New Trends in Applied Math and Machine Learning

June 8 - June 13, 2025
QingTian, Lishui, Zhejiang, China

PROGRAM

Conference Venue

5th Floor, SIENA NARADA GRAND HOTEL(青田西娜君澜大饭店)

Specific Conference Hall

June 9th Morning Session: WENLAN HALL(文澜厅)
The Rest Sessions: WENTAO HALL(文涛厅)

Dining Hall

Breakfast: ALL DAY DINING(澜悦全日制餐厅)，4th Floor, SIENA NARADA GRAND HOTEL
Lunch: ALL DAY DINING(澜悦全日制餐厅)，4th Floor, SIENA NARADA GRAND HOTEL
Dinner: WENZE HALL(文泽厅)，5th Floor, SIENA NARADA GRAND HOTEL

Banquet (For invited guests)

18:00 June 9th
WENLAN HALL(文澜厅)，5th Floor, SIENA NARADA GRAND HOTEL

Conference Schedule

Titles & Abstracts

JUNE 09 (MONDAY)

08:45-09:15 Opening Remarks

09:15-10:15 Zhigang Yao (National University of Singapore)

Title: Manifold Fitting Reveals Metabolomic Heterogeneity and Disease Associations in UK Biobank Populations
Abstract: The community studying the connection between statistics and geometry is growing rapidly in both size and scope. The idea of manifold fitting dates back to H. Whitney's work in the early 1930s. The Whitney extension problem has led to new approaches for data interpolation and inspired a set of questions now known as the Geometric Whitney Problems. One key question is: given a set of data, when can we find a smooth $d$-dimensional surface (or manifold) that approximates it well, and how accurately can we measure that fit in terms of distance and smoothness? In this talk, I will give an overview of the manifold fitting problem and discuss some recent insights and developments. I will focus on an application in NMR-based metabolomics, where we use manifold fitting to study metabolic variation in the UK Biobank population. Our method organizes 251 metabolic biomarkers into seven clusters, each representing a module of human metabolism. Within each cluster, we find low-dimensional structures that capture key patterns of metabolic variation linked to disease risk. In three of these clusters, the fitted manifolds clearly separate the population into two subgroups with distinct metabolic profiles. These subgroups are associated with a wide range of diseases, including metabolic, cardiovascular, and autoimmune disorders. This detailed stratification helps us better understand the links between metabolism and disease, and may inform personalized healthcare and preventive medicine. The discussion will draw on recent work by Yao, Yau, and collaborators, as well as ongoing research.

10:45-11:45 mil Saucan (Braude College of Engineering) online

Title: Curvature-based sampling - From manifolds to networks
Abstract: The problem of sampling and reconstruction of manifolds and, independently, of signals, has deep roots both in Differential Geometry as well as in Signal Processing. We overview the history of the subject and show that the two fields are not so widely divergent when sampling is concerned. Curvature, and more specifically Ricci curvature, represents the essential sampling tool. On the other hand, notions of discrete Ricci curvature have been proposed and applied with success in the study of Complex Networks. Thus, motivated by the methods and results of manifold sampling based on Ricci curvature, we are conduced to propose a similar sampling approach for networks. To this end we make appeal to three types of discrete curvature, namely the graph Forman-, full Forman- and Haantjes-Ricci curvatures for edge-based and node-based sampling. We present the results of experiments on real life networks, as well as for square grids arising in Image Processing. Moreover, we consider fitting Ricci flows, and we employ them for the detection of networks backbone. The relation between the Ricci curvature of the original manifold and that of a Ricci curvature driven discretization is also studied.

14:00-15:00 Shufei Ge (ShanghaiTech University)

Title: A spatial-correlated multitask linear mixed-effects model for imaging genetics
Abstract: Imaging genetics aims to uncover the hidden relationship between imaging quantitative traits (QTs) and genetic markers (e.g. single nucleotide polymorphism (SNP)), and brings valuable insights into the pathogenesis of complex diseases, such as cancers and cognitive disorders. However, most linear models in imaging genetics didn't explicitly model the inner relationship among QTs, which might miss some potential efficiency gains from borrowing information across brain regions. In this work, we developed a novel Bayesian regression framework for identifying significant associations between QTs and genetic markers while explicitly modelling spatial dependency between QTs, with the main contributions as follows. Firstly, we developed a spatial-correlated multitask linear mixed-effects model to model dependencies between QTs. We took full advantage of the dependent structure of brain imaging-derived QTs by introducing a population-level mixed effects term. Secondly, we implemented the model in the Bayesian framework and derived a Markov chain Monte Carlo (MCMC) algorithm to do model inference. Further, we incorporated the MCMC samples with the Cauchy combination test to examine the association between SNPs and QTs, which avoided computationally intractable multi-test issues. The simulation studies indicated improved power of our proposed model compared to classic models where inner dependencies of QTs were not modeled.

