During the project I and the other members of the project research group worked on applications of cohomological Donaldson-Thomas theory in cluster algebras, geometric representation theory and algebraic geometry.
Cluster algebras:
The idea of using scattering diagram techniques to prove positivity results in cluster algebras had already had spectacular success in the work of Gross, Hacking Keel and Kontsevich, who managed to prove both the positivity and strong positivity conjectures in cluster algebras using these techniques. By combining my earlier work with Sven Meinhardt on the integrality conjecture in cohomological Donaldson-Thomas theory with the quantum scattering diagram theory that Travis is an expert in, we were able to prove the quantum strong positivity conjecture in the preprint "Strong positivity for quantum theta bases of quantum cluster algebras", published in Inventiones Mathematicae in 2021
Nonabelian Hodge theory:
One of my main goals here was to prove a version of nonabelian Hodge theory for moduli stacks of objects, where we no longer identify non-isomorphic objects, and in addition we remember the data of automorphism groups of objects. This was finally achieved in joint work with team members Lucien Hennecart and Sebastian Schlegel Mejia, resulting in the preprint "BPS Lie algebras for totally negative 2-Calabi-Yau categories and nonabelian Hodge theory for stacks", uploaded to the Arxiv in 2022.
2-Calabi-Yau categories:
In the preprint "Purity and 2-Calabi-Yau categories", uploaded to the Arxiv in 2021, my main result is that the celebrated decomposition of Beilinson, Bernstein, Deligne and Gabber is true for the morphism from the moduli stack of objects in a 2-Calabi-Yau category to its coarse moduli space. This has deep implications for the study of the Borel-Moore homology of moduli stacks of objects in these categories, which I have exploited in a subsequent series of papers dedicated to geometric representation theory with team members Lucien Hennecart and Sebastian Schlegel Mejia.
Algebraic geometry:
With Francesca Carocci we have been working on proving the strong rationality conjecture via the approach outlined in my preprint "Refined invariants of finite-dimensional Jacobi algebras", published on the Arxiv in March 2019. So far we have succeeded in producing numerous constructions relating the vanishing cycle cohomology of Pandharipande-Thomas moduli spaces to the geometric representation theory of CoHAs, and in future work we intend to use these constructions to complete this project.
Somehow, even though there are no compact curves in complex three dimensional space A^3, the approach outlined above makes a strong prediction for the (affinized) BPS Lie algebra associated to the category of compactly supported sheaves on A^3, which controls the entire cohomological DT theory of this threefold. Defining what all this means, as well as verifying this prediction, was the result obtained in my paper Affine BPS algebras, W algebras, and the cohomological Hall algebra of A^2.
