Professor Jochen Heinloth who is visiting the School of Mathematics is giving a series of four lectures. The details of his lecture series can be accessed here mathweb.tifr.res.in/sites/default/files/conferences/lectures/ls-JochenH.pdf
Title: Finding good moduli spaces for algebraic stacks
Abstract: Algebraic stacks formalize the interesting geometric structure of moduli problems, i.e. universal parameter spaces for objects of a given type. For most such moduli problems the geometry of these objects quickly explains obstructions for the spaces to be varieties, but in many examples notions of stability have been found that cut out open pieces of the moduli problem that admit well-behaved good moduli spaces. Classically these stability conditions are often defined either through GIT or an explicit notion of Harder-Narasimhan-filtrations. In joint work with Alper and Halpern-Leistner we used a local structure theorem for algebraic stacks to give valuative criteria that characterize those stacks which admit good moduli spaces. In the lectures we will start by motivating the criteria in some classical examples and then try to explain the general result.
At the end I would like to explain how these can be used to prove a conjecture of Bialynicki-Birula and Sommese that predicted a classification of open subsets of quotients by torus actions that admit proper moduli spaces. Interestingly this classification contains many non-projective examples which cannot be explained by a stability condition coming from GIT.
Professor Emmanuelle Macri will give a series of three lectures via zoom on Tuesday-Wednesday-Thursday 2:30-3:30. The title and abstract of his lecture series is the following:
Title: Stable bundles on hyper-Kähler manifolds
Abstract: The aim of this course is to present some recent advances in the theory of stable sheaves on higher dimensional hyper-Kähler manifolds.
In the first part we will review Mukai's theory on K3 surfaces.
We will then consider hyper-Kähler fourfolds, by presenting in this setting general results due to O'Grady, Markman, Beckmann, and Bottini.