12 June 2018:
1) Today in our network theory seminar Joseph is talking about his latest paper:
Abstract. A network model is an algebraic formalism for modeling the construction of complex networks, in particular, communications networks. Previous constructions only produced relatively commutative network models. Certain types of networks demand a more expressive model. After reviewing the basic theory of network models, we construct the free network model on a given monoid. This construction utilizes various concepts, namely algebraic varieties, generalized graph products, and Kneser graphs. We will then realize graphs of bounded degree as an algebra of a noncommutative network model.
2) I forgot to mention this talk by Christian on May 4th in the grad student seminar! He also gave a related talk in our seminar, which I may also have forgotten to mention. Sorry!
- Categorical computation - form and content
Abstract. There is a duality of syntax and semantics – the form of a theory and the content of a model. This is a fundamental idea in category theory, which was introduced by William Lawvere in his 1963 PhD thesis. The notion of Lawvere theory provides an understanding of algebraic structures independent of presentation, improving upon the set-theoretic universal algebra. Soon after, these theories were proven equivalent to monads, the categorical manifestation of duality, through which the algebras of the monad correspond to models of the theory. Theories and monads provide complementary perspectives of algebraic structures, and both are becoming important to theoretical and practical computer science. We discuss the application to distributed computation, where enriched Lawvere theories can be used to create languages, programs, and data structures which have their operational semantics — the ways they can operate in context — integrated into their definition, effecting sound design of software.
3) I've been continuing my online course on applied category theory
. Apart from this nothing much has been finished, as far as I can tell. I spent the last two weeks doing massive revisions on my paper with Brendan, "A compositional framework for passive linear circuits". Brendan spent the last couple weeks writing a draft of a paper with David Spivak, "Constructing hypergraph categories" - and this week he's gonna help me finish our paper. I'm starting to work with the ACT2018 students on a paper "Biochemical coupling through emergent conservation laws". Also, Greg Egan and I solved a fun puzzle:
If P: Set -> Set is the covariant power set functor, is there a way to define a natural transformation m: P^2 P^2 => P^2 that is associative?
It's well-known that P is a monad. It's recently been shown that P^2 cannot be made into a monad, but the proof of that didn't settle the above puzzle.
I got interested in the puzzle partially just because it's sort of mind-boggling: if the answer were "yes", we’d have a natural way to take a set of sets of sets of sets and turn it into a set of sets in such a way that the two most obvious resulting ways to turn a set of sets of sets of sets of sets of sets into a set of sets agree!
You can see the answer in the discussion here:
It turns out to be quite pretty. BUT, I want to keep finishing up papers!