Abstract 

Surfaces in 3-space are one of the most fundamental objects to study in geometry, analysis, and physics. They appear naturally as optimal shapes under certain energy constraints such as surface tension or surface bending. Understanding and characterizing these shapes requires a plethora of methods from a wide ranging list of current mathematical activity, including the analysis of special partial differential equations, the study of energy minimizing as well as energy preserving flows, Riemann surface theory, discretization, computational and experimental methods, and computer visualizations. The workshop brings together key researchers from these areas for the first time to exchange techniques, concepts, ideas, to learn about their respective important questions and open problems, and to foster long-term collaboration. 

Description of the aim 

The focus of the proposed workshop lies almost exclusively on global problems in surface geometry. This area has seen significant progress in the interplay between geometric analysis, integrable systems, Higgs bundles, mathematical physics, and discretizations /experimentations. Recent examples of these profound cross-fertilizations include: 

1. The construction of new examples of compact higher genus constant mean curvature and minimal Lagrangian surfaces via methods from loop groups and surface group representations complementing the existing results obtained via non-linear gluing perturbation techniques. This approach also gives explicit expansions in the genus of the areas/energies of the con-structed surfaces proving, for instance, the conjectured monotonicity in the genus of the are as of Lawson's minimal surfaces.

2. The construction of special three dimensional Calabi-Yau metrics with singularities along aY-vertex using Higgs bundles and their relationship to hyperbolic afine spheres.

3. The resolution of the classical Bonnet problem-are there two non-congruent compact surfaces in

3- space with the same mean curvature and same induced metric by first applying the discrete theory of is other mic surfaces to find a discrete Bonnet pair using a construction first explained by Bianchi.

4. The constructions of surface geometries realizing certain Higgs bundle moduli spaces in higher Teichmueller theory.

5. The resolution of the (generalized)Lawson conjecture about embedded constant mean curvature tori in the 3-sphere using ideas which emerged in the study of mean curvature fow.

6. The classification of minimal planar domains in R3 using the KdV hierarchy

These developments strongly support the timeliness of the proposed workshop which will join key researchers from these various camps and bring ideas and techniques of one area to bear on another. Over the past decades researchers from the various groups-integrable systems; geometric analysis: numerics, discretization, visualization-have communicated individually and invited each other to conferences in their respective areas, Geometric-analytic existence proofs of minimizers of cer-tain functionals, such as the bending/Willmore energy, were complimented by viable candidates olthe minimizers found by integrable systems methods. This allowed for extensions of the analytical methods to investigate more general problems, such as the characterization of minimizers of the bending energy in a fixed conformal class. The wealth of examples of variational surfaces (minimal, constant mean curvature, Will more obtained by integrable systems methods allows for wide ranging computer experiments and visualizations. These provide credence to existing conjectures and often show novel, unexpected features giving rise to hypotheses and supporting evidence of much further reach, thereby stimulating new directions of inquiry.

It is fair to say that there are areas of commonality in the described approaches, but the challengeis to find bridges between the geometric-analytical techniques providing existenee and regular ityresults, the integrable systems methods providing a wealth of examples and sometimes even a complete description of all solutions, and the experimental camp exploring the space of solutions and predicting properties which, as of yet, are not within reach of theoretical investigations.

Areas where stronger links between the areas could be advantageous include：

1. Moebius invariance of the Willmore energy. A fact well known and taken advantage of in the geometric approach to integrable hierarchies but seemingly playing a less important role in geometric-analytical investigations and in the mathematical physics community.

2. Viewing surfaces as complex one dimensional curves, with its plethora of techniques such as Higgs bundles/algebraic geometry/delbar analysis, is consistently employed in the geometry community but has vet to fnd its way into the analvtical community.

3. Evidence of stability properties of stationary solutions, collected from explicit examples constructed by integrable methods and computer experiments, are less visible in geometric analytical approaches focusing on minimizers. Though recently it has been shown that all Willmore spheres, except the round sphere, are unstable.

4. Bringing together key researchers from these groups for the frst time will provide an ideal environment for participants of those communities to interact, to exchange techniques, concepts. ideas, to learn about their respective important questions and open problems, and to foster long term collaboration. There certainly is shared optimism that such a meeting is the first step for a much deeper understanding of major open problems in the theory of surfaces, including stability questions of stationary surfaces which could provide a more comprehensive view of the Will more conjecture and its resolution; the construction of higher genus compact constant mean curvature surfaces, especially of embedded examples, leading to a resolution of the higher genus Laws on conjecture; the determination of minimizing Willmore tori of fixed conformal type; and some inkling of a higher genus Willmore conjecture.

 The format of the workshop is designed to enable and enhance this dialogue. The conference will feature early overview lectures by prominent researchers with a reputation for accessible talks by each group. We have included in the list of invitees mathematicians who are known to build communities of researchers and stimulate collaboration. The initial lectures and discussions will provide a common backdrop for the subsequent, more research-oriented, talks and, at the same time. will provide junior researchers with a bird's eye view of the various subject areas. These will be followed by more traditional and narrowly focused talks on contemporary research topics and recent results. Even though researchers like to talk about problems they have solved, we will nevertheless ask speakers to begin their lectures with a fair amount of context and to end their presentations with a discussion of important open problems. Overall, we aim for an uncluttered schedule with ample time periods reserved for discussions, sharing and exchanges of ideas, and opportunities for building collaborations. We have contacted a fair number of more senior participants about our workshop and received strong encouragement and positive responses to participate. Communications and joint projects between the Chinese and western mathematical communities have seen a significant increase, especially in areas central to this proposal, lending itself to the choice of 'TSIMF for our workshop. Considering China's long history, its large mathematical community, and geographical expanse, this meeting will build upon and further connections, cooperations, and collaborations, by inviting Chinese researchers with less access to such meetings primarily held in the West. We aim for roughly half of the participants to work out of China and also invitere searchers from the far East, Japan, and Australia. We also expect to have a number of Ph.D students attend for whom there will be supplementary support by grants of senior participants.