This week's seminars
Pure Maths Colloquium: Keep calm and build objects in order
18 February 2019 16:00 in CM219
In the sixties, when Mumford introduced in algebraic geometry the so-called geometric invariant theory, he showed that, under the right (stability) conditions, some times we do not have to prove something for all objects, but only for those that are stables. Some years later, Harder and Narasimhan showed that every vector bundle of an algebraic curve can be built using the stable objects following the order induced by the slope function. Nowadays, the stability conditions have been adapted to multitude of branches of mathematics and often one can find a theorem which is an adaptation of the result of Harder and Narasimhan to each particular environment.
In this talk we will recall the definition of a torsion pair and we will introduce the indexed chains of torsion classes. Our main theorem is that every indexed chain of torsion classes induce a Harder-Narasimhan filtration. The result for stability conditions becomes then a particular case of our theorem.
After that, we will follow ideas of Bridgeland to show the existence of a (pseudo)metric in the set of indexed chain of torsion classes. Implying that all indexed chains of torsion classes form a topological space.
No previous knowledge on abelian categories nor stability condition will be assumed.
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