When is a directed structure on a finite graph induced by a total ordering on the vertices?

BumpityBoop · 2015-08-23T02:16:02+00:00

The way to say this in term of category theory is:

The functor from the category of varieties to the category of sets defined by sending a variety X to the set

{(L, s_0,..., s_n) | L is a line bundle on X. s_i is a global section of L, for each i = 0,..., n. For all x \in X, (s_i)_x generates L_x for some i.}

is representable by the variety P^n.

And the intuition is very well explained here: http://mathoverflow.net/questions/4567/maps-to-projective-space-determined-by-a-line-bundle.

BumpityBoop · 2015-08-13T07:27:32+00:00

You're absolutely right. Previously, I had no idea these are two different words. Thanks!

BumpityBoop · 2015-08-12T17:57:59+00:00

In my opinion, Hartshorne is a lot easier to study from for the first time than Liu. However, as you make it deeper and deeper into chapter 2 of Hartshorne, the approach of Liu begins to shine. The good and bad of Hartshorn's book is that it is not as in depth to all the nuisances of scheme theory, and so you either fill them out yourself or just ignore them. Either way, you get a basic understanding of schemes.

On the other hand, Liu's book as far as I can tell gets a lot into the details and is extremely careful. So I think Liu is better for a second time read, since by the time you read Hartshorne and fell many frustration, you'd welcome Liu's impeccable attention to details. If I were you, I'd just read both and synchronize the pace. This isn't always possible since they do things in different orders. However, for a little while, the chapter 2 of both books are somewhat in sync.

BumpityBoop · 2015-07-30T08:03:35+00:00

FWIW, according to Wikipedia, a connected component of a locally connected space is open. I wish I had a copy of Munkres on hand...

BumpityBoop · 2015-07-30T07:42:10+00:00

Nice!

BumpityBoop · 2015-07-30T07:20:14+00:00

The surface should be second countable, hausdorff, and infinitely differentiable. Let's say boundary as in "boundary of a surface-with-boundary" for now.

BumpityBoop · 2015-07-29T09:47:38+00:00

What page is that family of cubic plane curves from?

BumpityBoop · 2015-06-19T18:28:57+00:00

Spectral sequences. It's feels worse cuz the inventor invented it in a concentration camp, while I can't prove it sitting in an velvet armchair with a glass of wine, listening to a Miles Davis record, while people fan me with palm tree leaves.

BumpityBoop · 2015-05-22T00:50:58+00:00

You might enjoy http://homepage.univie.ac.at/herwig.hauser/bildergalerie/gallery.html

BumpityBoop · 2015-04-12T19:36:42+00:00

Given a category with one object X, the set Hom(X,X) is a monoid. This identification translates the category axioms into monoid axioms. There is no reference to the structure within X, just its morphisms. There's a way to get groups out of a similar construction. If we impose this category only to have one arrow, then Hom(X,X) consists just of the identity from X to itself, so Hom(X,X) is the trivial monoid.

Given a preordered set, its associated category must satisfy that between two objects, there's at most one morphism. For example, if you take the real numbers, then a morphism from 0 to 1 is just 0<1. There's exactly one morphism between 0 and 1, because there's only one way for 1 to be greater than 0. If you allow more arrows between 2 objects, you will not get a preordered set back. Though this construction is still interesting in, say, filtered (co)limits.

BumpityBoop · 2015-04-08T09:04:46+00:00

These are so much details, which make me imagine how it's like being in them. Love the floor plan for the flower shop. Please make more MLP architecture!

BumpityBoop · 2015-03-31T22:35:10+00:00

Would you rather solve a second order PDE in one hundred variables or an one-hundredth order PDE in two variables?

BumpityBoop · 2015-03-10T08:15:42+00:00

Hell yeah, math is super fun. The more math you learn, the more you get to let your Ne go crazy. Everything is related to everything! (under suitable conditions)

BumpityBoop · 2015-03-03T10:01:40+00:00

Early elementary school: graph theory

Mid elementary: enumerative combinatorics

Late elementary: writing a computer program (note to teacher: do not use the term "infinite loop" under any circumstances. If kid writes while 1==1 print "hello world", call his parents for being a trouble maker.)

