(Mathematical Physics and Probability Seminar; Math 402)
When
3 – 4 p.m., Oct. 1, 2025
Abstract: This talk is is in two parts.
The first part deals with functors that send label sets to sets of combinatorial structures. There is a calculus of such functors; one can add, multiply, compose, and differentiate. (First example: the functor X^n sends a label set with n elements to the set
of its linear orderings. The derivative of X^n is n X^(n-1).) This part is mainly extracted from the book of Bergeron, Labelle, and Leroux. The notes from the first part will be sent to participants to use as reference for the second part.
The second part explains generalizations of the calculus. This is analogous to going from the calculus of one variable to the calculus of several variables. In statistical physics one is interested in graph expansions. A central topic is cluster expansions,
that is, expansions indexed by connected graphs. The calculus leads to equations that are useful for cluster expansions. A convergence result due to Daniel Ueltschi will be stated but not proved.