What is a Hopf Algebra? 
And what do Hopf Algebras have to do with Combinatorics?

Frank Sottile

TWIGS seminar, University of Massachusetts at Amherst

Abstract: 

   A Hopf algebra is an algebra  whose linear dual is also an algebra,
with   some   compatibility  between   these   two  structures.   This
uninformative definition leads  to an object that pops  up quite often
in mathematics.  One such area  is in combinatorics, where  G.-C. Rota
pointed  out  how combinatorial  objects  naturally  posses some  Hopf
structure.  In fact,  under  this combinatorial  Hopf structure,  many
things that we  would like to count transform  naturally, yielding new
tools  in combinatorics.  This talk  will introduce  you to  some Hopf
algebras,  and explain  how they  arise in  combinatorics.  (And maybe
explain the more pressing question: what *is* combinatorics?)