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.

In this talk, I will first review a subsets- counting technique, and then present some applications in number theory and coding theory, with a focus on a partial result on the Borwein conjecture. This is based on joint work with Daqing Wan.

我们将给出密码货币体系中数字权益歧视性的定义，并分析歧视性与安全性、歧视性与效率之间的关系，并尽可能详细介绍相关的数学理论。

Exponential sums over finite fields play a fundamental role in number theory, arithmetic geometry and their applica- tions. This talk presents an expository introduction to the p-adic aspects of this subject. Various classical examp- les will be given to illustrate the general theory.