. it'll come in the fancy definition
[19:53] <@breeden> k sorry
[19:54] <@hochs> so you have (R^n, O), O is the functional structure on R^n
[19:54] <@hochs> where O(U) for any open subset R^n consists of all C^{\infty} functions U -> R^n
[19:54] <@hochs> and it's clear that this satisfies the sheaf axioms, so you have "standard" model of a manifold
[19:55] <@hochs> and the n-dimensional differentiable manifold is a second countable T_2 space endowed with functional structure, (X,O_X), which is locally isomorphic to (R^n, O)
[19:56] <@hochs> err i should explain what a morphism is in the category of functionally structured things (X,O_X)
[19:57] <@hochs> morphism (X,O_X) -> (Y,O_Y) is two things: 1) a continuous map f:X -> Y, and 2) a natural transformation f': O_Y -> f_{*}O_X
[19:58] <@hochs> so a natural transformation O_Y -> f_{*}O_X just means the data of ring homomorphisms O_Y(U) -> O_X(f^{-1}(U)) for each open subset U of Y
[19:58] <@hochs> which commutes with all restriction maps
[19:56] <@hochs> so what the fancy definition is saying is just that you take with your space
[19:56] <@hochs> X,
[19:56] <@hochs> the set of all smooth functions on U, U an open subset of X
[19:57] <@hochs> "smooth function" on U tacitly implies having a homeomorphism U -> R^n
[19:57] <@hochs> and your "smooth functions" on U are the ones you get by composing with this map all real smooth functions R^n -> R^n
[19:57] <@hochs> U -> R^n -> R^n
[19:58] <Capso> Nice. Can you give an example where we would make use of this algebraic definition of differentiable manifolds? Perhaps something that exploits the locally ringed structure of (R^n, O)?
[19:58] <@hochs> i don't know, a projective space?
[19:58] <@hochs> i think projective space is a good standard example maybe
[19:59] <@hochs> like in the P^2 case, you want to glue two R^1 and R^1 with transition the twist map x -> 1/x
[19:59] <@hochs> anad it's easy to compute the cohomology groups for this sort of things
[19:59] <@hochs> cech cohomology i think exploits locally ringed structure definition
[19:59] <@breeden> i see
[20:00] <@hochs> yes, cech cohomology is exactly what you do when you're given locally ringed definition
[20:00] <@breeden> where if you gave the definition using the homogenous charts, it would not be so evident
[20:00] <@hochs> i don't know, it's so similar to the classical definition to me
[20:00] <@hochs> and it's what you work with schemes anyway in AG
[20:00] <@hochs> you have affine schemes that come with structure sheaves (Spec A, O)
[20:01] <@hochs> and you glue these
[20:01] <Capso> well, you can also take C^n instead of R^n, with analytic maps.
[20:01] <@hochs> there we go
[20:02] <@breeden> brb
[20:03] <Capso> Here are some examples: http://www.math.brown.edu/~wgillam/diffspace06-03.pdf
[20:04] <@hochs> breeden: there's also all these business about sheaf cohomology. if you apply the theory on popular coherent sheaves like tangent sheaves, cotangent sheaves, and ofc the structure sheaf, you gain a lot of invariants
[20:04] <@hochs> you can define the genus of a curve as H^1 of the structure sheaf, for example
[20:05] <@hochs> nice, thanks for the link
[20:05] <Capso> no wozzies
[20:07] <Capso> And if you like algebra: http://arxiv.org/pdf/0905.0459v1.pdf
[20:08] <@hochs> kek that's the derived AG
[20:08] <@hochs> functor of points way of looking at things is something that i never got used to doing
[20:09] <@hochs> hmm i guess it lends itself nicely to rigid analytic space. everyone talks about rigid spaces now, and no longer schemes
[20:09] <Capso> depends where you are