Satellite Conference on Data Science and Algebraic Geometry

Overview:

    The conference will be held at Nanjing Center for Applied Mathematics from July 28 to 30, 2023, as a satellite conference on data science and algebraic geometry of the first International Congress of Basic Science (ICBS). This satellite conference is jointly organized by Nanjing Center for Applied Mathematics and Shing-Tung Yau Center of Southeast University.

Organizers:

Tiexiang Li (Southeast University & Nanjing Center for Applied Mathematics)

Jijun Liu (Southeast University & Nanjing Center for Applied Mathematics)

Haibing Wang (Southeast University & Nanjing Center for Applied Mathematics)

Shuanhong Wang (Southeast University & Shing-Tung Yau Center of SEU)

Invited Speakers (Confirmed):

Jingrun Chen (University of Science and Technology of China)

Yang Chen (Southeast University)

Jin Cheng (Fudan University)

Zhengyu Hu (Chongqing University of Technology)

Chung Pang MoK (Soochow University)

Xiaolong Qin (Hangzhou Normal University)

Sheng Rao (Wuhan University)

Guozhen Wang (Fudan University)

Xiang Wang (Nanchang University)

Chao Zhang (Southeast University)

Leihong Zhang (Soochow University)

Shuo Zhang (Chinese Academy of Sciences)

Venue:

Nanjing Center for Applied Mathematics, Nanjing

Offline location: Nanjing Center for Applied Mathematics.

4 Liye Zone, TusCity, 26 Zhishi Rd, Chi-Lin Innovation Park, Nanjing, Jiangsu.

Online live platform:

Contact:

Li Zhu, E-mail: njcam@njcam.org.cn, Tel & Wechat: 15951855511

Agenda (tentative)

Abstract

The Random Feature Method for Solving PDEs

Jingrun Chen

(Soochow University)

    In this work, we propose the random feature method (RFM) for solving PDEs, a natural bridge between traditional and machine learning-based algorithms. We will discuss the application of RFM for problems over complex geometries, interface problem, high-frequency problem, as well as time-dependent problem.

智能医学成像和处理

陈阳

（东南大学）

      报告以智能医学成像和处理为题，围绕临床任务驱动的智能医学成像中的基于特征学习的高质量成像技术、国产医学影像设备核心算法研发嵌入以及临床任务驱动的医学影像处理，主要讲述了智能医学成像、成像算法应用、智能影像处理及应用和医工交叉研究思考等四部分内容。

基于短期观测数据的复杂动力系统的参数辨识与趋势预测

程晋

（复旦大学 & 上海市现代应用数学重点实验室）

      动力系统描述了几何空间中的一个点随时间演化情况。具有重要的应用背景和理论研究价值。如何利用一些短期局部的观测数据，辨识复杂动力系统中的重要参数，进而预测系统的长期行为和趋势具有重要的实际意义和实用前景。在本报告中，我们提出可以通过观测动力系统的部分分量，来确定系统的重要参数的问题。由于实际观测通常带有较大的随机误差。为了克服这个困难，结合正则化方法和误差满足一定分布的随机特性，我们证明可以通过数据量来有效地获得高精度的观测值以及导数值，进而稳定地重构动力系统的重要参数。我们的想法是通过对经典Lorenz63系统进行分析和说明的。

Generic minimal model program

Zhengyu Hu

(Chongqing University of Technology)

    The theory of generalised pairs plays an important role in higher dimensional algebraic geometry, for example, in the proof of BAB conjecture. In many situations, the minimal model program(MMP) for generalised pairs can be reduced to that for usual pairs, which allows us to apply the classical minimal model theory. In this talk, I will discuss a more general class of geometric objects which also allows us to run MMP generically and some results. This is joint work with Caucher Birkar.

Pseudorandomness of Sato-Tate Distributions for Elliptic Curves

Chung Pang Mok

(Soochow University)

    Equidistribution is an important theme in number theory. The Sato-Tate conjecture, which was established by Richard Taylor et.al. in 2008, asserts that given an elliptic curve over Q without complex multiplication, the associated Frobenius angles are equidistributed with respect to the Sato-Tate measure. In this talk, we discuss refinements to the original Sato-Tate conjecture. In particular, we conjecture that the Frobenius angles are in fact statistically independently distributed with respect to the Sato-Tate measure, and satisfy a qualitative form of the Law of Iterated Logarithm for random numbers. Numerical evidences would be presented to support the conjectures. Joint work with Huimin Zheng.

