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sniffnoySo today was the first day of REU. And there was the discrete math class and there was pizza and there was the random walks class and then there was the, uh, last class.
J. Peter May was teaching it, and he starts off by just putting up on the board a definition and a theorem:
Definition: An n-TQFT (topological quantum field theory) is a symmetric monoidal functor n-Cob→VectK.
Theorem: The category of 2-TQFTs is equivalent to the category of commutative Frobenius K-algebras.
...meaning, of course, that in order to explain any of this he's going to have to start by teaching us all category theory. And so today was basic category theory.
Quotes:
After explaining how a monoid is a category with one object, he suggests that "a category is a monoid with many objects."
He mentions how some people do their function composition the other way around; Big-Haired Steve suggests we call this "co-standard notation".
Also, apparently, in category theory, the plural of "skeleton" is "skeleta".
Meanwhile, it turns out that while Grant has Smash, and Jim (!) has the N64, their apartment did not come with a TV. And none of Grant, Jim, or Amelia brought one. What will they do? We will see.
-Harry
J. Peter May was teaching it, and he starts off by just putting up on the board a definition and a theorem:
Definition: An n-TQFT (topological quantum field theory) is a symmetric monoidal functor n-Cob→VectK.
Theorem: The category of 2-TQFTs is equivalent to the category of commutative Frobenius K-algebras.
...meaning, of course, that in order to explain any of this he's going to have to start by teaching us all category theory. And so today was basic category theory.
Quotes:
After explaining how a monoid is a category with one object, he suggests that "a category is a monoid with many objects."
He mentions how some people do their function composition the other way around; Big-Haired Steve suggests we call this "co-standard notation".
Also, apparently, in category theory, the plural of "skeleton" is "skeleta".
Meanwhile, it turns out that while Grant has Smash, and Jim (!) has the N64, their apartment did not come with a TV. And none of Grant, Jim, or Amelia brought one. What will they do? We will see.
-Harry
no subject
Date: 2007-06-19 11:40 pm (UTC)Also - Laci Babai's class, the discrete math one, is also working towards a Big Theorem, but it's noticeably weirder. So we define, for a group G, α(G) to be the size of the largest product-free set in G (i.e. x,y∈A⇒xy∉A). And apparently, Babai asked, 25 years ago, if there is some c>0 st for any finite group G, α(G)≥c|G|. And apparently, last year, someone finally proved that the answer is no, because if you restrict to G=SL2(p), there's a c st α(G)<c|G|8/9. And that's what we're going to prove, apparently. :P