A gentle introduction to the Minimal Model Program
In every branch of mathematics, the classification of objects up to some sort of equivalence relation arises naturally as a guiding problem and an ultimate understanding of the subject. In algebraic geometry, the objects are the solutions of systems of polynomial equations, known as algebraic varieties, and relations are maps and morphisms between them. A very successful tool in the classification of varieties is a program started in dimension three by Shigefumi Mori (Fields medal in 1990), the so-called Minimal Model Program. I plan to give a very gentle introduction to the subject, focusing on recent exciting developments that were recognised with the 2017 Breakthrough prize to Hacon and McKernan and the 2018 Fields medal to Caucher Birkar. Both achievements revolve around the problem of understanding how varieties behave in families and whether these families are bounded. If time permits, I will explain what the term `bounded' means and why it is so important.
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