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.