第十届国际青年学者东湖论坛-数学与统计学院分论坛

时  间：2021年12月27日19：00-22:00

地  点：华中科技大学科技楼南楼706会议室

             腾讯会议室：656 963 275（密码3231）

主持人：王保伟

议 程：

1、王文重书记致开幕辞，刘斌院长介绍学院学科建设发展情况和人才政策。

2、学者分组做学术汇报。

第一组：主持人： 王湘君

              学 者： 程佳麒、杨 洋、黄 乔

              学院教师：吴付科、刘继成、杨美华

第二组：主持人： 柴振华

              学 者： 马世琪、孟旭辉、谈 进

              学院教师：李东方、明 炬、高华东

第三组：主持人： 王保伟

              学 者： 黄鹏飞、颜俊榕、张 庆

              学院教师：徐 剑、卢 文、张 宁

3、学院领导及相关学术骨干与学者互动交流。

学术汇报：

（一）

报告人： 程佳麒

个人简介：程佳麒，兰州大学基础数学专业本科, 现为奥本大学应用数学专业博士。长期从事随机微分方程及无穷维动力系统工作研究。发表4篇专著、论文，获得Mary Neal Mamie Hurt Baskerville and Margret Malone Baskerville Endowed Mathematics Fellowship，The Don and Sandy Logan Endowment for Fellowship in Mathematics and Statistics，Excellence in Research(Auburn University)奖项等。

报告题目：Dynamics of the lottery competition model in stochastic environments

报告摘要：In order to investigate the role of environmental fluctuation on the ecological competition, we study the lottery competition model. First a diffusion approximation for the fraction of sites occupied by each adult species is derived as the continuum limit of a classical discrete-time lottery model. As a result, a system of nonlinear stochastic differential equations (SDEs) are developed as the diffusion approximation for the discrete lottery model. Then, the existence and uniqueness of positive and bounded global solutions, as well as long term dynamics for the solution are investigated, from which sufficient conditions for the coexistence of species in the sense of stochastic persistence are established. Moreover, in the 2-D case, we have obtained a time-dependent limiting process under certain conditions. Finally, Numerical simulations are presented to illustrate the theoretical results.

（二）

报告人：杨洋

个人简介：杨洋，2015年7月在四川大学数学学院获理学学士学位, 以专业第一名保送直博；2020年7月在清华大学获统计学博士学位，导师为清华大学统计学研究中心执行主任邓柯副教授。现在在腾讯从事算法研究员工作。长期从事贝叶斯统计与统计计算、计算机实验设计、机器学习的工作研究，取得了多项创新成果。已完成的工作共计产出6篇论文，其中一作发表2篇论文，分别发表在应用统计顶刊Annals of Applied Statistics（AOAS）和Quantitative Biology；另外两篇一作还分别投稿到AOAS(二审小修)和arXiv上；共申请2项专利，参与面上及重点项目等共计5项。

报告题目：Efficient Bayesian Modelling and Optimization for Analyzing Complex Data

报告摘要：Analyzing data with complex structures is indispensable in scientific research and industry. In this talk, we proposed an efficient synergic design strategy based on Bayesian optimization to achieve the rapid design of metamaterials with multiple targets. Then we developed an additive truncated Gaussian process to integrate the multi-level training data to achieve efficient hyperparameter optimization in machine learning. We also proposed an innovative way to achieve efficient association pattern discovery via a probabilistic generative model with statistical inference in data mining. We provided the theoretical results of the proposed approaches. A wide range of simulation experiments and real-world datasets have verified the effectiveness and superiority of our proposed methods.

（三）

报告人：黄乔

个人简介：黄乔，华中科技大学数学与应用数学专业本科，华中科技大学统计专业博士，现为在葡萄牙里斯本大学从事博士后工作。长期从事随机分析工作研究，取得了多项创新成果。主持华中科技大学创新研究院2016年度创新项目，发表4篇论文等。

报告题目：From Second-order Differential Geometry to Stochastic Geometric Mechanics

报告摘要：Classical geometric mechanics, including symmetries, Lagrangian and Hamiltonian mechanics, and the Hamilton-Jacobi theory, are based on classical geometric structures such as jets, symplectic structures and contact structures. In this paper, we will continue the forgotten framework of the second-order (or stochastic) differential geometry developed originally by L. Schwartz and P.-A. Meyer, to construct second-order (or stochastic) counterparts of those classical structures. Based on these, we then study symmetries of SDEs, stochastic Lagrangian and Hamiltonian mechanics as well as their connections with the second-order Hamilton-Jacobi-Bellman equation. More precisely, stochastic prolongation formulae are established and symmetries of SDEs and mixed-order Cartan symmetries are studied. Stochastic Hamilton's equations are formulated via Nelson's mean derivatives. Canonical transformations for mixed-order contact structure lead to the second-order Hamilton-Jacobi-Bellman equation. When a stochastic variational problem is set up, a stochastic Euler-Lagrange equation is derived and its equivalence with a reduction of stochastic Hamilton's equations and a Hamilton-Jacobi-Bellman equation are proved. The latter equivalence suggests an alternative formulation to the Schr\"odinger problem equivalent to the viewpoint of optimal transport. A stochastic Noether's theorem is also established where the stochastic conserved quantities are martingales.

