The Eldritch Tomb of Unknowable Caffeine

Little Anonymous Me said:

On a lighter note, I turned in my last paper of the semester!

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Huzzah!

milkweed said:

Throws out calligraphered letters!

What is your thesis topic?

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Legendrian Contact Homology.

Hope your feet feel better soon, Milkweed!

asnys said:

Legendrian Contact Homology.

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That's some awfully salty language for the Cantina, dude.

asnys said:

Huzzah!



Legendrian Contact Homology.

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Little Anonymous Me said:

Hope your feet feel better soon, Milkweed!



That's some awfully salty language for the Cantina, dude.

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Thank you, sneakers with orthodics are now gracing my feet making my heal spur feel better.

Asnys, Congratulations I have no idea what you just said!

Good evening Cantina!

I would have come around earlier, but I busy this afternoon. After work I came home and dyed Easter eggs.

jallenecs said:

Stanley Ford gave me four lilac cuttings for my yard last spring, and my hubby was good enough to plant them for me. I can't wait until they're old enough to start flowering, because lilac is my favorite flower, my favorite scent, and the primary sign of spring for me, the one I look forward to.

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Lilacs smell nice too. We have a bunch of them around our yard.

Little Anonymous Me said:

People are dumb.

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Dumb and lazy. Doesn't surprise me though. The valedictorian of my high school cheated his way to the top. The jerk bragged about it too.

Little Anonymous Me said:

That's some awfully salty language for the Cantina, dude.

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milkweed said:

Asnys, Congratulations I have no idea what you just said!

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I am trying to decide if I should try to explain or not.

asnys said:

I am trying to decide if I should try to explain or not.

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Please do, because I looked it up and it confuses the heck out of me.

Lillith1991 said:

Please do, because I looked it up and it confuses the heck out of me.

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I will begin with manifolds.

A manifold is a space that locally looks like Euclidean space. Now, Euclidean space is simply an extension of the classical x-y graph from your high school Algebra course - in fact, that infinite x-y plane is two-dimensional Euclidean space. An infinitely-long straight line is one-dimensional Euclidean space. An infinite volume - like the one we live in - is three-dimensional Euclidean space. And you can extend it up to higher dimensions as well, to arbitrarily high dimension. The main thing is, a Euclidean space is one where, if you pick a fixed origin, any other point in the space can be uniquely and precisely described by a set of n coordinates, where n is the dimension of the space.

A manifold is usually not a Euclidean space, but it looks like it locally. That is, for any point on the manifold, there's a small neighborhood around the point where, if I zoom in far enough, it looks like a piece of Euclidean space. A classic example is the surface of a sphere. The sphere as a whole is not Euclidean because it's not the x-y plane. But if I pick a point on the sphere and zoom in on it really really close, it looks like a piece of Euclidean space. Another example is the torus, which is a fancy word for the surface of a donut. Another example - and an important one for Legendrian Contact Homology - is the rim of a circle. But there are lots and lots of manifolds. Euclidean spaces are themselves manifolds as well, since they locally look like themselves, after all.

Legendrian Contact Homology is part of the study of a particular kind of manifold. Imagine a three-dimensional Euclidean space. At each point in that space, there's a plane, and I pick these planes so that I cannot draw a two-dimensional manifold in the Euclidean space so that it is tangent to each plane it touches - but I can draw a one-dimensional manifold that is tangent to each plane. Then we call the three-dimensional Euclidean space with associated planes a contact manifold, and a one-dimensional manifold that is tangent to each plane it touches a Legendrian submanifold, or Legendrian for short. We can generalize to other types of contact manifolds, but this is already complicated enough, and this is itself a pretty interesting case. The study of this type of arrangement is called contact topology.

More coming in a bit.

I will abbreviate my three-dimensional Euclidean space by R3, and suppose I have two unspecified Legendrians which I will call L1 and L2. Suppose L1 can be deformed into L2, and that at each step along the deformation, that is a Legendrian. I say that L1 and L2 are Legendrian isotopic, which I often abbreviate by saying just isotopic (technically I shouldn't do that, because "isotopic" means something different, but I don't need the other meaning here).

If L1 and L2 are Legendrian isotopic, they have all the same properties. For all practical purposes, they are the same Legendrian. I say that the collection of all Legendrians that are isotopic to L1 is the Legendrian isotopy class of L1.

Now, the problem is that it's often very difficult to tell if L1 or L2 are isotopic. These Legendrians can get very complicated! So we calculate invariants of them. An invariant of a Legendrian is some number or other property we can easily calculate about it, and for which all of the Legendrians in its isotopy class have the same value. So if L1 and L2 have different invariants, they are not in the same isotopy class. If they have the same invariants, they may or may not be in the same isotopy class, we don't know. Most invariants are just numbers, e.g. the rotation class.

