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 $d\ge 3$.
Giulio Caviglia (PU)
Title: Generic Distractions and Initial Ideals
Abstract: Given a homogeneous ideal I in a polynomial ring over a field and a monomial order, we iteratively compute initial ideals and ad-hoc monomial distractions in order to construct a distinguished monomial ideal associated to I. We call it the distraction-generic initial ideal of I and denote it by D-gin(I). Such an ideal, in characteristic zero, agrees with the usual generic initial ideal of I. Furthermore, it is strongly stable in any characteristic and when computed with respect to the reverse lexicographic order has properties analogous to gin(I). As an application, we use it to extend to positive characteristic some work done by Mall in the nineties on strata of Hilbert schemes defined by upper bounds on the Castelnuovo-Mumford regularity.
[This is a joint work with Anna-Rose Wolff]
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.
Elisa Gorla (UniNE)
Title: Multiplicity-free prime ideals are glicci
Abstract: A central question in Gorenstein liaison asks whether every Cohen-Macaulay ideal is glicci, i.e., whether every Cohen-Macaulay ideal belongs to the Gorenstein liaison class of a complete intersection. After introducing Gorenstein liaison and motivating this question, I will report on a joint work with Rajchgot and Satriano, where we show that every multiplicity-free prime ideal is glicci.
Tai Huy Ha (Tulane)
Title: Binomial expansion and rational powers of sums of ideals
Abstract: Let $A$ and $B$ be algebra over a field $k$, and let $R = A \otimes_k B$. Let $I \subseteq A$ and $J \subseteq B$ be nonzero proper ideals. We shall discuss the binomial expansion formulas for powers of the sum $I+J$ in $R$. Particular attention will be paid toward rational powers of $I+J$, when $I$ and $J$ are monomial or classical invariant ideals.
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.
Martina Juhnke-Kubitzke (Osnabrück U)
Title: Chains of symmetric simplicial complexes
Abstract: Given an ascending chain $\Sym$-invariant squarefree monomial ideals, we study the corresponding chain of Alexander duals. We provide an explicit description of the minimal generating set up to symmetry in terms of the original generators and show that the cardinality of a minimial generating set is given by a polynomial inn for sufficiently large n. The same is true for the number of orbit generators of minimal degree, this degree being a linear function in n eventually. As an application, we show that, for each i the number of ii-dimensional faces of the associated Stanley-Reisner complexes is also given by a polynomial in n for large n. This is joint work with Uwe, Ayah Almousa, Kaitlin Bruegge, Uwe and Alexandra Pevzner. Building on this work, in ongoing work with Uwe, we ask which chains of symmetric simplicial complexes are Cohen-Macaulay or shellable and try to describe the behavior of their algebraic invariants, including depth, regularity and projective dimension.
Patricia Klein (TX A&M)
Title: Even Gorenstein liaison classes, elementary G-biliaison, and applications
Abstract: Gorenstein liaison was introduced by Schenzel in 1983, building off of the more classical study of complete intersection liaison, introduced by Peskine and Szpiro in 1974. The goal of liaison theories is to connect schemes one wants to study to schemes that are easier to study in a controlled enough manner to transfer informative invariants. We will focus in this talk on subschemes of projective space that can be Gorenstein linked in an even number of steps and describe information that can be passed from one to another in this case, leaning heavily on results due to Nagel, alone and with coauthors. We will pay particular attention to Hartshorne’s elementary G-biliaison, an isomorphism that induces a sequence of two Gorenstein links, and its past and current use in the theory of Grönber bases.
Linquan Ma (Purdue)
Title: The Stuckrad-Vogel constant and its variations
Abstract: Let (R,m) be a Noetherian local ring. The Stuckrad-Vogel constant is the supremum of the ratios between colength and multiplicity among all m-primary ideals. This invariant is finite if and only if the m-adic completion of R is equidimensional. We will sketch a proof of this fact. Then we will introduce some related invariants regarding colength and multiplicity, and discuss several open questions. The talk is based on joint work with Klein, Pham, Smirnov and Yao, as well as joint work with Pham and Smirnov.
