会议摘要（Abstract）

The interplay between mathematical analysis and quantum physics has led to spectacular advances in these disciplines. The topic of this symposium is on the recent advances in analysis and the mathematical aspects of quantum physics. Key research subjects include analysis of PDEs, microlocal analysis, spectral theory, and calculus of variations, especially as related to quantum dynamical systems and interacting particle systems.

数学分析与量子物理之间的交叉为这两个领域带来了巨大的推动。本次研讨会的主题是分析和量子物理数学方法的最新进展。重点研究课题包括偏微分方程、微局部分析、谱理论和变分法，特别是与量子动力系统和相互作用粒子系统相关的内容。

举办意义(Description of the aim)

The goal of this symposium is to bring together experts in the fields of mathematical analysis and quantum physics to share their recent results and enhance our understanding on the interplay of these subjects and related topics. It will provide an excellent platform for experts across disciplines to discuss potential collaboration as well as related open problems, e.g., many-body dynamical localization, rigorous theory of Bose-Einstein condensate, and many more.

本次研讨会的目标是汇聚数学分析和量子物理领域的专家，分享他们的最新科研成果，加深对这些学科及相关主题之间交叉领域的理解。本次会议将为跨学科专家提供了一个优秀的平台，以讨论潜在的合作以及相关的公开问题，例如多体动力学局部化、玻色-爱因斯坦凝聚态的严格理论等诸多议题。

会议摘要（Abstract）

Discrete and continuous Painlevé equations have attracted a lot of attention in recent decades, since they define (new) transcendental functions, and exhibit rich mathematical structures, in algebraic geometry, representation theory and asymptotic analysis. The higher analogues of the Painlevé equations, including the isomonodromic Garnier systems have an even richer structure, involving connections with multivariate special functions, including higher-genus Abelian functions, and multiple orthogonal polynomials and an expected higher-dimensional variant of the underlying algebraic geometry that was established for the usual Painlevé equations. The connections with integrable systems, via higher-order similarity reduction, may form a key to a further understanding of these more complicated systems, but so far the study of Garnier and higher Painlevé systems has been lagging behind. One of the aims of the workshop is to repair this inbalance. By bringing together experts as well as interested researchers, we aim at creating a platform where many of the open questions can be discussed and begin to be tackled. Thus, we hope the workshop can act as a launching pad for opening a systematic research program into these systems.

举办意义(Description of the aim)

Background:

In recent decades the Painlevé equations, and their discrete analogues have been studied extensively, both from the point of view of integrable systems as well as in physics (random matrix models and statistical mechanics) and in algebraic geometry (rational surfaces of initial conditions) and representation theory (affine Weyl groups). In contrast the higher order Painlevé equations have not attracted (yet) a similar level of attention. These higher order ordinary differential and difference equations emerged as multi-phase similarity reductions from integrable hierarchies (in the continuous case), as well as from constructions from integrable partial difference equations (in the discrete case). They are of interest, as they are expected to yield novel transcendental functions which asymptotically go to higher-genus Abelian functions, whereas the usual Painlevé equations tend to elliptic functions (genus one) in the long-time range.

Garnier in 1912 constructed a higher analogue of Fuchs’ isomonodromic deformation problem for Painlevé VI with multiple moving singularities and multiple dependent variables. This results in coupled systems of 2nd order ODE's, compatible through a system of linear PDEs, which can be viewed as the PVI hierarchy. Apart from the Painlevé the property of this higher order system, the limiting behaviour of the solutions lead to hyper-elliptic integrals. The Garnier system remained relatively unexplored until the work by Okamoto in the 1970s, who focused on its Hamiltonian aspects. It has special solutions in terms of multivariate hypergeometric functions (Lauricella, etc.) (see e.g. the monograph "From Gauss to Painlevé" by Iwasaki et al.). More recently a q-Garnier system (Sakai, 2005), and an elliptic variant (Ormerod & Rains, 2015) were established, while reductions from KdV and Boussinesq systems to discrete and continuous Garnier systems were given (Nijhoff & Walker, 2001; Tongas & Nijhoff, 2005), and the isomonodromy problem was studied (Dubrovin & Mazzocco , 2000). Some classification results on algebraic solutions of Garnier systems were obtained (Diarra & Loray, 2015) and in recent years some quantum Painleve and Garnier systems have been investigated (Nagoya et al, 2004 and 2008, Novikov & Sukeimanov, 2016). Apart from these isolated results, the study of the Garnier systems and higher rank Schlesinger systems (isomonodromic matrix systems) has remained relatively sparse.

