会议摘要（Abstract）

The topic of this workshop is the Hilbert module approach in operator theory, with an emphasis on the techniques of analytic function theory, complex geometry, and algebraic geometry. Operator theory, and more specifically, multivariable operator theory, and the aforementioned subjects share intimate connections. These subjects are closely related to a variety of disciplines, including PDE, operator algebras, linear analysis, and harmonic analysis, to mention only a few. The theory of operators evolved from the study of normal operators, Toeplitz operators, the Volterra operator, and index theory. A milestone is the Nagy-Foias analytic operator model theory developed in the 1950s and 60s, which states that every bounded linear operator can be represented as a compression of the shift operator to a certain Hilbert space of holomorphic functions. This fact reaffirms the algebraic link between traditional operator theory and function theory. Around the same time, index theory was given a topological, geometrical, and analytic framework in the deep work of Brown-Douglas-Fillmore, which in part motivated the development of noncommutative geometry. In light of these and other intriguing theories, it became clear that a more general framework is required to unify all the pertinent concepts. R. G. Douglas' introduction of Hilbert modules in the 1980s came at an opportune time. The theory makes available diverse tools and techniques from a wide range of fields, such as commutative algebra, complex geometry, several complex variables, and algebraic geometry, to name a few, for the study of operator systems and multivariable spectral theory. Indeed, with the passage of time, a great number of achievements have been made along this line, for example in the study of

 1) Cowen-Douglas operators,

 2) distinguished varieties (a concept introduced by Rudin),

 3) the interpolation problem,

 4) extension of analytic functions from algebraic variety,

 5) bounded symmetric domains,

 6) the Chevalley-Shephard-Todd theorem,

 7) the Riemann zeta function in terms of infinite polydisc,

 8) characteristic spaces.

Additional success has been recorded in the study of Samuel multiplicity, analytic K-homology, projective spectrum, bounded analytic functions, etc.

The theory of Hilbert modules has been very actively pursued since the last meeting on this subject, which took place at TSIMF shortly before the Covid pandemic. This proposed workshop aims to analyze the subject's evolution in recent years and outline possible future directions for growth. Particular attention will be given to early-career researchers. Indeed, a significant percent of the proposed participants are PhD students, postdocs, and tenure-track assistant professors. The proposed workshop will provide a good opportunity for them to communicate in-person and foster collaborations.

举办意义(Description of the aim)

The primary aim of the workshop is to provide a platform for the dissemination of recent research discoveries and to highlight the key challenges in the field of operator theory after reformulating them in the language of Hilbert modules. Over the past three decades, it has been amply clear that this reformulation is not merely a choice of language. It in fact creates a new landscape. For instance, submodules in several variable have complicated structure, and classifications of them become an appealing and yet rather challenging task. On the other hand, the study of quotient modules amounts to develop a multivariable Nagy-Foias model theory, which provides a fertile ground for the growth of multivariable operator theory.

The main themes of this workshop are as follows:

(A) Submodules and quotient modules over function algebras and the corresponding resolutions of Hilbert modules.

(B) Function theory on an infinite polydisc.

(C) The theory of bounded analytic functions in several variables and its connection to Hilbert modules.

(D) Multivariable spectral theory.

In terms of scientific goals, it is understood that function theory, multivariable operator theory, and the Hilbert module structure that goes with them depend heavily on the specific domain. The function theory of the open unit ball and that of the open unit polydisc, for example, differ significantly. Another emerging theory, as far as domain-related studies are concerned, is the Riemann zeta function in terms of function theory on an infinite polydisc. Indeed, the manner in which function theories vary across distinct domains is remarkable and fascinating. How these differences play a role in the study of Hilbert modules is worth serious investigation. It is undoubtedly a part of the allure of Hilbert modules and multivariable operator theory.

Another important subject concerns with essentially normal quotient modules. On infinite-dimensional Hilbert spaces, compact operators are the ``small'' objects that introduce numerical invariants, such as the index of a Fredholm operator. Self commutators [T, T*] and cross commutators [T_1, T_2*] of multiplication operators$T, T_1, and T_2 on Hilbert modules are often ``small'', prompting the question whether they are in fact ``smaller'', i.e., whether they belong to the so-called Schatten p-class. For homogeneous submodules in the Drury-Arveson space over the unit ball, this question is framed as the Arveson-Douglas conjecture, which has been a tantalizing open problem for some time. In fact, it is anticipated that the answer depends on the geometry of the zero variety of such submodules. This connection facilitates the study of analytic K-homology in an appropriate way. This workshop shall review the current status of the conjecture and investigate on recent novel approaches.

In addition to the aforementioned topics, the workshop also concerns with the following list of problems.

