Interests meme
Meme, as invited by michiexile.
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.
The seven that Michi picked for me were autechre, b/h=(d+f)/(d+e), grim meathook future, punting, spinnwebe, universal algebra, and category theory.
autechre
Like a lot of the entries in my interests list, Autechre are a band. Specifically, they're a British electronica act signed to the legendary Warp records, and they're considered (according to Wikipedia, at least) to be one of the driving forces behind the development of Intelligent Dance Music (IDM). They've been known to say that given the incredible range of tools available to modern composers, especially in the electronic genres, it is incomprehensible that any band should "sound like" any other band. Apart from the music, I love the names they give their tracks and albums, which hover on the edge of meaning - "chiastic slide", "kalpol introl", "incunabula1", "ipacial section". Like a lot of great music, I encountered them as part of a large batch that dynix lent me back in 2004. As I recall, there were several CDs of "R&B", "general chart stuff" and the like, and a couple simply labelled "weirdness". "You might not like those ones," she said, "they're not to most people's tastes." Needless to say, I loved almost everything on them.
I went to see Autechre at the Glasgow School of Art bar with wormwood_pearl a couple of years ago. This was on the tour accompanying the release of their album "Untilted". Hearing them through a proper venue sound-sysem was... physically painful. A couple of my friends suffer from tinnitus and hearing problems owing to overexposure to loud music, so I'm a big fan of earplugs, but in this case it didn't help: the bass was transmitted up through the floor and resonated up your skeleton. Great gig, though. I particularly liked the way they used the treble range to carry the rhythm and the bass to carry the melody. I still have the black T-shirt saying "autechre" in black writing that I bought there, and most of the plastic shot glasses we stole from the bar :-)
Interestingly enough, my supervisor is also a fan...
b/h=(d+f)/(d+e)
This is Shannon's First Theorem (yes, that Shannon). It's the first result to be discovered in the mathematical theory of juggling. b is the number of balls; h is the number of hands; d is the "dwell time", the time the ball spends in the hand between catching and releasing; f is the "flight time", the time the ball spends in the air between between being thrown and being caught again; and e is the time each hand spends empty between throwing one ball and catching the next. The theorem applies to so-called "uniform juggling", in which h, b, d, e and f are all constant - this is true for the cascades and fountains, and many other patterns, but certainly not for all jugglable siteswaps. There's a nice duality between (b,f) and (h,e) - in fact, swapping "balls" and "hands" and "flight" with "dwell" in any true statement about uniform juggling will yield another true statement. I first came across this theorem when it was put on the T-shirt for the 2005 Scottish Juggling convention: the idea of listing it as an interest was shamelessly stolen from azrelle.
grim meathook future
I've blogged about this before. To quote Joshua Ellis:The upshot of all of this is that the Future gets divided; the cute, insulated future that Joi Ito and Cory Doctorow and you and I inhabit, and the grim meathook future that most of the world is facing, in which they watch their squats and under-developed fields get turned into a giant game of Counterstrike between crazy faith-ridden jihadist motherfuckers and crazy faith-ridden American redneck motherfuckers, each doing their best to turn the entire world into one type of fascist nightmare or another.
punting
The sport of kings. Take a flat-bottomed boat and a long pole. Stand at the end (on the sloping bit, no matter what those degenerates at Cambridge think). Fill the middle of the boat with cushions and your friends. Using the pole, push the boat (or "punt") forward, sliding the pole along the side of the punt to ensure straightness. Hang the pole in the river for a while and use it as a rudder. Flick the pole up so your wrist's near the middle, swiping the water off the pole with your loosely-circling fingers as you do so. Position the pole, drop into the river through the aforementioned circled fingers, catch before it drops out of your hands, push. Repeat. Serve with a dose of sunshine, good company, and Pimm's no. 1. I did almost no punting in my first three years at Oxford, but in my fourth year my Finals finished half-way through term, so I had nearly four weeks and very little better to do with my time. I got quite good at punting that year :-)
spinnwebe
A website run by Greg "spinn" Galcik. Home of the late lamented Dysfunctional Family Circus.
universal algebra
category theory
My PhD topic lies somewhere in the middle of these two. Both are attempts to discover the underlying structure of mathematics: category theory is the younger, more surprising, and more general of the two. There's a certain amount of overlap, and a lot of interplay: most modern approaches to universal algebra make use of categorical constructions like Lawvere theories, monads and operads, and many important theorems of category theory are inspired by universal algebra, or use universal-algebraic ideas in their proofs.