Objectives:

The workshop aims at stimulating the research into the higher Painlevé equations and Garnier systems, by bringing together experts as well as interested researchers and bring to the fore open problems, challenges and possible new directions. This is meant to be a mostly explorative venture in the hope that some synergies can bring about progress in this largely not yet developed area. The following directions will be highlighted:

● Reduction from integrable PDEs and PΔEs: While continuous and discrete Painlevé equations often arise as reductions from integrable partial difference and differential equations, for Garnier systems this remains mostly to be established. Isolated precedents comprise Garnier systems derived from higher-order similarity reduction of continuous & discrete KdV and Boussinesq hierarchies.

● Lagrangian multiform aspects: While the second order discrete and continuous Painlevé equations possess a conventional Lagrangian description, the newly established Lagrangian multiform theory (Lobb & Nijhoff, 2009), which provides a natural variational formalism for multi-time integrable systems, is directly applicable to the case of Garnier systems.

● Connection with higher-genus abelian functions: `Garnier transcendents’ tend to higher-genus Abelian functions in asymptotic limits. The study of the singularity structure of those solutions may help to link the isomonodromy theory to algebra-geometric techniques on Riemann surfaces.

● Special solutions in terms of multivariate hypergeometric functions: For special parameter values of the Garnier systems solutions exist in terms of multivariate hypergeometric functions. This remains to be done for discrete Garnier systems leading possibly to multivariate elliptic hypergeometric functions;

● Algebraic geometry of spaces of initial conditions; Lifting the celebrated work by Sakai on the classification of discrete and continuous Painlevé equations within the context of the algebraic geometry of rational surfaces and affine Weyl groups, to the case of Garnier systems. Some work has begun (e.g. Takenawa, 2024), but requires further developement.

● Applications in random matrix theory and physics: Continuous and most distinctly discrete Painlevé equations have played an important role in random matrix ensembles and in the theory of semi-classical orthogonal polynomials. These relations were often motivated from physics, e.g. in 2D quantum gravity and string theory. So far, there have been little appearance of Garnier systems in this context, but the structures are ready to be explored for such connections.

Abstract

This workshop is the second edition of the Tsinghua-Tokyo Workshop, which was started in 2024 to promote cooperation between China and Japan.  The topics cover various types of partial differential equations, especially those related to mathematical physics, such as Navier-Stokes equation, nonlinear Schrödinger equation, dispersive equations, reaction-diffusion equations, nonlinear elliptic equations on the PDE side and stochastic partial differential equations, random media, statistical mechanics-type models on the Probability side.  These are important topics which attract a lot of attention internationally and show a rapid progress.

Description of the aim

Tsinghua-Tokyo Workshop was launched in January 2024 highlighting on Calabi-Yau as its main theme (https://indico.ipmu.jp/event/422/). This is the second edition of the workshop and the themes of this time are nonlinear Partial Differential Equations (PDEs), Probability and interactions between these two fields.  The workshop will bring together experts from Japan, mainly from the Tokyo area, and from China.  China and Japan are geographically close and have important achievements and recent growth in mathematical study, respectively.  It is important to maintain and develop a cooperative relationship.  We expect that this workshop will provide an opportunity to establish a close relationship between these two countries. The aim of the workshop is to discuss recent developments in the field of nonlinear partial differential equations from the viewpoints of both PDE Theory and Probability Theory.

会议摘要（Abstract）

本次会议以“几何拓扑及相关主题”为主题，致力于促进几何拓扑领域的前沿研究与跨学科交流。几何拓扑作为数学中一个重要而活跃的研究方向，涉及多维空间的复杂结构与性质，涵盖从低维拓扑到高维拓扑的丰富内容。此次研讨会旨在为学者们提供一个高水平的交流平台，汇聚各地的顶尖研究人员与新兴学者，共同探讨几何拓扑中的最新进展，分享研究成果，并推动学术合作。