15:30-16:30 Shijia Wang (ShanghaiTech University)

Title: An adaptive approximate Bayesian computation MCMC with Global-Local proposals
Abstract: We address the challenge of Markov Chain Monte Carlo (MCMC) algorithms within the approximate Bayesian Computation (ABC) framework, which often get trapped in local optima due to their inherent local exploration mechanism. We propose a novel Global-Local ABC-MCMC algorithm that combines the ``exploration" capabilities of global proposals with the ``exploitation" finesse of local proposals. We integrate iterative importance resampling into the likelihood-free framework to establish an effective global proposal distribution, and select the optimum mixture of global and local moves based on a relative version of expected squared jumped distance via sequential optimization. Furthermore, we propose two adaptive schemes. The first adaptive scheme is a normalizing flow-based probabilistic distribution learning model to iteratively improve the proposal of importance sampling. The second adaptive scheme is optimizing the efficiency of the local sampler by utilizing Langevin dynamics and common random numbers. We numerically demonstrate that our method is able to improve sampling efficiency and achieve more reliable convergence for complex posteriors.

JUNE 10 (TUESDAY)

09:00-10:00 Guowei Wei (Michigan State University) online

Title: Commutative Algebra Learning
Abstract: Recently, we have introduced commutative algebra as an emerging paradigm for machine learning and data science. Specifically, we developed persistent Stanley-Reisner theory to bridge commutative algebra, algebraic topology, machine learning, and data science. We proposed persistent h-vectors, persistent f-vectors, persistent graded Betti numbers, persistent facet ideals, and facet persistence modules for data analysis. We show that commutative algebra predicts protein-ligand binding affinity, uncovers phylogenetic evolution, and reveals the genetic origins of diseases.

10:30-11:30 Shuyang Ling (NYU Shanghai)

Title: Local geometry determines global landscape in low-rank factorization for synchronization: theory and statistical bounds
Abstract: The orthogonal group synchronization problem, which focuses on recovering orthogonal group elements from their corrupted pairwise measurements, encompasses examples such as high-dimensional Kuramoto model on general signed networks, $\mathbb{Z}_2$-synchronization, community detection under stochastic block models, and orthogonal Procrustes problem. The semidefinite relaxation (SDR) has proven its power in solving this problem; however, its expensive computational costs impede its widespread practical applications. We consider the Burer-Monteiro factorization approach to the orthogonal group synchronization, an effective and scalable low-rank factorization to solve large scale SDPs. Despite the significant empirical successes of this factorization approach, it is still a challenging task to understand when the nonconvex optimization landscape is benign, i.e., the optimization landscape possesses only one local minimizer, which is also global. In this work, we demonstrate that if the degree of freedom within the factorization exceeds twice the condition number of the ``Laplacian" (certificate matrix) at the global minimizer, the optimization landscape is absent of spurious local minima. Our main theorem is purely algebraic and versatile, and it seamlessly applies to all the aforementioned examples: the nonconvex landscape remains benign under almost identical condition that enables the success of the SDR. Finally, we will discuss the statistical sides of group synchronization by quantifying the uncertainty of both MLE and spectral estimators, and introduce a few open problems in this area.

14:00-15:00 Manuel Mellado Cuerno (CUNEF University)

Title: Pattern-type clustering of surveys: construals, bipolar items, and applications
Abstract: Empirical knowledge about construals—social affinity groups that share similar patterns of meaning to navigate social life—has grown significantly in recent years. This progress is largely due to the development of Construal Clustering Methods (CCMs), which analyze representative survey data to classify respondents into construal classes based on the similarity of their response patterns. In this talk, I will review existing CCMs and highlight their limitations, introducing a novel CCM called Bipolar Class Analysis (BCA), designed specifically for bipolar question data. Additionally, I will compare all CCMs through simulations, demonstrating the strengths of BCA as well as its limitations, which point to directions for future research. The talk will conclude with applications to real-world survey data, where BCA has provided novel or more robust insights into social construals.

