(no subject)
Oct. 8th, 2007 10:21 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
michiexileMeme, as invited by thette
Comment on this post and I will pick seven of your interests. You then explain them in your journal and re-post. ...if you want to, of course. No pressure. And you can ask me about my interests, too.
abstract nonsense
Abstract nonsense is one of the most common ways to USE homological algebra and category theory outside these disciplines (or, occasionally, within them as well). The idea behind an argument using abstract nonsense is that you recognize that a bunch of statements do hold for a situation (most often, a diagram of maps commutes), and therefore, using arguments about commuting diagrams of maps, you can conclude more or less surprising results from it.
One of the more "applied", as it were, abstract nonsense results I know of is the Mayer-Vietoris long exact sequence in homology, which describes how to figure out the homology of a space using a decomposition into two subspaces, and the inclusion of the space into the union of the components.
Also commonly used for the study of these disciplines as such. And given that they belong to my absolute favourite bits and pieces of mathematics, the inclusion into my LJ interests is obvious.
category theory
It occurs almost painfully often that a single result is reproven for all sorts of settings in algebra. The Noether isomorphism theorems are a prime example - Im(f) = G/Ker(f) is a kind of result that holds for groups, rings, fields, vector spaces, modules, et c. et c. and every time you encounter it, you need a new proof of the fact.
So, what would be the obvious thing to do? Generalize. So, instead of studying groups, rings, fields, et c. all separately, we sit down and study the entity formed by "ALL Xs and nice maps between Xs". This forms a new algebraic entity called a category. And category theory is the theory of these entities.
Now, there is a REAL expert on this floating around here - yes,pozorvlak, I am talking about you. He actually does HIS PhD on these beasts, while I just use them in my homological algebra PhD quite a lot.
Oh, and it turns out that once you nailed down what a category is, surprisingly much turns out to be categories. The big charm with Haskell for me is that you actually program by constructing maps in a specific category of Haskell types. And the programmers crowd uses this metaphor extensively. I like!
conlangs
Constructed languages. Such as Esperanto, Quenya, Sindarin, Klingon, Zanaquen (speling?) et c. I have constructed a few, including some conscripts to join up with them, in my days. My wife did one for her senior year, and I helped with the random-syllable-generation.
At one point, helping Anders Sandberg with an RPG-world construction, I constructed a handful to be spoken by alien races. One based on a heavy arabicized english language fusion. And one constructed for an alien kind of speech apparatus, with a calligraphic script that reflected the oddities of this anatomy.
I used to hang out on the relevant mailing lists too - but that was one or two email adresses ago.
creative destruction
destructive creation
These belong together, really. I think I saw at least one of these inkrfsms interest list, and figured that yeah, it is kinda interesting. The other, obviously, follows by dualizing.
koszul duality
Now this is one of the main areas of study in mathematics at Stockholm University. The idea is that for a quadratic graded algebra k[x,...]/I with I in k[x,...]2, you can figure out a different algebra, such that projective resolutions work nicely and map down to Nice Things. In the end, you get all sorts things Just Working if the algebra has good properties (the map from the Koszul dual algebra to the trivial module should be a quasiisomorphism iirc). I have been away for too long to give a great explanation of the intricacies, but the kind of good stuff happening is similar to what A-infinity brings to group cohomology rings - all of a sudden you can recover everything you needed from the extra information provided.
And boy, is this a horribly horribly bad explanation.
tetrapyloctomy
One of four departments of the School of Comparative Irrelevance that Umberto Eco introduces in Foucaults Pendulum. These are tetrapyloctomy (to cut a hair four ways), adynata (also known as impossibilia), potio-section (the art of slicing soup) and oxymoronics. By writing this, I further notice that I had forgotten to add adynata and oxymoronics to the interest list. Hereby fixed.
thetteComment on this post and I will pick seven of your interests. You then explain them in your journal and re-post. ...if you want to, of course. No pressure. And you can ask me about my interests, too.
abstract nonsense
Abstract nonsense is one of the most common ways to USE homological algebra and category theory outside these disciplines (or, occasionally, within them as well). The idea behind an argument using abstract nonsense is that you recognize that a bunch of statements do hold for a situation (most often, a diagram of maps commutes), and therefore, using arguments about commuting diagrams of maps, you can conclude more or less surprising results from it.
