MFO-RIMS Tandem Workshop: Symmetries on Polynomial Ideals and Varieties

The study of symmetry as a structural property of algebraic objects is one of the fundamental pillows of the developments of modern mathematics, most prominently beginning with the work of Abel and Galois. The focus of the workshop was on permutation actions of the symmetric group on polynomial rings and algebraic and semi-algebraic sets. More concretely, it was centered around recent developments in the asymptotic setup of symmetric ideals in the polynomial ring in infinitely many variables. Mathematics Subject Classification (2010): 13A99, 13F20, 13P10,20B30. Introduction by the Organizers The MFO-RIMS Tandem Workshop Symmetries on Polynomial Ideals and Varieties organised by Gunnar Fløystad (Bergen), Satoshi Murai (Tokyo), Cordian Riener (Tromsø), and Kohji Yanagawa (Osaka) took place between September 5– 11 2021 as a joint hybrid workshop hosted at the Mathematical Research Institute Oberwolfach and the Research Institute for Mathematical Sciences in Kyoto. Unfortunately, the ongoing Covid-19 pandemic made it impossible to have a physical meeting in Kyoto and the workshop was therefore attended by 14 participants physically at MFO and another 24 participants who joint in virtually. The main focus of the workshop was on the study of properties of ideals in the infinite polynomial ring which are invariant with respect to actions of the 2 Oberwolfach Report 43/2021 symmetric group. As has been observed by various authors, many properties of classical commutative algebra, which fail for the infinite polynomial ring, can be restored when considered in an equivariant setup, up to symmetry. Most notably, it had been already observed by Cohn in the 1980s that although the infinite polynomial ring is not Noetherian symmetric, ideals can always be generated by finitely many orbits of polynomials. In recent years renewed motivation to study such ideals stemmed from the observation that one can view such ideals as limits of sequences of algebraic objects which arise in the area of algebraic statistics and algebraic chemistry. One main goal of the joint workshop between the two research institutes was to bring together different mathematicians working on various aspects of this thematic area and, in particular, build new research connections between Europe and Asia around this topic. The technical facilities in the lecture halls allowed recording of the talks and interactions between remote participants. We would say this worked well, with some minor technical problems. On the Japanese side they had the disadvantage that each participant was in their home or office, so all interaction was digital. Nevertheless, questions after talks, and problem discussions worked well interactively. Also, the time difference between the two groups of participants made it necessary to distinguish between talks, which could followed by all participants live and talks which were recorded because their time slot was falling into very early hours in Europe or very late hours at night in Japan. But all in all, the hybrid setup allowed for a rich program of diverse talks, joint problem sessions as well as discussions. The workshop had talks on the following topics: • Stability of incand symchains of ideals. • Hilbert functions and free resolutions of chains of ideals. • SAGBI basis approach to invariant rings • Polynomial representations. • Specht ideals. • Equivariant modules and sheaves. • Cohomology of symmetric semi-algebraic sets. The presentations featured both recent developments on these topics as well introductory expositions. On Monday we had a shorter problem session with spontaneous volunteers. For Wednesday we took a more active approach. On Tuesday we requested and approached participants to prepare and inform on problems. The whole Wednedsay morning we then had a planned session of problem presentations. First, two hours from the Oberwolfach side, and then one hour from the RIMS side. This worked very well. Both more systematic programs of problems were presented, for instance on asymptotic sequences of symmetric ideals, as well as more concrete specific problems. The workshop was on a timely topic with recent surging interest. We believe it was very inspiring and informative to both the Oberwolfach and RIMS participants. From the Oberwolfach side it was especially Symmetries on Polynomial Ideals and Varieties 3 useful to be informed on the diversity of activity in Japan, and similarly the talks in Oberwolfach were a strong inspiration for participants in Japan. We would like to especially thank the staff of MFO for their excellent additional efforts which ensured to provide perfect working conditions also under the special circumstances of the Covid-19 pandemic and allowed for collaboration between on site and remote participants of the workshop. Symmetries on Polynomial Ideals and Varieties 5 MFO-RIMS Tandem Workshop (hybrid meeting): Symmetries on Polynomial Ideals and Varieties