15:30-16:30 Mengmeng Zhang (Beijing Institute of Mathematical Sciences and Applications)

Title: Calculating higher digraph homotopy groups
Abstract: In 2014, Grigor'yan, Lin, Muranov and Yau introduced the homotopy theory of digraph and defined the digraph homotopy groups by taking the fundamental group of the (n-1)-fold iterated based loop digraph. Then inspired by the cubical approach to defining the higher homotopy groups of a based space, a more direct approach to defining higher digraph homotopy groups, which exhibits properties similar to those of the homotopy groups of based topological space, was given. At this point two questions naturally arise. Is there a digraph whose higher homotopy group is non-trivial? What other properties of higher homotopy groups for topological spaces have digraph analogues? The purpose of this paper is to address these questions. This is a joint work with Stephen Theriault, Jie Wu and Shing-Tung Yau.

16:30-17:30 Peng Luo (Shanghai Jiao Tong University)

Title: Linear-quadratic extended mean field games with common noises
Abstract: In this paper, we consider a class of linear quadratic extended mean field games (MFGs) with common noises where the state coefficients and the cost functional vary with the mean field term in a nonlinear way. Based on stochastic maximum principle, solving the mean field game is transformed into solving a conditional mean field forward-backward stochastic differential equation (FBSDE). We first establish solvability for a type of (more general) conditional mean field FBSDEs under monotonicity conditions. We further provide some regularity results which lead to classical solutions for the associated master equations. In particular, the linear quadratic extended mean field game is solved and classical solution for (extended mean field game) master equation is obtained.

JUNE 11 (WEDNESDAY)

09:00-10:00 Kelin Xia (Nanyang Technological University)

Title: Mathematical AI for Molecular Sciences
Abstract: Artificial intelligence (AI) based Molecular Sciences have begun to gain momentum due to the great advancement in experimental data, computational power and learning models. However, a major issue that remains for all these AI-based learning models is the efficient molecular representations and featurization. Here we propose advanced mathematics-based molecular representations and featurization. Molecular structures and their interactions are represented by high-order topological and algebraic models (including Rips complex, Alpha complex, Neighborhood complex, Dowker complex, Hom-complex, Tor-algebra, Rhomboid tiling, etc). Mathematical invariants (from persistent homology, Ricci curvature, persistent spectral, Analytic torsion, algebraic variety, etc) are used as molecular descriptors for learning models. Further, we develop geometric and topological deep learning models to systematically incorporate molecular high-order, multiscale, and periodic information, and use them for analysing molecular data from chemistry, biology, and materials.

10:30-11:30 Tobias Wöhrer (Technical University Vienna)

Title: Global stability for McKean-Vlasov equations on large networks
Abstract: This talk investigates the mean-field dynamics of stochastic McKean (or Kuramoto-type) differential equations with particle interaction patterns described by large network/graph structures. These dynamics are fundamental in wide range of natural processes such as synchronization phenomena, opinion formation and biological mechanisms. We formulate the limiting McKean-Vlasov equation with a Vlasov interaction term that incorporates the recently developed graph limit theory of graph operators (or graphops). This allows us to cover a wide range of graph structures including dense, sparse and various intermediately dense cases. In this rather general setting, we prove global stability of the homogeneous steady state via entropy methods and provide explicit graph-structure dependent stability conditions and decay rates.

JUNE 12 (THURSDAY)

09:00-10:00 Tao Luo (Shanghai Jiao Tong University)

Title: The Dynamics of Condensation in Neural Networks
Abstract: In this talk, we present a rigorous mathematical framework for analyzing the condensation phenomenon in neural networks—a process wherein weights of different neurons align with one another during training. First, we establish a phase diagram characterizing the hyperparameter regimes under which condensation emerges. Next, we investigate the structure of global minima, offering a comprehensive geometric analysis of their properties and convergence behavior. Finally, we generalize our results to saddle points, extending the theoretical understanding of condensation dynamics in non-convex landscapes.

