τ-tilting theory was introduced by Adachi, Iyama and Reiten, which completes the classical tilting theory from the viewpoint of mutation. They constructed a class of Λ modules named (support) τ-tilting modules, where Λ is a finite dimensional basic algebra over an algebraically closed field. We call Λ a τ-tilting finite algebra if there are finitely many isomorphism classes of basic τ-tilting Λ-modules. In this talk, we explain τ-tilting theory by discussing the τ-tilting finiteness of two-point algebras. More precisely ,we will show the poset structure on the set of support τ-tilting modules, the bijection between support τ-tilting modules and two-term silting complexes, etc,.

在本报告中，我将给出九维流形上存在切触结构的充分必要条件。该结果是与墨尔本大学Diarmuid Crowley合作完成的。

Hamilton's conjecture in the Ricci flow says that the scalar curvature of the finite-time Ricci flow blows up. In this talk under a curvature condition we give a proof of this conjecture.

Kahler tensor calculus could be formulated in terms of directed graphs. We use to it explore the algebraic relations of invariant differential operators on bounded symmetric domains. I will also talk about the Feynman diagram formulas for the coefficients in the asymptotic expansion of Bergman and heat kernels.

We develop a Fulton-style intersection theory in motivic homotopy categories, which is expressed in terms of the fundamental class in twisted bivariant groups. By introducing a general notion of Euler classes of vector bundles, we prove a formula of excess intersection in this framework; as an application, we deduce a motivic Gauss-Bonnet formula computing the Euler characteristic. This is a joint work with F. Déglise and A. Khan.

In this talk, we report the results of robust tensor completion using tubal singular value decomposition, and its applications. Several applications and theoretical results are discussed. Numerical examples are also presented for demonstration.

xVA is a collection of valuation adjustments made to the classical risk-neutral valuation of a derivative or derivatives portfolio for pricing or for accounting purposes, and it has been a matter of debate and controversy. This presentation is intended to clarify the notion of xVA as well as the usage of the xVA items in pricing, accounting or risk management. Based on replication pricing using shares and credit default swaps, we attribute the P/&L of a derivatives trade into the compensation for counterparty default risks and the costs of funding. The expected present values of the compensation and the funding costs under the risk-neutral measure are defined to be the bilateral CVA and FVA, respectively. The latter further breaks down into FCA, MVA, ColVA and KVA. We show that the market funding liquidity risk, but not any idiosyncratic funding risks,can be bilaterally priced into a derivative trade, without causing price asymmetry between the counterparties.

We investigate the optimal sample complexity of recovering a general high dimensional smooth and sparse function based on point queries. Our result provides a precise characterization of the potential loss in information when restricting to point queries as opposed to the more general linear queries., as well as the benefit of adaption. In addition, we also developed a general framework for function approximation to mitigate the curse of dimensionality that can also be easily adapted to incorporate further structure such as lower order interactions, leading to sample complexities better than those obtained earlier in the literature.