Mathematics Colloquium | Amritanshu Prasad
September 14 @ 4:30 pm - 5:30 pm
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Title: Generating Functions Associated to Species of Structures
Abstract: Species of structures were introduced by André Joyal and his group in Québec in the 1980s. They provide a way of organizing classes of labeled combinatorial objects that elevate the art of studying their generating functions to a science.
Combinatorial relationships realized bijectively among such classes are transformed into functional relationships of their generating functions. For example, from the combinatorial interpretation of a set partition as a set of non-empty sets, the exponential generating function for Bell numbers exp(exp(z)-1) becomes blindingly clear; exp(z) is the generating function of sets and exp(z)-1 that of non-empty ones.
I will discuss species of structures and some generating functions that are associated with them. I will explain how algebraic operations on generating functions can be seen to arise from set-theoretic operations on species. I will introduce the Frobenius characteristic generating function of a species of structures, which is a simple variation of the cycle index generating function, landing us in the world of symmetric polynomials.
About the Speaker: Amritanshu Prasad is a Professor of Mathematics in The Institute of Mathematical Sciences (IMSc), Chennai. His research interests are in Combinatorics and Representation theory. He is the recipient of many honors, including the Swarnajayanti Fellowship, DST, (2014-15), and the Young Scientist Medal, Indian National Science Academy (2010). He is a Fellow of the Indian Academy of Sciences. He is the author of the book Representation Theory: A Combinatorial Viewpoint, Cambridge University Press (2015).
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