Differential and topological problems in modern theoretical physics

s of the minicourses Kenichi Konishi — Università di Pisa Advent of non-Abelian vortices Generalizing the well-known Abrikosov-Nielsen-Olesen vortex solution in the Abelian Higgs model to non-Abelian gauge theories, vortex solutions carrying continuous zeromodes (vortex moduli space) were constructed in the year 2003 inspired by supersymmetric SU(N) gauge theories, triggering an intense research activity. The vast set of new results obtained since then cover the issues of non-Abelian monopoles, vortex dynamics and their relation to the gauge dynamics in 4 dimensions, study of higher-winding solutions and their moduli space, semilocal vortex solutions in theories with larger number of matter multiplets, stability of non-BPS vortex solutions, vortex solutions in gauge theories based on a more general gauge groups such as SO(N), USp(2N), etc., and the fractional vortex and sigma-model lump. Some of the most salient features of this exciting development will be discussed in these lectures. Paul Michael Sutcliffe — University of Durham Topological solitons and nuclei Skyrmions are topological soliton solutions of a generalized harmonic map equation from 3-dimensional Euclidean space into SU(2). Physically the model describes a nonlinear theory of pions in which the soliton is interpreted as the nucleon. Nuclei are then modelled by multi-soliton solutions, with an identification between the number of solitons and nucleons. Results will be present on multi-soliton solutions and their symmetries: which play an important role in quantization. Rational maps between Riemann spheres will be used to provide some understanding of these results. If time permits, a connection between Skyrmions and Yang-Mills instantons will be discussed.