Speaker(s):

Speaker(s):

We discuss recent developments on ultra-violet completeness of holomorphic quantum field theories. We discuss two applications. One is about elliptic trace formula for chiral deformations of 2d CFT, and another one on holomorphic Chern-Simons theory in the large N limit.

In the first half of the talk, I shall discuss some new incompressible finite symmetric tensor categories in prime characteristic. These appeared in joint work with Etingof and Ostrik. In the second half, I shall discuss the theory of finite group schemes over a symmetric tensor category, and show how to view the tangent space at the identity as a restricted Lie algebra in a suitable categorical sense.

We propose a new framework of monoidal categorifications of finite ADE types cluster algebras involving triangulated categories instead of abelian categories. Let Q be any Dynkin quiver and let A_Q denote the corresponding finite type cluster algebra (with a suitable choice of frozen variables). We define a certain additive symmetric monoidal category out of the Auslander-Reiten theory of Q and consider its (bounded) homotopy category K_Q. Using some iterated mapping cone technique, we construct a family of chain complexes in bijection with almost positive roots that are characterized by natural exactness conditions. We then prove that the Euler charasteristics of the chain complex associated to any positive root β coincides with the truncated q-character of the simple module categorifying the cluster variable x[β] in A_Q via Hernandez-Leclerc’s monoidal categorification. We also conjecture that all exchange relations in A_Q are categorified by distinguished triangles in K_Q. If time allows, we will discuss potential extensions of this construction to other cluster structures.

TBA

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