Algebra/Topology seminar

Speaker: Takuji Kashiwabara

Title: Generating series method in algebraic topology

Abstract: In algebraic topology, quite often a family of equalities can be expressed as an equality between formal series, called generating series.
Best known example include
Bullett-Macdonald identity for Adem relations, Bisson-Joyal Q-ring formulation
for Adem relations, Ravenel-Wilson main relations in the Hopf ring for complex
oriented cohomology, Turner's relations in the Hopf ring for $QS^0$,
recently generalized to odd prime case by Chon.  It turns out
in many cases, the "formal indeterminate" allows a geometric interpretation,
and in some cases, the generating series themselves have geometric meanings.  In this talk, we discuss such interpretations, and how they make things simpler.

As an original application, we give a (complete) set of relations among the mod 2 unoriented analogue of Miller-Morita-Mumford classes, or $\mu$-classes defined by
Randal-Williams, by an explicit computation.  This last result is taken from
a joint work with Hadi Zare.