Abstract: Semialgebraic and subanalytic sets have become ubiquitous in mathematics since their introduction by Lojasiewicz in the 1960s, following the famous Tarski-Seidenberg theorem on quantifier elimination. I will discuss two long-standing questions in real-analytic geometry, on global smoothing of a subanalytic set (an analogue of resolution of singularities), and on transformation of a proper real-analytic mapping to a mapping with equidimensional fibres by global blowings-up of the target (a classical result in the complex-analytic case).

These questions are related: a positive answer to the second can be used to reduce the first to the simpler semianalytic case. It turns out that the second question has a negative answer, in general, and the first nevertheless has a positive solution.

Speaker's web page: http://www.math.toronto.edu/bierston/