On the K-theory of rigid tensor categories

• Maxime Ramzi (Muenster)

Room: Amphithéâtre Yvonne Choquet-Bruhat A theorem of Deligne guarantees that under some finiteness assumptions, rigid tensor categories over an algebraically closed field admit a fiber functor and are therefore (super-)Tannakian. This, in turn, guarantees that they are relatively close to categories of modules over commutative rings. Beyond the Tannakian case, there is also a general feeling that rigid tensor categories behave "more" like categories of modules over commutative rings than arbitrary tensor categories. In this talk, I will discuss a K-theoretic failure of this "feeling". More precisely, I will give examples to show that the K-theory of rigid tensor categories lacks one key structural property of the K-theory of commutative rings, by exhibiting failures of the so-called redshift principle (which holds for the K-theory of commutative rings). In the first half of the talk, I will focus on describing the context and discuss examples based on Deligne's category Rep(GL_t), and in the second half, I will discuss a general result that fully computes the K-theory of certain "algebraically closed" rigid tensor categories.

Computing quadratic Donaldson-Thomas invariants

• Anna Viergever (Hannover)

Room: Amphithéâtre Yvonne Choquet-Bruhat (Zero-dimensional) Donaldson-Thomas-invariants "count" things like ideal sheaves of a given length which have zero-dimensional support on a smooth projective complex threefold. Maulik, Nekrasov, Okounkov and Pandharipande have proven a formula for the generating series of these Donaldson-Thomas invariants in terms of the MacMahon function in the toric case. We discuss a conjectural quadratically enriched analogue of this result for smooth projective real threefolds satisfying an orientation condition, using a quadratic version of Donaldson-Thomas invariants taking values in Witt rings which are constructed using work of Levine. We provide evidence for the conjecture coming from computations for $\mathbb{P}^3$ and $(\mathbb{P}^1)^3$. This talk is based on my thesis and on joint work with Marc Levine.