The basic idea behind universal algebra is that most of the usual examples of algebraic structures (groups, rings, Lie algebras, monoids...) have a standard form: some collection of sets, some set of operations taking n-tuples of elements of your sets to other members of your sets, and some equations that these operations have to obey. We can study these (sets, operations, equations) triples as objects in their own right, and (given this definition), we can define what we mean by a model for a structure, a homomorphism of models, submodels, products, etc. The Noether isomorphism theorems (im f = G/ker f, etc) can all be proved at this level of generality.
Category theory is more subtle. The essential idea is that the transformations between objects are (at least) as important as the objects themselves. A category is a directed graph with a notion of "associative composition": a path of arrows A → B → ... Z can be "composed" into a single arrow A → Z in a unique way. The motivating example is the case where the vertices of your graph are the objects of interest in some mathematical field (models for an algebraic structure, possibly), and the edges are the appropriate structure-preserving transformations. But there are plenty of other examples: sets, groups and posets are all examples of categories. Taking our idea that transformations are as important as objects seriously, we should think about transformations between categories, called functors (which are the obvious thing: graph morphisms that preserve composition), and this gives us a way of thinking about connections between different branches of mathematics. And transformations between functors give us a way of comparing different connections between different branches of mathematics...
Category theory has shown itself useful in some unexpected places: though it's about as abstract as it's currently possible for mathematics to get, it's indispensable in much of computer science and some physicists (notably John Baez) are starting to apply it to quantum gravity. There have also been applications of category theory to neuroscience, the study of ecosystems, and to philosophy. In mathematics, it's allowed us to discover some fundamental structures (such as adjunctions, limits and colimits) that we couldn't see before because of the mass of obscuring detail and the lack of bridges between different subjects. The lack of obscuring detail also means that purely categorical proofs can be wonderfully simple - it's sometimes said that the real aim of category theory is to make all theorems not only trivially true, but also trivially trivial.
1 Yes, I know what "incunabula" means. But that's just it: most people don't (I certainly didn't when I first heard the album), and it's wonderful as a semi-nonsense sound. When you learn what it means, you see what a fantastic name it is for a band's first album :-)
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.
The seven that Michi picked for me were autechre, b/h=(d+f)/(d+e), grim meathook future, punting, spinnwebe, universal algebra, and category theory.
autechre
Like a lot of the entries in my interests list, Autechre are a band. Specifically, they're a British electronica act signed to the legendary Warp records, and they're considered (according to Wikipedia, at least) to be one of the driving forces behind the development of Intelligent Dance Music (IDM). They've been known to say that given the incredible range of tools available to modern composers, especially in the electronic genres, it is incomprehensible that any band should "sound like" any other band. Apart from the music, I love the names they give their tracks and albums, which hover on the edge of meaning - "chiastic slide", "kalpol introl", "incunabula1", "ipacial section". Like a lot of great music, I encountered them as part of a large batch that dynix lent me back in 2004. As I recall, there were several CDs of "R&B", "general chart stuff" and the like, and a couple simply labelled "weirdness". "You might not like those ones," she said, "they're not to most people's tastes." Needless to say, I loved almost everything on them.
I went to see Autechre at the Glasgow School of Art bar with wormwood_pearl a couple of years ago. This was on the tour accompanying the release of their album "Untilted". Hearing them through a proper venue sound-sysem was... physically painful. A couple of my friends suffer from tinnitus and hearing problems owing to overexposure to loud music, so I'm a big fan of earplugs, but in this case it didn't help: the bass was transmitted up through the floor and resonated up your skeleton. Great gig, though. I particularly liked the way they used the treble range to carry the rhythm and the bass to carry the melody. I still have the black T-shirt saying "autechre" in black writing that I bought there, and most of the plastic shot glasses we stole from the bar :-)
Interestingly enough, my supervisor is also a fan...
b/h=(d+f)/(d+e)
This is Shannon's First Theorem (yes, that Shannon). It's the first result to be discovered in the mathematical theory of juggling. b is the number of balls; h is the number of hands; d is the "dwell time", the time the ball spends in the hand between catching and releasing; f is the "flight time", the time the ball spends in the air between between being thrown and being caught again; and e is the time each hand spends empty between throwing one ball and catching the next. The theorem applies to so-called "uniform juggling", in which h, b, d, e and f are all constant - this is true for the cascades and fountains, and many other patterns, but certainly not for all jugglable siteswaps. There's a nice duality between (b,f) and (h,e) - in fact, swapping "balls" and "hands" and "flight" with "dwell" in any true statement about uniform juggling will yield another true statement. I first came across this theorem when it was put on the T-shirt for the 2005 Scottish Juggling convention: the idea of listing it as an interest was shamelessly stolen from azrelle.