Some results on mean value algorithms and their applications

Xiaolong Qin

(Hangzhou Normal University)

    Mean value algorithms play an important role in fixed point theory, optimization theory, and data science. In this talk, we recall some classical (Bruck, Halpern, Mann, and Isihkawa algorithms) and also some new mean value algorithms. Some new convergence theorems of solutions are established in various framework of spaces. Some applications are considered for variational inequalities and variational inclusion problems with the aid of revolvent methods.

Projective, Moishezon and Kahler loci of family II

Sheng Rao

(Wuhan University)

    This talk mainly concerns the projective, Moishezon and Kahler loci of a family. We first review our recent results on deformation limit and invariance of plurigenera of Moishezon manifolds, based on several joint works with I-Hsun Tsai, Yi Li and Runze Zhang. Then we talk about an in-process joint work with Mu-Lin Li and Mengjiao Wang, on various loci of a family and their applications. Among them are a Chow-type Lemma and a reverse side of one theorem of Zhiwei Wang on the modification of some special complex structure.

Structures and computations in the motivic stable homotopy categories

Guozhen Wang

(Fudan University)

    Motivic homotopy theory is an application of abstract notions of homotopy in the world of algebraic varieties. It turns out tht motivic homotopy theory gives us powerful tools in understanding classical homotopy theory. In this talk, we will show how structures in the motivic stable homotopy categories can be used to compute both classical and motivic stable homotopy groups.

Solving the quadratic eigenvalue problem expressed in non-monomial basis by the tropically scaled CORK linearization

Xiang Wang

(Nanchang University)

    In this talk, the quadratic eigenvalue problem (QEP) expressed in various commonly used bases, including Taylor, Newton, and Lagrange basis functions will be introduced. We propose to investigate the backward errors of the computed eigenpairs and condition numbers of eigenvalues incurred by the application of the recently developed and well-received compact rational Krylov (CORK) linearization. To improve the backward error and condition number of QEP expressed in a non-monomial basis, we combine the tropical scaling with the CORK linearization. We then establish upper bounds for the backward error of an approximate eigenpair of the QEP relative to the backward error of an approximate eigenpair of the CORK linearization with and without tropical scaling. Moreover, we get bounds for the normwise condition number of an eigenvalue of the QEP relative to that of the CORK linearization.We unify both bounds and these bounds suggest the tropical scaling to improve the normwise condition number for the CORK linearization and the backward errors of approximate eigenpairs of the QEP obtained from the CORK linearization. Our investigation is accompanied by adequate numerical experiments to justify our theoretical findings.

Smooth deformations of p-divisible groups in characteristic p

Chao Zhang

(Southeast University)

    I will explain some of my recent attempts in understanding smooth deformations of p-divisible groups with additional structure in char p>0. This could be viewed as a char p counterpart of Faltings's deformation theory. Our main observation is that it is possible to develop systematically an infinitesimal deformation theory, when a certain group theoretic invariant is fixed.

Eigenvector-dependent nonlinear eigenvalue problems and the SCF iteration in data science

Leihong Zhang

(Soochow University)

    In many applications of multivariate statistical analysis in data science, certain trace-related objective functions over the orthogonal constraints need to be minimized. In this talk, we shall first present some recent applications in data science and show that solving the optimization problems can be converted to eigenvector-dependent eigenvalue problems (NEPv) for which the self-consistent filed  (SCF) iteration can be effectively applied. We then discuss recent developments of the general SCF on the local convergence rate and the level-shifted technique.

Preservation of Poincaré-Alexander-Lefschetz type Dualities in Numerical Discretizations

Shuo Zhang

(Academy of Mathematics and Systems Science, Chinese Academy of Sciences)

    Structural properties in the form of Poincaré-Alexander-Lefschetz(P-A-L) type dualities can be found here and there in applied mathematics. Their preservation is an important issue in the process of numerical discretizations. In this talk, a theoretical framework is given how P-A-L type dualities can be preserved in numerical discretizations. Some important properties are thus inherited for the discretizations.

    The main ingredient to design discretizations to preserve P-A-L type dualities is to design them by preserving P-A-L type dualities. A new kind of finite element scheme, called accompanied-by-conforming discretization (ABCD), supports the implementation of the theoretical framework. Examples are given for the validity of the theory.