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Expansions and Quantum Groups
Abstract. I don't understand quantum groups. My talk will be about
my struggle to understand them as "expansions". So I will start with the
definition of an expansion (an isomorphism of the unipotent completion
of a group or another algebraic object with its associated graded space). I
will explain how expansions may be useful (they are about "polynomial
functions" on anything, and they allow for iterative investigation of
"definable properties"). I will give a couple of examples (free groups,
braids, knotted trivalent graphs). Finally I will tell you why I think
the coolest explanation of quantum groups (deformations of universal
enveloping algebras) ought to be given using expansions of virtual knots.
If you don't know what quantum groups are, you'll come out of my
talk still not knowing. If I don't understand quantum groups, neither
shall you. But you will appreciate expansions.
"God created the knots, all else in topology is the work of mortals."