Juan Migliore (ND)
Title: A brief overview of Uwe Nagel’s work
Abstract: We describe some of Uwe’s many contributions in commutative algebra, algebraic geometry and combinatorics.
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.
Claudiu Raicu (ND)
Title: Cohomology of line bundles on the incidence correspondence
Abstract: A fundamental problem at the confluence of algebraic geometry, commutative algebra and representation theory is to understand the structure and vanishing behavior of the cohomology of line bundles on (partial) flag varieties. Over fields of characteristic zero, this is the content of the Borel-Weil-Bott theorem and is well-understood, but in positive characteristic it remains wide open, despite important progress over the years. In my talk I will describe recent developments in the case of the incidence correspondence – the partial flag variety consisting of pairs of a point in projective space and a hyperplane containing it.
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)
Tim Römer (UOsnabrück)
Title: From extended degree functions to commutative algebra up to symmetry
Abstract: Over the last two decades, Uwe and I have worked on various topics in commutative algebra.
A summary of the joint work with Uwe Nagel is presented from a very personal point of view.
Hal Schenck (Auburn University)
Title: Syzygies of permanental ideals
Abstract: We describe the minimal free resolution of the ideal of 2×2 subpermanents of a 2×n generic matrix M. In contrast to the case of 2×2 determinants, the 2×2 permanents define an ideal which is neither prime nor Cohen-Macaulay. We combine work of Laubenbacher-Swanson on the Gröbner basis of an ideal of 2×2 permanents of a generic matrix with our previous work connecting the initial ideal of 2×2 permanents to a simplicial complex. The main technical tool is a spectral sequence arising from the Bernstein-Gelfand-Gelfand correspondence. Joint work with F. Gesmundo, H. Huang, J. Weyman
Alexandra Seceleanu (UNL)
Title: Artinian Gorenstein algebras having binomial Macaulay dual generator
Abstract: One of the areas in which Uwe Nagel has made major contributions is the study of graded Artinian rings. In this talk, we focus on a special family of Artinian Gorenstein rings, which exhibit connections to his work. Every graded Artinian Gorenstein ring corresponds via Macaulay-Matlis duality to a homogeneous polynomial, called a Macaulay dual generator. In this way, Macaulay dual generators which are monomials correspond to monomial complete intersection rings. Monomial complete intersections have particularly nice properties: their natural generators form a Gröbner basis, their minimal free resolutions are well understood, and they satisfy the strong Lefschetz property in characteristic zero. In this talk we consider Macaulay dual generators which are the difference of two monomials and we seek to understand to what extent the properties listed above still persist for the corresponding Artinian Gorenstein algebras. This is joint work with Nasrin Altafi, Rodica Dinu, Sara Faridi, Shreedevi K. Masuti, Rosa M. Miró-Roig, and Nelly Villamizar.
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.
Junzo Watanabe (TokaiU)
Title: A new example of a principal radical system.
Abstract: Let R = K[x1, x2, . . . , xn] be the polynomial ring. Let h be an integer n ≥ h ≥ 1. Let b(h, n − h) = ∩ σ∈Sn σ(x1, x2, · · · , xh), where Sn is the symmetric group acting on R by permutation of variables. b(n, h) is the ideal generated by all square free monomials of degree n−h+1. Let n ≥ h > [n/2]. Let a(h, n − h) = ∩ σ∈Sn σ(x1 − x2, x2 − x3 · · · , xh − xh+1). This is an intersection of linear primes generated by h binomial linear elements of the form xi − xj. We prove that the ideal I = a(h, n − h)∩(b(n, h) : xnxn−1 · · · xn−k) can be generated by products of Specht polynomials and square-free mono-mials. (k is an integer k = 0, 1, 2, . . . , n.) (If k < h, then I = a(n, h).) This is a new proof for the fact which says that the Specht ideals corresponding to two rowed Young diagrams are radical. Also this may be regarded as a new example of a “principal radical system” introduced by Hochster and Eagon in 1971 for a proof for the perfection of determinantal ideals.