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: Balanced+balanced splitting loci have rational singularities
Abstract: Whenever there is a vector bundle on a P^1-bundle, the base is stratified by how the vector bundle splits when it is restricted to the fibers. The strata that arise this way are called splitting loci. In this talk, I will explain how splitting loci are defined and explain why they arise naturally. Then I will give an outline of the proof that certain splitting loci have rational singularities. The key ingredient is the construction of a modular resolution of singularities for all splitting loci, and cohomology vanishing for certain tautological bundles on Quot schemes on P^1.
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 \(\left(\mathbb C^\perp\right)^{\perp} = \mathbb C\).
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: TBA
Abstract: TBA
Closing Remarks