会议将重点讨论以下几个方面：

1. 泰希米勒理论与高阶泰希米勒理论：研究将聚焦于二维几何拓扑，特别是对泰希米勒理论、二次微分、映射类群、模空间、随机双曲面、高阶泰希米勒理论的研究。

2. 分类问题与理论框架：三维和四维几何拓扑将作为会议的另一重要议题，讨论这些维度中的拓扑空间分类问题及其相关的数学理论框架，探索新的分类方法与算法。

3. 几何拓扑与同伦论的联系：高维几何拓扑与同伦论之间的关系仍是一个重要的研究方向。此次会议将进一步探讨同伦论在高维拓扑中的应用，特别是动机同伦论的拓扑视角。

The conference, themed "Geometric Topology and Related Topics," aims to promote frontier research in the field of geometric topology and interdisciplinary communication in general geometry and topology. As a vibrant and significant area of mathematical research, geometric topology deals with the complex structures and properties of multi-dimensional spaces, spanning a wide range of topics from low-dimensional to high-dimensional topology. This workshop provides a platform for researchers and students to come together, discuss the latest developments in geometric topology, share research findings, and foster academic collaboration.

The conference will focus on the following key topics:

Teichmüller Theory and higher Teichmüller Theory: Research will center on two-dimensional geometric topology, particularly the study of Teichmüller theory, quadratic differentials, mapping class groups, moduli spaces, random hyperbolic surface, higher Teichmüller theory.

Classification Problems and Theoretical Frameworks: Three-dimensional and four-dimensional geometric topology will be another central theme, addressing classification problems of topological spaces in these dimensions, and the mathematical theoretical frameworks involved.

Connections Between Geometric Topology and Homotopy Theory: The relationship between high-dimensional geometric topology and homotopy theory remains a key area of research. This session will further investigate the application of homotopy theory in high-dimensional topology, with a particular focus on the topological perspectives of motivic homotopy theory.

Geometric topology is a quite active research field in modern mathematics. In the current research, the two dimensional geometric topology studies about the Riemann surfaces, Teichmüller theory, moduli spaces and their dynamic systems, the 3 and 4 dimensional geometric topology concerns the classification problems, while the higher dimensional geometric topology relates to the motivic homotopy theory. This workshop will bring together experts to talk about recent developments in these related fields, discuss further questions of mutual interest, and seek possible new cooperation. More importantly, to encourage young mathematicians to participate more in these communities, we would like to make enough rooms for young scholars, including Postdocs and PhD students, to present their recent works or even work-in-progress. We believe TSIMIF is a perfect location for this purpose.

举办意义(Description of the aim)

This conference aims to bring together leading scholars in contemporary geometric topology to provide a high-level platform for exchanging ideas and discussing the latest advancements in the field. It will gather top researchers and emerging scholars from around the world to explore new developments in geometric topology. Our main objectives include:

1. Exploring new Developments
Geometric topology has undergone significant transformations in recent years, with breakthrough methods emerging in higher-dimensional spaces. This workshop aims to provide a comprehensive discussion of these innovative methods, with a focus on:

Recent developments in Teichmüller space

Recent developments in higher Teichmüller space

Classification problems in low-dimensional topology

Motivic homotopy theory and its geometric interpretation

2.  Interdisciplinary Dialogue
We are committed to creating a powerful knowledge-sharing platform that allows researchers to:
Share recent research findings: Through presentation sessions, participants will have the opportunity to present their work in detail and receive feedback from their peers.
Identify potential collaborative research opportunities: The conference will provide spaces for informal interactions, group discussions, and networking, where attendees can explore complementary aspects of their research and uncover potential cross-disciplinary and cross-team collaborations. This will promote synergistic innovation in the field of geometric topology.
Connect different subfields of geometric topology: By structuring the academic program, we aim to create dialogue platforms for researchers working in different branches of geometric topology, such as 2-dimensional, low-dimensional, and high-dimensional geometric topology. This will encourage participants to share the latest progress in their respective areas and explore potential pathways for cross-disciplinary research.

