Registration/Opening Remarks
Speaker: Xianglong Ni
Title: Perfect ideals with fixed deviation and type
Abstract: Two important parameters associated to a perfect ideal are its deviation and type, which are reflected in the "ends" of its minimal free resolution. Does fixing these two ends give an upper bound on the intermediate Betti numbers? This question is trivial in small codimension, but the situation is subtle in general and reveals a surprising connection to representation theory.
Speaker: Anastasia Nathanson
Title: Permutation action on chow rings of matroids
Abstract: Given a matroid with a symmetry group, we study the induced group action on the Chow ring of the matroid with respect to symmetric building sets. This turns out to always be a permutation action. Work of Adiprasito, Huh and Katz showed that the Chow ring satisfies Poincaré duality and the Hard Lefschetz theorem. We lift these to statements about this permutation action, and suggest further conjectures in this vein.
Turbo Talks
Conference Photo
Lunch
Speaker: Karthik Ganapathy
Title: Stability patterns in free resolutions of symmetric ideals
Abstract: The Le--Nagel--Nguyen--Römer conjectures predict that for chains of symmetric ideals in polynomial rings with a growing number of variables, both the projective dimension and regularity are eventually linear functions in the number of variables. In this talk, I will discuss recent progress toward resolving these conjectures, which remain open.
Speaker: Feiyang Lin
Title: Resolving the singularities of splitting loci
Abstract: A vector bundle on a scheme B x P^1 gives rise to a stratification of the base B based on how the vector bundle splits on each fiber. The strata that arise are called splitting loci. In this talk, I will construct a modular resolution of singularities for each splitting locus. Using Hurwitz-Brill-Noether theory, this construction recovers the classical Gieseker-Petri theorem that for a general curve of genus g, G^r_d is smooth. If time permits, I will also outline how to use this construction to prove that certain tame splitting loci have rational singularities. As a corollary, we prove that components of the Brill-Noether variety W^r_d for a general k-gonal curve have rational singularities.
Speaker: Jesus Ruiz Bolanos
Title: The algebraic structure of \(\mathbb F_q\)-linear skew cyclic \(\mathbb F_{q^m}\)-codes
Abstract: In \({\it error-correcting codes}\), we are interested in designing strategies to detect and correct errors that arise from communication and storage problems. The ring of \({\textit {skew polynomials}}\) is a non-commutative collection of polynomials with very different properties when compared with the usual ring of polynomials. This structure has motivated more than one strategy to solve the \({\textit {error correction problem}}\).
In this work, we analyze a variety of \(\mathbb{F}_q\)-linear evaluation codes over \(\mathbb{F}_{q^m}\), where the evaluation is realized in the classical sense. These codes exhibit properties similar to those found in the skew cyclic and skew quasi-cyclic codes. This connection motivates the study of \(\mathbb{F}_q\)-linear skew quasi-cyclic \(\mathbb{F}_{q^m}\)-codes, where \(q\) is a prime and \(m\) is a positive integer. We determine their module structure and propose a definition of the dual space that satisfies \(\left(\mathbb C^\perp\right)^{\perp} = \mathbb C\).
Speaker: Naufil Sakran
Title: Unipotent numerical monoids
Abstract: In this talk, I will introduce the theory of unipotent numerical monoids. This theory serves as the generalization of the theory of numerical semigroups - complement-finite submonoids in \(\mathbb{N}\) - in the setting of unipotent linear algebraic groups. I will then discuss the theory of relative ideals in this context and give the classification of irreducible unipotent numerical monoids with respect to symmetric and pseudo-symmetric classes. I will conclude by outlining future directions and potential applications of this theory.
Coffee
Speaker: Caitlin Davis
Title: Weighted rational curves and the (nonstandard) Koszul property
Abstract: (Joint work with Ola Sobieska.) The rational normal curve is a well-understood projective variety whose coordinate ring has many nice algebraic properties. We consider an analogous family of curves in weighted projective space, and we show that this family of curves has many of the same algebraic properties as the rational normal curve. In this talk, I will introduce the weighted rational curves as a case study for the (nonstandard) Koszul property.
Speaker: Bek Chase
Title: The Lefschetz properties for some modules over polynomial rings
Abstract: The Lefschetz properties is an active research area in commutative algebra with strong connections to many other areas of math, including algebraic geometry, topology, and combinatorics. In this talk, I will introduce the Lefschetz properties, starting with a bit of the history and geometric motivation for the problem, then discuss the setting of modules over polynomial rings. Using a result in combinatorics on lattice path enumerations, I will demonstrate one fun approach to determining when modules have the strong Lefschetz property.
Speaker: Nawaj KC
Title: Modules of finite length and finite projective dimension
Abstract: If R is a local ring of dimension \(d\) and \(x_1, .., x_d\) is a maximal regular sequence, then \(M = R/(x_1, .., x_d)\) is an R-module of finite length and finite projective dimension. A guiding philosophy in this area is that such quotients of regular sequences are the "simplest" or "smallest" modules amongst all modules of finite length and finite projective dimension. I will discuss recent new evidence supporting this philosophy due to Josh Pollitz and myself. Over strict Cohen-Macaulay rings, we prove that if the m-adic filtration of a module is "too small", then the module must be of infinite projective dimension.
Closing Remarks