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Abstract

The study of totally positive matrices, i.e., matrices with positive minors, dates back to 1930s. The theory was generalised by Lusztig to arbitrary split reductive groups using canonical bases, and has significant impacts on the theory of cluster algebras, higher teichmuller theory, etc. In the first talk, we survey basics of total positivity and explain its generalization to general semifields for Kac-Moody groups. In the second talk, we shall focus on totally positive flag manifolds and discuss various topological properties.