(i) The classification of submodules and quotient modules of analytic Hilbert modules.

(ii) Analytic and algebraic invariants of the submodules and quotient modules with respect to unitary equivalence, similarity, and quasi-similarity.

(iii) Investigating the essential normality of Hilbert modules, including the Douglas-Arveson conjecture mentioned above.

(iv) The problem of holomorphic imprimitivities as the restrictions of  imprimitivities in the sense of Mackey and its close connection with the subnormality of Cowen-Douglas modules.

(v) The problem of finding a complete set of invariants for holomorphic hermitian sheaves.

(vi) Interpolation on various domains and its consequences for Hilbert modules.

(vii) Spectral theory in the setting of Hilbert modules.

(viii) Beurling-Wintner dilation problem.

(ix) Szego's problem in infinite polydisc.

(x) Determinantal point process from multivariable operator theory. Undoubtedly, some of the aforementioned problems are complex and necessitate a long-term approach. We hope that this workshop will also help identify rising stars who might lead future efforts in addressing these challenges. Therefore, it will be in our best interest to invite a large number of young researchers to this workshop and foster their growth for the advancement of mathematics.

Previous Workshops: Some of the events that have taken place in the past few 15 years:

(i) Hilbert Modules in Analytic Function Spaces workshop,  Tsinghua Sanya International Mathematics Forum (TSIMF), Sanya, China, Dec 30, 2019 - Jan 3, 2020.

(ii) Hilbert Modules in Analytic Function Spaces workshop,  Tsinghua Sanya International Mathematics Forum (TISMF), Sanya, China, May 22-26, 2017.

(iii) Multivariate Operator Theory, BIRS, Banff, Canada, April 6 - 10, 2015.

(iv) Hilbert Modules and Complex Geometry, Oberwolfach, Germany, Apr 20- 26, 2014.

(v) Multivariate Operator Theory, BIRS, Banff, Canada, August 15 - 20, 2010.

(vi) Hilbert Modules and Complex Geometry, Oberwolfach, Germany, Apr 12- 18, 2009.

Abstract

The topic of this conference is Geometric Analysis, with emphasis on the research topics related to conformal geometry and CR geometry and their geometric flows. This includes existence and uniquesness of Poincare-Einstein metric, Paneitz operator, Q-curvature, Q’-curvature in CR geometry, Yamabe problem, Yamabe flow, CR Yamabe problem, related fully nonlinear partial differential equations. Other topics in Geometric Analysis will also be covered, such as, minimal surface, mean curvature flow, and geometric partial differential equations. The purpose of this meeting was to bring together international experts working on various aspects of conformal and CR geometry so that research ideas may be exchanged, and our understanding in conformal and CR geometry may be deepened.

Description of the aim

In this workshop, we plan to cover a wide variety of topics in Geometric Analysis, with emphasis on conformal geometry and CR geometry and their geometric flow. We plan to invite speakers to talk about the most recent topics in this field. Topics include CR Yamabe problem, minimal surface, mean curvature flow, Paneitz operator, Poincare-Einstein metric, Q-curvature, Q’-curvature in CR geometry, Yamabe problem, Yamabe flow, related fully nonlinear partial differential equations, and geometric partial differential equations.

There will be mathematicians from all over the world participating in the workshop, including Brazil, Hong Kong, Japan, Korea, Taiwan, USA, and Vietnam. Through this, we hope that participants from different research areas could learn from each other, and ideas and thoughts could be exchanged during the workshop, and new research results could be generated through it.

There will be a significant number of young mathematicians, who are PhD student, postdoc or assistant professor, attending the workshop. We hope that this will foster the young generation in the area of Geometric Analysis, especially those who are working in conformal geometry and CR geometry.

There will be 19 speakers: 2 speakers in the morning and 3 speakers in the afternoon every day from Monday to Friday except Wednesday; there will be 2 speakers in the morning of Wednesday, and a free discussion in the afternoon of Wednesday. 

Here is a tentative list of speakers: Jun-Cheng Wei, Pak Yeung Chan, Jih-Hsin Cheng, Eric Chen, Fang Wang, Kazuo Akutagawa, Yi Wang, Andrew Waldron, Keomkyo Seo, John Man Shun Ma, Yuya Takeuchi, Hung-Lin Chiu, Xingwang Xu, Sergio Almaraz, Seunghyeok Kim, Wei Yuan, Mijia Lai, Quoc Anh Ngo, Zongyuan Li.

This will bring together experts from different mathematical backgrounds with the goals of furthering and deepening our understanding of conformal and CR geometry and of strengthening the connections and analogies between the two fields.

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.