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.

The Yamabe equation is an important class of equations in the geometric analysis and nonlinear elliptic equations. It arises from the conformal geometry and describes the scalar curvatures of the conformal metrics. In this talk, we study asymptotic behaviors of solutions of the Yamabe equation with isolated singularities and discuss some important results by Caffarelli, Schoen, Spruck, Korevaar, Mazzeo and Pacard. We will also present some recent results that solutions near isolated singularities are well approximated by series that decay at discreet orders.

In this talk, I will review a few existing results about Picard groups of moduli spaces of sheaves over curves, as well as the rank 2 case over surfaces. Based on the related methods, I will discuss how to obtain similar results for high rank cases over surfaces. This is work in progress, conducted jointly with Jun Li, Howard Nuer, Xiaolei Zhao and Yi Xie.