MiGraRe

MiGraRe aims at using homotopical methods to attack several recent and very recent conjectures in geometric representation theory. Part of these homotopical methods are well established, and involve the use of derived geometry and stratified homotopy theory to study Yangians of curves and surfaces, new exciting objects that are currently emerging at the research frontiers of representation theory. In this respect, the goal is use homotopical methods to settle several conjectures introduced in Schiffmann’s ICM 2018, among which shines the analog of Kac’s positivity conjecture for Higgs bundles. A different set of homotopical methods will instead have to be developed from scratch, starting from the recent theory of derived foliations developed by Toën-Vezzosi, and extending it to a new, far-reaching generalisation of derived algebraic geometry. Concretely, we will enlarge the class of test schemes, passing from commutative differential graded algebras to mixed graded algebras. In doing so, we aim at discovering new algebraic structures on classical moduli spaces, and in particular on the one parametrising 2-periodic complexes of representations of a fixed quiver. This example is fundamental in Bridgeland’s work on the realisation of the whole quantum group of Dynkin quivers, and one of MiGraRe’s main objectives is to lift this result from classical Hall algebras to cohomological ones. This would in particular lead to a completely new way of describing the whole Maulik-Okounkov Yangians and provide a new approach to attack a conjecture voiced by Okounkov, predicting a Lie theoretic interpretation for the coefficients of the Kac’s polynomial of a quiver.