3. Nurturing Emerging Talent

A major highlight of this workshop is the special support for early-career mathematicians. The conference will provide focused opportunities for postdoctoral researchers and PhD students to present their latest research, including ongoing projects. This approach not only highlights emerging talent but also promotes knowledge exchange and potential future collaborations. Our plans include:

Providing speaking opportunities for postdoctoral and PhD researchers Offering opportunities for postdoctoral and PhD researchers to show their new work

本次会议旨在汇集当代几何拓扑研究最前沿的学者提供一个高水平的交流平台，汇聚各地的顶尖研究人员与新兴学者，共同探讨几何拓扑中的最新进展。我们的主要目标包括：

1. 探索前沿的新发展发展
几何拓扑近年来经历了重大变革，在多维空间中出现了突破性方法。本工作坊旨在全面讨论这些创新方法，重点关注：
- 泰希米勒理论的最新进展
- 高阶泰希米勒理论的最新进展
- 低维拓扑中的分类问题
- 动机同伦论及其几何学意义

2. 跨学科对话
我们致力于创造一个强大的知识分享交流平台，让研究者能够：
- 分享最新研究发现：通过报告环节，为参会学者提供详细阐述自身学术成果的机会。
- 识别潜在的合作研究机会：为参会者提供互动空间，使得参会者可以通过非正式交流、小组讨论等形式，发现彼此研究中的互补性，探索跨学科、跨研究团队的合作可能性，从而推动几何拓扑领域的协同创新。
- 连接几何拓扑的不同子领域：通过设计学术日程，为二维、低维、高维几何拓扑等不同分支的研究者创造对话平台，鼓励他们分享各自领域的最新进展，并探讨跨领域研究的潜在路径。

3. 培养新兴人才

本次研讨会的一大亮点是对早期职业数学家的特别支持。会议将特别关注为博士后研究人员和博士生提供展示其最新研究成果的机会，包括正在进行的研究项目。这种做法不仅突出了新兴人才，还促进了知识的交流与潜在的未来合作。我们计划：
- 为博士后、博士生研究者提供演讲机会
- 为博士后、博士生研究者提供展示新工作的机会

会议摘要（Abstract）

The stable homotopy groups of spheres have been a central and enduring topic in algebraic topology, playing a key role in our understanding of the deep structure of spaces. Over the decades, substantial progress has been made in both theoretical and computational aspects of these groups. Classical tools such as spectral sequences, especially the Adams and Adams-Novikov spectral sequences have provided a robust framework for advancing the theoretical understanding of stable homotopy groups. However, many questions remain unanswered.

This workshop will focus on the problem of computing stable homotopy groups, including both history and the recent advances in this subject. We will begin by providing an overview of the history and foundational concepts underlying the study of stable homotopy groups of spheres. We will also explore how these groups are connected to broader areas in algebraic topology and mathematics, such as cobordism theory, K-theory, and motivic homotopy theory.

The computational aspect has seen recent breakthroughs, particularly through the use of modern tools like motivic and equivariant homotopy theory, for example, the recent result by Lin, Wang and Xu. We will survey some of the latest results in these areas, highlighting the role of computational techniques, including the use of motivic tools, higher chromatic tools and spectral sequences, which have pushed the boundaries of what can be computed.

Despite these advancements, significant challenges remain in understanding the full structure of the stable homotopy groups of spheres. We will address some open problems in the field. Finally, we will outline possible future directions: where does the future of stable homotopy theory lie, and how might we move closer to a complete understanding?

举办意义(Description of the aim)

This workshop aims to gather leading experts and early-career researchers to explore the latest advances in the field of stable homotopy theory, with a particular focus on the calculation and structure of stable homotopy groups of spheres. As one of the most fundamental problems in algebraic topology, understanding these groups has led to deep insights across multiple areas of mathematics, including cobordism theory and K-theory.

Thematic Focus:

The workshop will revolve around the following thematic areas:

1. Computational Techniques and theoretical developments: Recent progress in computational methods, particularly the use of motivic and equivariant homotopy theory, has led to new insights into the stable homotopy groups of spheres. Sessions will focus on motivic homotopy theory, computation-assisted spectral sequences computation, and other advances in computational tools, offering a comprehensive overview of the current state of the art.

2. Interactions with Other Fields and Open Problems: The workshop will emphasize the intersections of stable homotopy theory with other mathematical areas. We will address some open problems in the field and outline possible future directions.

Distinguished Lectures (tentative):

The workshop will feature special lectures by Weinan Lin, Guozhen Wang and Zhouli Xu. whose pioneering work on Adams spectral sequence computation has largely extend our understanding of the stable homotopy groups.

In addition to these distinguished lectures, the workshop will include interactive problem-solving sessions, collaborative working groups, and open discussions on new computational challenges and conceptual developments. The goal is to foster collaboration, encourage the exchange of ideas, and inspire further research in stable homotopy theory and its related fields.