Dear participants, While planning our ISM School on Mutations, we tried to employ the successful experiences we had observed in other similar academic events. We are hoping that through the following short survey we could receive your valuable feedback on this school, which allows us to improve the events we will organize in the future. This survey consists of only 3 questions. You can write your response in the comment box below, for which you should first log in. If you wish to fill in the survey anonymously, you can use this Google survey: https://forms.gle/LZqLPtDX2pxnC7vh8 Should you wish to communicate your response with the organizers, please send your comments to the school's email address at rep.mutation.school@gmail.com [1] What do you think about the content and level of the mini-courses? In particular, do you feel that the level of the mini-courses was suitable to learn about the recent developments of the notion of mutation from different perspectives? [2] What do you think about the Discussion Sessions, in which the in-person participants were divided into smaller groups? Do you feel these sessions helped with a better understanding of the new subjects? Did they spark your interest in studying a problem related to this school? [3] What do you think about the Poster Session? Did you get the chance to present your work or communicate your thoughts with the other participants/poster presenters? Any further comments? Please share them with in the comment box or via an email.

As announced earlier, the in-person participants will have the opportunity to do collective research on some problems that can lead to new results. Among the problems posted on the Forum, the following 4 subjects are chosen as the main directions of collaborative research, to be conducted by 4 smaller subgroups. Every in-person participant is kindly asked to select at least 2 of these problems which are closer to their interests. Then, based on your choices, we split the in-person participants into smaller sugroups.

Attached, please see the posters submitted by some of the participants. By clicking on each poster, it opens in a new page and you can view it in full-screen mode. All participants are invited to exchange their questions/comments via the comment box below. On Thursday afternoon (July 7th), there will be a Poster Session for the in-person participants, where the presenters will give a short presentation and explain further details of their work. For the exact time, please see the schedule.

For a better format of the mathematical expressions, see the attached Pdf. file! This problem was suggested by Francesco Sentieri, based on a question posed by Haruhisa Enomoto on https://haruhisa-enomoto.github.io/problems/ Let A be a finite dimensional algebra. In the lattice of torsion classes tors(A), the completely meet-irreducible classes are in bijection with finitely generated bricks. More precisely, for a torsion pair (u,v) in mod-A, the class u is completely meet-irreducible if and only if there is a unique brick B in mod-A such that u = {M in mod-A | Hom_A(M;B) = 0} (see [2, Theorem 3.4]). Is there a similar characterization of meet irreducible torsion classes, maybe involving infinite dimensional bricks? Let us fix a torsion pair (u,v) in mod-A. We will see in my lecture series that (u,v) is associated to a cosilting torsion pair (U,V) = (direct limit of u, direct limit of v) in Mod-A, and to a non-empty semibrick consisting of the modules which are torsion-free, almost torsion with respect to (U,V). The modules in this semibrick may be infinite dimensional, the finite-dimensional ones correspond to torsion classes covering u. For details we refer to [2, 1, 3]. Question: Is it true that u is meet-irreducible if and only if there is precisely one torsion-free, almost torsion module with respect to (U,V)? As a first step, one should observe that if the class u is meet-irreducible but not completely meet-irreducible, then all modules which are torsion-free, almost torsion with respect to (U,V) are infinite dimensional. Then the question becomes: Question: Is it true that u is meet-irreducible but not completely meet-irreducible if and only if there is precisely one torsion-free, almost torsion module with respect to (U,V) and this module is infinite dimensional? Here are some more specific problems. Questions: (1): Is there a torsion pair (u,v) in mod-A such that u is not meet-irreducible and all torsion-free, almost torsion module with respect to (U,V) are infinite dimensional? (2) Is there a torsion pair (u,v) in mod-A such that u is not meet-irreducible and there are only a finite number of infinite dimensional torsion-free, almost torsion module with respect to (U,V)? References: [1] E. Barnard, A. Carroll and S. Zhu, Minimal inclusions of torsion classes, J. Algebraic Combin. 2 (2019) 879901. [2] L. Demonet, O. Iyama, N. Reading, I. Reiten and H. Thomas, Lattice theory of torsion classes, preprint (2017), arXiv:1711.01785. [3] F. Sentieri, A brick version of a theorem of Auslander, preprint (2020), arxiv:2011.09253.