We prove that the (homotopy) hypercommutative algebra structure on the de Rham cohomology of a Poisson or Jacobi manifold defined by several authors is (homotopically) trivial, i.e. it reduces to the underlying (homotopy) commutative algebra structure. We do so by showing that the DG operads which codify the algebraic structure on the de Rham complex of Poisson and Jacobi manifolds, generated by the exterior product and the interior products with the structure polyvector fields, are quasi-isomorphic to the commutative suboperad. Hence, there is no hope to endow the de Rham cohomology of such manifolds with any (higher) structure beyond the well-known (homotopy) commutative algebra structure, which exists for any smooth manifold. We proceed similarly with the commutative Batalin-Vilkovisky-infinity algebra structure on the de Rham complex of a generalized Poisson supermanifold.

@misc{arXiv:2312.07321,
     author = {Guan, Ai and Muro, Fernando},
     title = {Operations on the de {Rham} cohomology of {Poisson} and {Jacobi} manifolds},
     year = {2023},
     howpublished = {Preprint, {arXiv}:2312.07321 [math.{DG}] (2023)},
     url = {https://arxiv.org/abs/2312.07321},
     }

We work over a perfect field. Recent work of the third-named author established a Derived Auslander Correspondence that relates finite-dimensional self-injective algebras that are twisted $3$-periodic to algebraic triangulated categories of finite type. Moreover, the aforementioned work also shows that the latter triangulated categories admit a unique differential graded enhancement. In this article we prove a higher-dimensional version of this result that, given an integer $d\geq1$, relates twisted $(d+2)$-periodic algebras to algebraic triangulated categories with a $d\mathbb{Z}$-cluster tilting object. We also show that the latter triangulated categories admit a unique differential graded enhancement. Our result yields recognition theorems for interesting algebraic triangulated categories, such as the Amiot cluster category of a self-injective quiver with potential in the sense of Herschend and Iyama and, more generally, the Amiot-Guo-Keller cluster category associated with a $d$-representation finite algebra in the sense of Iyama and Oppermann. As an application of our result, we obtain infinitely many triangulated categories with a unique differential graded enhancement that is not strongly unique. In the appendix, B. Keller explains how -- combined with crucial results of August and Hua-Keller -- our main result yields the last key ingredient to prove the Donovan-Wemyss Conjecture in the context of the Homological Minimal Model Program for threefolds.

@misc{arXiv:2208.14413,
     author = {Jasso, Gustavo and Keller, Bernhard and Muro, Fernando},
     title = {The {Derived} {Auslander}-{Iyama} {Correspondence}},
     year = {2022},
     howpublished = {Preprint, {arXiv}:2208.14413 [math.{RT}] (2022)},
     keywords = {18G80,18N40},
     url = {https://arxiv.org/abs/2208.14413},
     }

Several model structures related to the homotopy theory of locally constant factorization algebras are constructed. This answers a question raised by D. Calaque in his habilitation thesis. Our methods also solve a problem related to cosheafification and factorization algebras identified by O. Gwilliam - K. Rejzner in the locally constant case.

@misc{arXiv:2107.14174,
     author = {Carmona, Victor and Flores, Ramon and Muro, Fernando},
     title = {A model structure for locally constant factorization algebras},
     year = {2021},
     howpublished = {Preprint, {arXiv}:2107.14174 [math.{AT}] (2021)},
     url = {https://arxiv.org/abs/2107.14174},
     }