There are currently 7 publicly available research papers linked to the project. Preliminary work for 3 publications on topics of the fellowship has been carried out and the PI will work on finishing these in the coming months.
(1) Regarding construction of NCRs, the PI considered 1-dimensional commutative local rings. The main results are a characterization of certain of these rings in terms of trace ideals, and a characterization of the curves for which NCRs of global dimension 3 exist.
In collaboration with G. Muller and K.E. Smith the PI studied NCRs for toric rings R. They prove with elementary methods that the endomorphism ring of the sum of all (isomorphism classes of) conic R-modules has finite global dimension equal to the dimension of R. It is shown that this is a NCCR if and only if the toric variety Spec(R) is simplicial. For toric varieties over a perfect field of prime characteristic, it is shown that the ring of differential operators has finite global dimension.
(2) The main result of the collaboration with Buchweitz and Ingalls is that if the finite group G is generated by reflections of order 2 (i.e. G is a true reflection group), then a quotient of the skew group ring is isomorphic to the endomorphism ring of the hyperplane arrangement over the discriminant of G. This yields a NCR of the discriminant of G. The indecomposable projective modules over the quotient of the skew group ring are in bijection with the non-trivial irreducible representations of G and also with maximal Cohen-Macaulay-modules over the discriminant. This gives a McKay correspondence for true reflection groups.
As a byproduct, Buchweitz, the PI, and Ingalls, studied the magic square of reflections and rotations in dimension 2 and they show how to interpret it in terms of Clifford algebras.
In order to understand the geometry for pseudo-reflection groups better, the PI studied McKay quivers, together with C. Ingalls and M. Lewis. The main results are: a complete combinatorial description of the McKay quivers of the infinite family of pseudo-reflection groups G(r,p,n), and the definition of Lusztig algebras, which are path algebras of the McKay quiver.
(3) In collaboration with K. Baur, S. Gratz, K. Serhiyenko, and G. Todorov, the PI constructed SL_k-friezes from Plücker coordinates,
using the cluster structure on the homogeneous coordinate ring of the Grassmannian of k-spaces in n-space A(k,n). One of the main results is that when the cluster algebra A(k,n) is of finite type, then SL_k-friezes are in bijection with mesh friezes of the corresponding Grassmannian cluster category.
Together with J. August, M.-W. Cheung, S. Gratz, and S. Schroll, the PI constructs Grassmannian categories of infinite rank, providing an infinite analogue of Grassmannian cluster categories. They show that generically free modules of rank 1 in a Grassmannian category of infinite rank are in bijection with the Plücker coordinates in an appropriate Grassmannian cluster algebra. This work combines the study of matrix factorizations and cluster categories.
Transfer-of-knowledge: During the fellowship, the PI started mentoring a Ph.D. student, who is working on problems related to topic (2). The fellowship supported the PI through travel money and time that she could spend on the organisation of 5 scientific events (3 workshops, 1 conference, and 1 special session).
Dissemination: the PI gave 25 talks at international conferences, workshops, and research seminars. We mention as highlights 3 talks at workshops in Oberwolfach and CMO Oaxaca, and talks at conference on singularity theory at TSIMF in Sanya and an online conference on the McKay correspondence hosted by Kavli-IPMU.
