Explore a 51-minute lecture on vertex algebras and quantum groups with large centers, presented by Simon Lentner at the Centre de recherches mathématiques (CRM). Delve into joint work with B. Feigin, focusing on G-graded tensor categories arising from finite-dimensional semisimple Lie algebras and integers. Examine explicit answers for cases p=1 and g=sl2, p=2. Investigate deformable families of vertex algebras related to quantum geometric Langlands correspondence, their large centers at deformation limits, and their relationship to affine Lie algebras at critical level. Learn about the Feigin-Tipunin vertex algebra and its twisted modules. Discover the quantum group associated with G at a 2p-th root of unity, its infinite center version, and the small quantum group. Understand the logarithmic Kazhdan-Lusztig conjecture and its implications for braided tensor categories of representations.