10:30-11:30 Shixiao Jiang (ShanghaiTech University)

Title: Generalized Moving Least-Squares Methods for Solving PDEs on Manifolds
Abstract: In this talk, we introduce the Generalized Moving Least-Squares (GMLS) method to solve the vector-valued PDEs on smooth 2D manifolds without boundaries embedded in R^3, identified with randomly sampled point cloud data. The approach formulates tangential derivatives on a submanifold as the projection of the directional derivative in the ambient Euclidean space onto the tangent space of the submanifold. One challenge of this method is that the discretization of vector Laplacians yields a matrix whose size relies on the ambient dimension. To overcome this issue, we reduce the dimension of vector Laplacian matrices by employing an appropriate projection so that the complexity of the method scales well with the dimension of manifolds rather than the ambient dimension. We also present supporting numerical examples, including eigenvalue problems, linear Poisson equations, and nonlinear Burgers' equations, to examine the numerical accuracy of the proposed method on various smooth manifolds.

14:00-15:00 Zhongyuan Cao (NYU Shanghai)

Title: Convergence Analysis of the Deep BSDE Method for McKean-Vlasov FBSDEs
Abstract: McKean-Vlasov equations model systems where dynamics depend on the distribution of the state, with applications in mean field games, systemic risk, and interacting particle systems. Their nonlocal and high-dimensional nature makes them challenging to solve numerically. The deep BSDE method offers a scalable approach to solving high-dimensional FBSDEs by combining time discretization, neural networks, and sampling. In this talk, I will present a convergence analysis of an adaptation of this method for McKean-Vlasov FBSDEs. The analysis covers time discretization, distribution approximation via particle method, and the convergence of the shooting method for the backward equation. This provides a first step toward rigorous guarantees for deep learning-based solvers in the McKean-Vlasov setting.

15:30-16:30 Xiaoli Wei (Harbin Institute of Technology)

Title: Continuous-time q-learning for mean-field control problems with common noise
Abstract: This paper investigates the continuous-time entropy-regularized reinforcement learning (RL) for mean-field control problems with common noise. We study the continuous-time counterpart of the Q-function in the mean-field model, coined as q-function in Jia and Zhou (2023) in the single agent's model. It is shown that the controlled common noise gives rise to a double integral term in the exploratory dynamic programming equation, rendering the policy improvement iteration intricate. The policy improvement at each iteration can be characterized by a first-order condition using the notion of partial linear derivative in policy. To devise some model-free RL algorithms, we introduce the integrated q-function (Iq-function) on distributions of both state and action, and an optimal policy can be identified as a two-layer fixed point to the soft argmax operator of the Iq-function. The martingale characterization of the value function and Iq-function is established by exhausting all test policies. This allows us to propose several algorithms including the Actor-Critic q-learning algorithm, in which the policy is updated in the Actor-step based on the policy improvement rule induced by the partial linear derivative of the Iq-function and the value function and Iq-function are updated simultaneously in the Critic-step based on the martingale orthogonality condition. In two examples, within and beyond LQ-control framework, we implement and compare our algorithms with satisfactory performance.

16:30-17:30 Yuqing Li (Shanghai Jiao Tong University)

Title: Universality of Condensation in Neural Networks Trained by Various Algorithms
Abstract: In this talk, we delve into three primary areas of research. First, we explore the dynamics of the Neural Tangent Kernel (NTK) for finite-width Deep Residual Networks (ResNet) using the neural tangent hierarchy (NTH) framework introduced by Huang and Yau, and we obtain that for ResNet with smooth and Lipschitz activation function, the requirement on the layer width $m$ with respect to the number of training samples $n$ can be reduced from quartic to cubic. Secondly, we focus on the intriguing observation that neural networks exhibit distinct behaviors depending on the scales of initialization. Drawing from previous research, notably the work by Luo et al., we present a phase diagram characterizing the phenomenon of initial condensation in two-layer neural networks and convolutional neural networks. Condensation refer to a situation where weight vectors in neural networks tend to concentrate on isolated orientations during the training process. This phenomenon is a key feature of nonlinear learning processes and contributes to enhancing the generalization capabilities of neural networks. Finally, we shift our focus to the Dropout algorithm, a widely adopted regularization technique in neural network training. We embark on a rigorous theoretical derivation of stochastic modified equations, with the primary objective of providing an effective approximation for the discrete iterative process of dropout. Moreover, our empirical findings substantiate that Dropout indeed facilitates the phenomenon of condensation at the end of training stage of neural networks.