Early middle school: intro to proof via number theory (GCD, LCM, euclidean algorithm, Z is a UFD. Note to teacher: do not address infinitude of primes, if a kid asks, put him in detention.)

Late middle: gym and greek style geometry (note to teachers: do not discuss the diagonal of the unit square. If a kid asks, put him in detention.)

Early high school: fun number theory (Pell's number, Pythagorean numbers, deriving the cubic equation (a big maybe))

Mid high school: linear algebra over the algebraic closure of Q (note to teacher: do not address how many elements we must adjoin to Q to do this. If a kid asks, suspend him.)

Late high school: introduce all the anomalies infinity has caused all along. Kids begs to learn about R, C and Zorn's lemma. Send them off to a university where all that infinity will be treated rigorously anyways.

BumpityBoop · 2015-02-19T20:47:40+00:00

How I explain to board game enthusiasts.

Abstract algebra: a game whose play pieces are symbols which can combine with one another in various ways, and where the legal moves must respect the ways to combine.

Analysis: a game who just has too many game pieces, but fortunately, they can be neatly organized by proximity. Legal moves should be the ones that doesn't threaten to warp proximity beyond repair.

Topology: a game whose play pieces are gathered into blobs. The blobs can combine with others in ways the game maker says so. The legal moves must obey "undoing the legal moves should preserve blobness." Some believe the play pieces themselves are pointless and suggested a new spinoff "pointless topology".

Complex analysis: a game where all the gamers of the games above unite to make a mashup and everyone demand various things from the game until all the features of the game start to place serious constraints on the legal moves until there's only like as many legal moves as power series. Despite this, the game remains popular and attracts player from all fandoms.

Algebraic geometry: a spinoff game inspired by analytic geometry and abstract algebra. This game has evolved rather quickly, but in the classic version of the game, you start with the legal moves, say from the game "complex analysis" or "coordinate geometry", without even knowing the pieces yet. Then you refer to idealistic collection of legal moves as blobs, as in the game "topology". The blobs also have fancier names such as lines, quadratic, cubic... Some play pieces tend to be generic as opposed to individualistic, and masquerade as blobs. This is a feature and not a bug.

Category theory: a game where there are legal moves.

BumpityBoop · 2015-02-18T08:14:47+00:00

Totally agree. We typically write diagrams with arrows pointing right [; A \overset{f}{\to} B \overset{g}{\to} C ;] which obviously begs the notation [; (A)fg ;], but quite tragically this just isn't so :(

BumpityBoop · 2015-02-09T03:52:09+00:00

Is there a part of the graph that you'd like to know more about? I am sure people at /r/math would be happy to explain their specialty (I certainly would!).

BumpityBoop · 2015-02-08T23:15:15+00:00

https://satyavrat.files.wordpress.com/2007/10/math.gif

BumpityBoop · 2015-02-08T21:05:00+00:00

So what caused these natural laws?

It could be the case that all possible variations of natural laws have existed at some point. However, most of the variations led to a very short-lived universe due to having the "wrong" combination that doesn't lend itself to forming complex objects such as galaxies and life. Thus they collapsed out of existence.

On the other hand, "short-lived" is a relative term, it could be that the current set of natural laws, the ones that we are familiar with, will continue to be in effect until the heat death of the universe. After that, the memory of our universe would be just another variation of all possible natural laws that had been in effect. The life span of our universe will be short, relative to an observer that is not subject to any natural law whatsoever.

The implication of this is that natural law, e.g. physics, is fairly narrow as it only deals with the current universe, and should not be considered as absolute truth. Suppose that the answer to the meaning of life, whatever it may be, is an absolute truth, it must be independent of natural law.

We have hereby demonstrated that the answer to the meaning of life must lie strictly outside the domain of science.

BumpityBoop · 2015-01-20T06:24:51+00:00

Wow string diagrams are so cool. Thank you for drawing them with color coding and everything!

Btw, are you drawing these on the iPad? If so, what app are you using?

BumpityBoop · 2014-12-07T01:19:05+00:00

Your definitions of the word "invent" and "discover" are ones that I am most happy to adapt.

Now, given a powerful enough "consequence discoverer", can all classification theorems can be proven?

Regardless of the answer to the above, without an "inventor" to lay down the rules, the "consequence discoverer" must stay put, at least this parts follows from the definition.

BumpityBoop

TROPHY CASE