Even though it's 8:30 in the morning here, I'd kill for that burger.
which green hallucinogenic nectar of mathematics would this be?
+1 - curious as well.
The course is on category theory, but we'll be emphasizing abelian categories and introducing (n,k)-categories and (∞,1)-categories early on. (Category theory is notable for many things, including being called 'general, abstract nonsense' by other pure mathematicians and even many of its own practitioners.)
The subject is very new - category theory was first developed in a 1945 paper and developed heavily, mainly by algebraic geometers and algebraic topologists, over the next couple of decades. At some point one geometer (Grothendieck) came up with a sufficiently general redefinition of *practically everything* that another pair of mathematicians (Lawvere and Tierney, who seemed to be algebro-topologically minded but with mathematical logic training) generalized further into a concept that can encode just about everything we would expect a 'universe' to have.
For a quick summary of just how absurdly powerful (and scary) the language can be... algebraic geometry began as the study of the simplest possible invariants (relatively simple algebraic quantities) of the simplest possible curves and surfaces (those defined by polynomial equations - the circle, for example, which is cut out by x^2+y^2=1 in the plane), and Grothendieck turned it into a monster that seems bent on (and capable of) taking over almost all other mathematics by force. He threw out the curves and surfaces themselves, replacing them with purely algebraic devices representing them (ideals of a ring, which can be thought of as 'generalized numbers' or 'generalized functions') and then generalizing even that viewpoint by considering an individual object in terms of *every mapping from any other object into it*. (If this does not sound insane, you might just be a potential algebraic geometer. Run.) In order to handle the new invariants that were now within easy reach, he also had to replace the usual notion of a space with what is now called a Grothendieck topos, and these were the first examples of what are called elementary topoi - things that have enough structure and depth to act as universes in which mathematics can be done. From what I understand, Grothendieck is now living in a small hut in the Pyrenees, where he alternates between sanity and coming up with ideas like 'eating nothing but dandelion soup' (fortunately prevented by the villagers near him) and selling his own... erm, fertilizer.
Jeff
The course is on category theory, but we'll be emphasizing abelian categories and introducing (n,k)-categories and (∞,1)-categories early on. (Category theory is notable for many things, including being called 'general, abstract nonsense' by other pure mathematicians and even many of its own practitioners.)
The subject is very new - category theory was first developed in a 1945 paper and developed heavily, mainly by algebraic geometers and algebraic topologists, over the next couple of decades. At some point one geometer (Grothendieck) came up with a sufficiently general redefinition of *practically everything* that another pair of mathematicians (Lawvere and Tierney, who seemed to be algebro-topologically minded but with mathematical logic training) generalized further into a concept that can encode just about everything we would expect a 'universe' to have.
For a quick summary of just how absurdly powerful (and scary) the language can be... algebraic geometry began as the study of the simplest possible invariants (relatively simple algebraic quantities) of the simplest possible curves and surfaces (those defined by polynomial equations - the circle, for example, which is cut out by x^2+y^2=1 in the plane), and Grothendieck turned it into a monster that seems bent on (and capable of) taking over almost all other mathematics by force. He threw out the curves and surfaces themselves, replacing them with purely algebraic devices representing them (ideals of a ring, which can be thought of as 'generalized numbers' or 'generalized functions') and then generalizing even that viewpoint by considering an individual object in terms of *every mapping from any other object into it*. (If this does not sound insane, you might just be a potential algebraic geometer. Run.) In order to handle the new invariants that were now within easy reach, he also had to replace the usual notion of a space with what is now called a Grothendieck topos, and these were the first examples of what are called elementary topoi - things that have enough structure and depth to act as universes in which mathematics can be done. From what I understand, Grothendieck is now living in a small hut in the Pyrenees, where he alternates between sanity and coming up with ideas like 'eating nothing but dandelion soup' (fortunately prevented by the villagers near him) and selling his own... erm, fertilizer.
Jeff
It's nice thinking super clearly and actually being able to explain things the way I want to and in a coherent, organized manner. Not 'grabbing' odd, or the wrong words in an attempt to make my point (which projects an "I'm stoopid!" image, which I am not, I have had a great education thanks to my employer). Things actually make sense to others when I'm talking with people about science, religion, psychology and philosophy now, and not just to myself because I'm trying to find the words and phrases I need to explain something cogently.