Guest lectures and seminars - Page 92

Kurusch Ebrahimi-Fard (NTNU) will give a talk with title: Moment-cumulant relations in noncommutative probability and shuffle-exponentials

Abstract: In this talk we consider monotone, free, and boolean moment-cumulant relations from the shuffle algebra viewpoint. Cumulants are described as infinitesimal characters over a particular combinatorial Hopf algebra, which is neither commutative nor cocommutative. As a result the moment-cumulant relations can be encoded in terms of shuffle and half-shuffle exponentials. These shuffle exponentials and the corresponding logarithms permit to express monotone, free, and boolean cumulants in terms of each other using the pre-Lie Magnus expansion. If time permits we will revisit additive convolution in monotone, free and boolean probability and related aspects. Based on joint work with F. Patras (CNRS).

In this second talk I will prove the local slice theorem and give examples of applications, discuss compactness properties of instanton moduli spaces, and explain the definition and some properties of instanton homology.

In their book "Riemann-Roch Algebra", Fulton and Lang give an account of Chern classes in lambda-rings and a general version of Grothendieck's Riemann-Roch theorem. Their definition of Chern classes is based on the additive formal group law.  In work on connective K-theory, Greenlees and I have given an account of Chern classes in lambda-rings based on the multiplicative formal group law.  This account has an evident generalization to any formal group law.  The course will be an attempt to carry out Fulton and Lang's program in this more general setting.  Hoped for applications include generalizations of results relating rational lambda-modules to twisted Dirichlet characters. ---