Abstract.We will discuss some relatively recent work concerning asymptotic and exact formulas for the irreducible characters of the symmetric group. Two different asymptotic results, due to Vershik-Kerov and Biane, will be given. The work of Biane is related to Kerov's "universal character polynomial" which Kerov defined in a lecture shortly before his death. A new formula for irreducible characters of rectangular shape yields some information about Kerov's universal character polynomial.Abstract: Almost a year ago, studying phenomenologically interesting Yang-Mills theory in four dimensions, M. Atiyah, J. Maldacena and C. Vafa conjectured existence of a certain non-compact manifold of G2 holonomy whose geometry encodes dynamics of SU(N) gauge theory at large N.  Later, this conjecture was extended by Vafa et al to a large class of G2 metrics.  In the seminar, we explicitly construct cohomogeneity one G2 metrics with the proposed properties, and explain a relation to variational problems in classical mechanics, which play an important role in this construction.  The approach can be also extended to the newly discovered metrics of Spin(7) holonomy.