One of the more "applied", as it were, abstract nonsense results I know of is the Mayer-Vietoris long exact sequence in homology, which describes how to figure out the homology of a space using a decomposition into two subspaces, and the inclusion of the space into the union of the components.
Also commonly used for the study of these disciplines as such. And given that they belong to my absolute favourite bits and pieces of mathematics, the inclusion into my LJ interests is obvious.
category theory
It occurs almost painfully often that a single result is reproven for all sorts of settings in algebra. The Noether isomorphism theorems are a prime example - Im(f) = G/Ker(f) is a kind of result that holds for groups, rings, fields, vector spaces, modules, et c. et c. and every time you encounter it, you need a new proof of the fact.
So, what would be the obvious thing to do? Generalize. So, instead of studying groups, rings, fields, et c. all separately, we sit down and study the entity formed by "ALL Xs and nice maps between Xs". This forms a new algebraic entity called a category. And category theory is the theory of these entities.
Now, there is a REAL expert on this floating around here - yes,
pozorvlak, I am talking about you. He actually does HIS PhD on these beasts, while I just use them in my homological algebra PhD quite a lot.Oh, and it turns out that once you nailed down what a category is, surprisingly much turns out to be categories. The big charm with Haskell for me is that you actually program by constructing maps in a specific category of Haskell types. And the programmers crowd uses this metaphor extensively. I like!
conlangs
Constructed languages. Such as Esperanto, Quenya, Sindarin, Klingon, Zanaquen (speling?) et c. I have constructed a few, including some conscripts to join up with them, in my days. My wife did one for her senior year, and I helped with the random-syllable-generation.
At one point, helping Anders Sandberg with an RPG-world construction, I constructed a handful to be spoken by alien races. One based on a heavy arabicized english language fusion. And one constructed for an alien kind of speech apparatus, with a calligraphic script that reflected the oddities of this anatomy.
I used to hang out on the relevant mailing lists too - but that was one or two email adresses ago.
creative destruction
destructive creation
These belong together, really. I think I saw at least one of these in
krfsms interest list, and figured that yeah, it is kinda interesting. The other, obviously, follows by dualizing.koszul duality
Now this is one of the main areas of study in mathematics at Stockholm University. The idea is that for a quadratic graded algebra k[x,...]/I with I in k[x,...]2, you can figure out a different algebra, such that projective resolutions work nicely and map down to Nice Things. In the end, you get all sorts things Just Working if the algebra has good properties (the map from the Koszul dual algebra to the trivial module should be a quasiisomorphism iirc). I have been away for too long to give a great explanation of the intricacies, but the kind of good stuff happening is similar to what A-infinity brings to group cohomology rings - all of a sudden you can recover everything you needed from the extra information provided.
And boy, is this a horribly horribly bad explanation.
tetrapyloctomy
One of four departments of the School of Comparative Irrelevance that Umberto Eco introduces in Foucaults Pendulum. These are tetrapyloctomy (to cut a hair four ways), adynata (also known as impossibilia), potio-section (the art of slicing soup) and oxymoronics. By writing this, I further notice that I had forgotten to add adynata and oxymoronics to the interest list. Hereby fixed.
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no subject
Date: 2007-10-08 08:08 am (UTC)no subject
Date: 2007-10-08 08:09 am (UTC)no subject
Date: 2007-10-08 08:17 am (UTC)autechre
b/h=(d+f)/(d+e)
grim meathook future
punting
spinnwebe
universal algebra
category theory
with the last two in order to give you license to prove the nice things I said about you. :)