JUNE 13 (FRIDAY)

9:00-10:00 Qi Meng (Chinese Academy of Mathematics and Systems Science, CAS)

Title: AI for Solving SPDE
Abstract: Stochastic Partial Differential Equations (SPDEs) driven by random noise play a central role in modelling physical processes whose spatio-temporal dynamics can be rough, such as turbulence flows, superconductors, and quantum dynamics. To efficiently model these processes and make predictions, machine learning (ML)-based surrogate models are proposed, with their network architectures incorporating the spatio-temporal roughness in their design.In this talk, I will introduce a ML-based surrogate model named DLR-Net, which is specially designed for solving SPDEs, and then, I will introduce SPDEBench, which includes typical datasets, benchmark ML models and unified evaluation for solving regular and singular SPDEs.

10:30-11:30 Siran Li (Shanghai Jiao Tong University)

Title: Restricted Path Characteristic Function Determines the Law of Stochastic Processes
Abstract: A central question in rough path theory is characterising the law of stochastic processes. It is established in [I. Chevyrev & T. Lyons, Characteristic functions of measures on geometric rough paths, Ann. Probab.(2016), 4049--4082] that the path characteristic function (PCF), i.e., the expectation of the unitary development of the path, uniquely determines the law of the unparametrised path. We show that PCF restricted to certain subspaces of sparse matrices is sufficient to achieve this goal. The key to our arguments is an explicit algorithm -- as opposed to the nonconstructive approach in [I. Chevyrev & T. Lyons, op. cit.] -- for determining a generic element $X$ of the tensor algebra from its moment generating function. Our only assumption is that $X$ has a nonzero radius of convergence, which relaxes the condition of having an infinite radius of convergence in the literature. As applications of the above theoretical findings, we propose the restricted path characteristic function distance (RPCFD), a novel distance function for probability measures on the path space that offers enormous advantages for dimension reduction. Its effectiveness is validated via hypothesis testing on fractional Brownian motions, thus demonstrating the potential of RPCFD in generative modeling for synthetic time series generation. Joint work with Hao Ni (UCL), Zijiu Lyu (Oxford), and Jiajie Tao (UCL).

Popular Science talks

Talk 1:

Title: 大模型时代的数学学习与研究
Speaker: Xiaoyang Chen (同济大学)
Abstract: 人工智能大模型正重塑数学学习与研究，实现辅助证明、解题及教育个性化，我们将讨论这方面的进展以及局限性。

Talk 2:

Title: 黑暗料理遇上贝叶斯： “真凶”计算手册
Speaker: Shijia Wang (上海科技大学)
Abstract: 假设学校食堂发生了一起"黑暗料理"事件，我们将以侦探破案的方式带同学们认识贝叶斯思维，锁定黑暗料理“真凶”。

Talk 3:

Title: 如何用数学知识玩好抓娃娃机
Speaker: Chong Liu (上海科技大学)
Abstract: 假设在游乐场里有多台抓娃娃机，每一台机器能抓出娃娃的可能性都不同，并且我们在玩之前并不知道任何一台机器能抓出娃娃的可能性，因此我们必须用游戏币在这些机器上进行多次测试，目的是尽快找出回报最高的那台机器。那么我们应该用何种策略对这些机器进行探索使得我们可以用最少的游戏币找出最好的那台机器呢？实际上这个问题的解决对应了一个非常经典的强化学习算法，里面涉及到很多有趣的数学知识。在这次报告里我将会对这个有趣的问题和算法做简单介绍。

Talk 4:

Title: 从河内寺庙传说到汉诺塔游戏
Speaker: Rongji Li; Yiyao Zhang; Yuyang Tao (上海科技大学)
Abstract: 传说在一座神庙的穹顶之下有三根柱子, 创世神在其中一根上放置了64片纯金圆盘。僧侣们的使命是将它们按规则移动到另一根柱子上。究竟僧侣们多久才能完成使命呢? 让我们用数学的思维来寻找答案。