Mats Boij (KTH)
Title: The weak Lefschetz property for artinian Gorenstein algebras of small Sperner number
Abstract: For artinian Gorenstein algebras in codimension four and higher, it is well known that the Weak Lefschetz Property (WLP) does not need to hold. For Gorenstein algebras in codimension three, it is still open whether all artinian Gorenstein algebras satisfy the WLP when the socle degree and the Sperner number are both higher than six. In recent joint work with Juan C. Migliore, Rosa M. Miró-Roig and Uwe Nagel we show that all artinian Gorenstein algebras with socle degree $d$ and Sperner number at most $d+1$ satisfy the WLP, independent of the codimension. This is a sharp bound in general since there are examples of artinian Gorenstein algebras with socle degree $d$ and Sperner number $d+2$ that do not satisfy the WLP for all $d\ge 3$.
Sara Faridi (Dalhousie)
Title: Extremal ideals
Abstract: Extremal ideals are square-free monomial ideals whose powers achieve maximal multigraded/graded/total betti numbers when compared to powers of any square-free monomial ideal. If $\mathcal{E}_q$ is the extremal ideal with $q$ generators, then all of its powers ${\mathcal{E}_q}^r$ are conjectured to have Scarf resolutions in the sense that every monomial in the minimal free resolution appears only once in the lcm lattice. The Scarf complex of ${\mathcal{E}_q}^r$ is conjectured to support a free resolution of $I^r$ when $I$ is any square-free monomial ideal with $q$ generators.
Despite a relatively straightforward combinatorial description, understanding the powers extremal ideals has turned out to be a challenging problem.
In this talk we will introduce extremal ideals, what is known about them, and the combinatorial nature of the Scarf complexes of their powers. I will be reporting on past an ongoing work with various subsets of the following coauthors: Trung Chau, Susan Cooper, Art Duval, Sabine El Khoury, Thiago Holleben, Hasan Mahmood, Sarah Mayes-Tang, Susan Morey, Liana \c{S}ega, Sandra Spiroff.
Brian Harbourne (UNL)
Title: Uwe Nagel’s impact on some recent research
Abstract: There is an expanding body of work on point sets in projective space whose projection from a general point has special properties. This work grew out of two papers for which Uwe is a co-author. I will briefly describe this foundational work and how it led to the notion of geproci sets. We will see that this notion has connections to algebraic geometry, commutative algebra, representation theory and combinatorics. I will also describe some recent work on geproci sets, including work of my students Jake Kettinger and Allison Ganger.
Rosa M. Miro-Roig (University of Barcelona)
Title: LAGRANGIAN SUBSPACES OF THE MODULI SPACE OF SIMPLE SHEAVES ON K3 SURFACES
Abstract: Let X beasmooth connected projective K3 surface over the complex numbers and let Spl(r;c1,c2) be the moduli space of simple sheaves on X of fixed rank r and Chern classes c1 and c2. In 1984, Mukai proved that Spl(r;c1,c2) is a smooth algebraic space of dimension 2rc2− (r − 1)c2 1 − 2r2 + 2 with a natural symplectic sstricture, i.e., it has a non-degenerate closed holomorphic 2-form. In my talk, I will present a useful method to construct isotropic and Lagrangian subspaces of Spl(r; c1, c2).
This is joint work with Barbara Fantechi Facultat de Matem` atiques i Inform` atica, Universitat de Barcelona, Gran Via des les Corts Catalanes 585, 08007 Barcelona, Spain
Email address: miro@ub.edu, ORCID 0000-0003-1375-6547
Chris Peterson (CSU)
Title: Two problems involving groups, graphs, and combinatorics
Abstract: I have been involved in several joint projects with Uwe and have taken inspiration from a large number of papers of Uwe, Juan, and others (many present in the audience). However, apart from the body of work alluded to in the first sentence, there is quite different body of Uwe’s work that has also provided inspiration to me; one that is more in the direction of combinatorics and graphs. In this talk I will present a pair of problems that are more aligned with this second body of work. In particular, we will see two problems related to groups, graphs, and combinatorics. One resembles Hilbert functions of Artinian algebras while the other involves an interaction more in the direction of Polya enumeration. Both problems are accessible in that they are elementary to describe. It is my hope that Uwe (and the audience) find these problems interesting and intuitive. The second problem involves joint work with Alissa Romero.
Victor Reiner (UMN)
Title: Macaulay inverse systems and graded Ehrhart theory
Abstract: For any configuration Z of finitely many points in space, commutative algebra provides a natural q-analogue for its cardinality |Z|: the Hilbert series for the associated graded ring of the coordinate ring of Z, thought of as an affine variety. We use this to study a q-deformation of the Ehrhart series of a lattice polytope P, by viewing the lattice points inside each integer dilation mP as a finite point configuration, and tracking the q-analogues of their cardinalities.
For any configuration Z of finitely many points in space, commutative algebra provides a natural q-analogue for its cardinality |Z|: the Hilbert series for the associated graded ring of the coordinate ring of Z, thought of as an affine variety. We use this to study a q-deformation of the Ehrhart series of a lattice polytope P, by viewing the lattice points inside each integer dilation mP as a finite point configuration, and tracking the q-analogues of their cardinalities.
(based on arXiv:2407.06511, with Brendon Rhoades)
Adam Van Tuyl (McMaster)
Title: Vertex decomposable simplicial complexes: Uwe’s contributions and influence
Abstract: In this talk, I will survey some of Uwe Nagel’s contributions to the study of vertex decomposable simplicial complexes and their Stanley-Reisner ideals.