Legendrian Contact Homology is a new invariant of Legendrians, and a very powerful one. It allows us to distinguish Legendrians that cannot be distinguished by earlier, simpler invariants. Unfortunately, it's also harder to compute.

I can continue if there's interest, but I need to go get dinner. I will take questions over the break.

asnys said:

I am trying to decide if I should try to explain or not.

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Lillith1991 said:

Please do, because I looked it up and it confuses the heck out of me.

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Yes, please do explain!

Good morning Cantina. Something of a flying visit as I'll be off to Eastercon in a couple of hours - flying up to Scotland, returning Monday. Should be fun. I have quite a few friends from Glasgow and Edinburgh whom I first met at Eastercons in the early 1990s, and one of them is having his first novel launched this weekend.

Hope everyone is well.

asnys said:

I will abbreviate my three-dimensional Euclidean space by R3, and suppose I have two unspecified Legendrians which I will call L1 and L2. Suppose L1 can be deformed into L2, and that at each step along the deformation, that is a Legendrian. I say that L1 and L2 are Legendrian isotopic, which I often abbreviate by saying just isotopic (technically I shouldn't do that, because "isotopic" means something different, but I don't need the other meaning here).

If L1 and L2 are Legendrian isotopic, they have all the same properties. For all practical purposes, they are the same Legendrian. I say that the collection of all Legendrians that are isotopic to L1 is the Legendrian isotopy class of L1.

Now, the problem is that it's often very difficult to tell if L1 or L2 are isotopic. These Legendrians can get very complicated! So we calculate invariants of them. An invariant of a Legendrian is some number or other property we can easily calculate about it, and for which all of the Legendrians in its isotopy class have the same value. So if L1 and L2 have different invariants, they are not in the same isotopy class. If they have the same invariants, they may or may not be in the same isotopy class, we don't know. Most invariants are just numbers, e.g. the rotation class.

Legendrian Contact Homology is a new invariant of Legendrians, and a very powerful one. It allows us to distinguish Legendrians that cannot be distinguished by earlier, simpler invariants. Unfortunately, it's also harder to compute.

I can continue if there's interest, but I need to go get dinner. I will take questions over the break.

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See, maybe if you'd been my linear algebra professor, then I could've actually understood half the stuff he taught.

This is also why my quantum mechanics professor said he teaches abstinence by the associated Legendre polynomials.

asnys, you make me feel unworthy to write science fiction, and I'm fairly certain you're not even talking about science, but math. Possibly. I don't even know anymore.

asnys said:

I'm running out of letters!

Seriously, I'm writing the outline for my thesis and I'm running out of letters to symbolize different things! Send more alphabets, quick!

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Have you considered cryillic? Or possibly shapes? I used smiley faces once.

I understood the first two paragraphs perfectly. Then things got wobbly.
Thank you Asnys for making me feel very small in this world. (yes it is a good thing)

*Dons protective equipment. Hangs out with the dust bunnehs in the corner.*

*Lurks*

skkkt-ello? Hel-ssskkt-ear me? I'm-skkkttt-connection tr-skkkkkttttttttttttttt (Internet is doing this weird up/not up thing)

tjwriter said:

*Dons protective equipment. Hangs out with the dust bunnehs in the corner.*

*Lurks*

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No more lurks!

Says the lurker...

Hello! I'm currently taking a break from murderizing a chocolate bunneh to melt down into drizzle for pumpkin-pie perogies.

*returns into the shadows holding a very large and chocolate-stained knife*

KWEEI *TACKLE* Stop lurkiiiinnngg.

Wait, did Lily say chocolate and pumpkin pie? *TACKLE*

TJ, what are you doing with Don's protective equipment? And who's Don anyway?

TAMaxwell said:

KWEEI *TACKLE* Stop lurkiiiinnngg.

Wait, did Lily say chocolate and pumpkin pie? *TACKLE*

TJ, what are you doing with Don's protective equipment? And who's Don anyway?

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Well, Don Juan of course!

I'm having a crazy hard time staying focused today. -.-

Pandora played this dubstep song called "Dragon Pirates" and all I could think was, "This is so Cantina."

tjwriter said:

Pandora played this dubstep song called "Dragon Pirates" and all I could think was, "This is so Cantina."

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...Yeaaaahhh. Yeah...

...I just put a different word before "pirate".

Thank you for the inspiration! *runs off*

I don't need to say what came to mind, do I?

lilyWhite said:

...I just put a different word before "pirate".

Thank you for the inspiration! *runs off*

I don't need to say what came to mind, do I?

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Just a random guess here, but was it... siren?