（四）

报告人：马世琪

个人简介：马世琪，电子科技大学通信工程专业本科, 香港浸会大学数学专业博士，现为在芬兰于韦斯屈莱大学从事博士后工作。长期从事反问题工作研究，取得了多项创新成果。主持发表4篇论文，获得2020年香港数学会最佳论文奖。

报告题目：反散射问题：随机参数与固定角度入射波情形

报告摘要：首先进行自我介绍。然后分别介绍随机薛定谔方程的反问题和固定角度入射波的反演问题。最后介绍其他相关的偏微分方程的工作。

（五）

报告人：孟旭辉

个人简介：孟旭辉，华中科技大学能源与动力工程学院热能工程专业本科, 华中科技大学能源与动力工程学院热能工程专业博士，现为美国布朗大学应用数学系博士后。主要研究兴趣为介观数值方法(格子Boltzmann方法)及求解PDE正反问题的深度学习方法，目前已在Journal of Computational Physics, Computer Methods in Applied Mechanics and Engineering等期刊发表SCI论文12篇，google scholar总引用796次。

报告题目：Deep learning for solving forward and inverse PDE problems: multi-fidelity data fusion, and uncertainty quantification

报告摘要：Deep learning algorithms have emerged recently for solving partial differential equations (PDEs), especially in conjunction with sparse data. In particular, the recently developed physics-informed neural networks (PINNs) have shown their effectiveness in solving both forward and inverse PDE problems. Different from the classical numerical methods in which the differential operators are approximated by the data on certain discrete lattices (meshes), PINNs compute all the differential operators of a PDE using the automatic differentiation technique involved in the backward propagation. Consequently, no mesh (structured mesh or unstructured mesh used in the classical numerical methods) is required for the PINN to solve PDEs, which saves a lot of effort in grid generation. Another attractive feature is that PINNs are capable of solving the inverse PDE problems effectively and with the same code that is used for forward problems. However, the vanilla PINNs (1) require lots of expensive high-fidelity data for inverse PDE problems, and (2) do not predict uncertainty. In this talk, I will introduce two newly developed PINNs to address these two issues: (1) multi-fidelity physics-informed neural networks (M-PINNs), which are capable of utilizing a very small set of high-fidelity data and plenty of inexpensive lower fidelity data to achieve good accuracy for inverse PDE problems, and (2) Bayesian physics-informed neural network (B-PINN), which can solve both forward and inverse PDE problems with noisy data, and also provides uncertainty quantification in predictions.

（六）

报告人：谈进

个人简介：谈进，湖北理工学院信息与计算科学专业本科, 武汉大学基础数学硕士专业学习两年，巴黎-东大学 克雷特伊分校 数学专业博士，现在法国 巴黎-赛尔吉大学 从事博士后工作。长期从事流体力学和磁流体力学方程组在临界空间的适定性研究。

报告题目：Well-posedness in critical spaces for the  Hall-MHD system

报告摘要：We are concerned with the incompressible Hall-magnetohydrodynamic system (Hall-MHD). Our first aim is to provide an elementary proof of a global well-posedness result for small data with critical Sobolev regularity, in the spirit of Fujita-Kato’s theorem for the Navier-Stokes equations. Next, we investigate the long-time asymptotics of global solutions of the Hall-MHD system that are in the Fujita-Kato regularity class.

（七）

报告人：黄鹏飞

个人简介：黄鹏飞，华中科技大学材料成型及控制工程专业本科(2014年6月), 法国尼斯大学(现称“蓝色海岸大学”)和中国科学技术大学基础数学专业博士(2020年7月)，现在在德国海德堡大学从事博士后工作。长期从事代数几何和微分几何相关工作研究，更具体地说，研究与非阿贝尔霍奇理论(nonabelian Hodge theory)相关的模空间。目前已经在CMP, JGA, JMP, JGP, manus. math.等杂志上发表SCI论文9篇。曾获得博士生国家奖学金,中科院院长奖学金，欧洲数学会Kovalevskaya Travel Grant(参加ICM 2022)等奖项。

报告题目：Geometry of moduli spaces in nonabelian Hodge theory

报告摘要：Over a base variety X, nonabelian Hodge theory provides a correspondence among representations of the fundamental group of X, flat bundles, and Higgs bundles over X. The corresponding three moduli spaces (called Betti, de Rham, and Dolbeault) are also related to each other. In this talk, firstly I will give a brief introduction to this theory as the background setting, then I will report some recent work on exploring the geometry of these moduli spaces, more precisely:

(1) Stratification of the de Rham moduli space (some conjectures of Simpson);

(2) Dynamical systems on the Dolbeault moduli space;

(3) Generalization of Deligne’s twistor construction;

(4) Geometry of the base manifold which parametrizes a family of Higgs bundles.

（八）

报告人：颜俊榕

个人简介：颜俊榕，南开大学数学专业本科, University of California, Santa Barbara, 数学专业博士，现博士五年级。从事微分几何与数学物理研究，目前发表1篇论文。

报告题目：Witten Deformation on noncompact manifolds and Landau-Ginzburg B-models

报告摘要：In 1987, Witten introduced a deformation of the de Rham complex by considering the new differential d + df, where f is a (Morse) function. Since then, Witten deformation has found important applications such as Bismut-Zhang/Cheeger-Muller theorem as well as being instrumental in the development of Floer homology theory. To understand the mathematics of the Landau-Ginzburg (LG) B-model, we study the Witten deformation for noncompact manifolds. We will discuss the L^2 cohomology of d+df, the index theorem, and analytic torsion of Witten Laplacian as well as their relations with the Mirror symmetry for the LG models.

（九）

报告人：张庆

个人简介：张庆，浙江大学数学专业本科, 俄亥俄州立大学数学专业博士，现为在韩国科学技术院从事访问教授工作。从事p-adic群的表示论工作研究。发表论文13篇。

报告题目：representations of p-adic groups

报告摘要：The theory of representations of p-adic groups is one important branch in the study of modern number theory, in particular, in Langlands program. In this talk, I will briefly introduce my work on this subject, including certain local converse theorems, and geometric construction of certain Arthur packets.