grim meathook future
I've blogged about this before. To quote Joshua Ellis:The upshot of all of this is that the Future gets divided; the cute, insulated future that Joi Ito and Cory Doctorow and you and I inhabit, and the grim meathook future that most of the world is facing, in which they watch their squats and under-developed fields get turned into a giant game of Counterstrike between crazy faith-ridden jihadist motherfuckers and crazy faith-ridden American redneck motherfuckers, each doing their best to turn the entire world into one type of fascist nightmare or another.
punting
The sport of kings. Take a flat-bottomed boat and a long pole. Stand at the end (on the sloping bit, no matter what those degenerates at Cambridge think). Fill the middle of the boat with cushions and your friends. Using the pole, push the boat (or "punt") forward, sliding the pole along the side of the punt to ensure straightness. Hang the pole in the river for a while and use it as a rudder. Flick the pole up so your wrist's near the middle, swiping the water off the pole with your loosely-circling fingers as you do so. Position the pole, drop into the river through the aforementioned circled fingers, catch before it drops out of your hands, push. Repeat. Serve with a dose of sunshine, good company, and Pimm's no. 1. I did almost no punting in my first three years at Oxford, but in my fourth year my Finals finished half-way through term, so I had nearly four weeks and very little better to do with my time. I got quite good at punting that year :-)
spinnwebe
A website run by Greg "spinn" Galcik. Home of the late lamented Dysfunctional Family Circus.
universal algebra
category theory
My PhD topic lies somewhere in the middle of these two. Both are attempts to discover the underlying structure of mathematics: category theory is the younger, more surprising, and more general of the two. There's a certain amount of overlap, and a lot of interplay: most modern approaches to universal algebra make use of categorical constructions like Lawvere theories, monads and operads, and many important theorems of category theory are inspired by universal algebra, or use universal-algebraic ideas in their proofs.
The basic idea behind universal algebra is that most of the usual examples of algebraic structures (groups, rings, Lie algebras, monoids...) have a standard form: some collection of sets, some set of operations taking n-tuples of elements of your sets to other members of your sets, and some equations that these operations have to obey. We can study these (sets, operations, equations) triples as objects in their own right, and (given this definition), we can define what we mean by a model for a structure, a homomorphism of models, submodels, products, etc. The Noether isomorphism theorems (im f = G/ker f, etc) can all be proved at this level of generality.
Category theory is more subtle. The essential idea is that the transformations between objects are (at least) as important as the objects themselves. A category is a directed graph with a notion of "associative composition": a path of arrows A → B → ... Z can be "composed" into a single arrow A → Z in a unique way. The motivating example is the case where the vertices of your graph are the objects of interest in some mathematical field (models for an algebraic structure, possibly), and the edges are the appropriate structure-preserving transformations. But there are plenty of other examples: sets, groups and posets are all examples of categories. Taking our idea that transformations are as important as objects seriously, we should think about transformations between categories, called functors (which are the obvious thing: graph morphisms that preserve composition), and this gives us a way of thinking about connections between different branches of mathematics. And transformations between functors give us a way of comparing different connections between different branches of mathematics...
Category theory has shown itself useful in some unexpected places: though it's about as abstract as it's currently possible for mathematics to get, it's indispensable in much of computer science and some physicists (notably John Baez) are starting to apply it to quantum gravity. There have also been applications of category theory to neuroscience, the study of ecosystems, and to philosophy. In mathematics, it's allowed us to discover some fundamental structures (such as adjunctions, limits and colimits) that we couldn't see before because of the mass of obscuring detail and the lack of bridges between different subjects. The lack of obscuring detail also means that purely categorical proofs can be wonderfully simple - it's sometimes said that the real aim of category theory is to make all theorems not only trivially true, but also trivially trivial.
1 Yes, I know what "incunabula" means. But that's just it: most people don't (I certainly didn't when I first heard the album), and it's wonderful as a semi-nonsense sound. When you learn what it means, you see what a fantastic name it is for a band's first album :-)