Confirmed speakers:
Michael DeBellevue, Syracuse University
Title: Structure of Syzygies over Rings of High Burch Index via the Bar Resolution
Abstract: The Burch index Burch(R) is a numerical invariant of rings recently introduced by Dao and Eisenbud. They showed that if Burch(R)>1, then every high syzygy of every module has a direct k summand. The bar resolution provides a universal means to transfer resolutions along a map of local or graded rings. We discuss how this resolution is used to provide a structural proof of Dao and Eisenbud's result, as well as slightly improve it. We discuss how in the Golod case, we use the bar resolution to show that the number of k summands of syzygies grows exponentially.
Eleonore Faber, University of Graz (Austria)
Title: Singularities arising from cluster algebras of finite cluster type
Abstract: Cluster algebras were introducted by Fomin and Zelevinsky in the context of total positivity in Lie theory. Since then, cluster structures have appeared in many different contexts, ranging from representation theory and combinatorics to mirror symmetry.
In this talk, I will first give a short introduction to cluster algebras and then focus on the class of cluster algebras of finite cluster type, which are classified by Dynkin diagrams. We study these commutative algebras from the point of view of singularity theory: we classify their singularities and develop constructive resolutions of these singularities over fields of arbitrary characteristics. We also discuss how the so-called coefficients of cluster algebras affect their singularity types. This is joint work with Angélica Benito, Hussein Mourtada, and Bernd Schober.
Brian Harbourne, University of Nebraska--Lincoln
Title: Geproci sets: a new perspective on classification in algebraic geometry
Abstract: The classification of geproci subsets of projective space is a special instance of
carrying an inverse scattering perspective over to a classification problem in
algebraic geometry. A geproci set is a finite set of points in projective space whose
image, when projected from a general point to a hyperplane, is a complete intersection.
This concept turns out to have not-yet-fully understood connections to combinatorics and
representation theory, and possibly even quantum mechanics. The notion of a geproci
set is only a few years old. This talk will introduce the concept, establish some basic
facts, show some connections to combinatorics and representation theory, and state some
open problems.
Selvi Kara, Bryn Mawr College
Title: Barile-Macchia Resolutions
Abstract: In this talk, I will introduce Barile-Macchia (BM) resolutions of monomial ideals. These resolutions are constructed using ideas from discrete Morse theory. Specifically, we will discuss an algorithm that produces homogeneous acyclic matchings, and these matchings induce BM resolutions, a class of cellular resolutions. I will discuss various classes of ideals for which BM resolutions are minimal, and we will explore results related to the independence of directions for edge ideals of weighted oriented paths and cycles. Additionally, we will discuss a generalization of BM resolutions and demonstrate that they minimally resolve path ideals of paths and cycles. This is joint work with Trung Chau.
Josh Pollitz, Syracuse University
Title: Frobenius pushforwards and generators for the derived category
Abstract: By now it is quite classical that one can understand singularities in prime characteristic local algebra/algebraic geometry, through properties of the Frobenius endomorphism. A foundational result illustrating this is the celebrated theorem of Kunz characterizing the regularity of a noetherian scheme (in prime characteristic) in terms of whether a Frobenius pushforward on that scheme is flat. In this talk, I'll discuss a structural explanation of, that also recovers, the theorem of Kunz and other theorems of this ilk. Namely, I’ll discuss recent joint work with Ballard, Iyengar, Lank, and Mukhopadhyay where we show that over an F-finite noetherian scheme of prime characteristic high enough Frobenius pushforwards generate the bounded derived category.
Hans Schoutens, CUNY Graduate Center
Title: A differential-algebraic criterion for obtaining a small maximal Cohen-Macaulay module in positive characteristic
Abstract: I will show how for a three-dimensional complete local ring in positive characteristic, the existence of an F-invariant, differentiable derivation implies Hochster's small MCM conjecture. As an application we get that any three-dimensional pseudo-graded ring in positive characteristic satisfies Hochster's small MCM conjecture.
Adela Vraciu, University of South Carolina
Title: Exact zero divisors and Hilbert functions
Abstract: We explore the possible Hilbert functions of standard graded Artinian algebras that admit a pair of exact zero divisors of fixed degrees. For a fixed pair of positive integers d_1, d_2, we describe a necessary and sufficient condition for a sequence of positive integers to be realizable as the Hilbert function of a standard graded algebra that admits a pair of exact zero divisors of degrees d_1, d_2. We also discuss the Hilbert function of a standard graded algebra with the property that generic linear forms are part of a pair of exact zero divisors.
Poster titles and abstracts:
Sudipta Das, Arizona State University
Title: The rational powers of sum of ideals.
Abstract: In this poster we introduce a special classes of ideals, called ideals with Rees packages, e.g. monomial ideals in normal semi group ring, determinantal ideals of a generic matrix, Pfaffians of a skew symmetric matrix, minors of a Hankel matrix, and proof binomial sum formula for rational powers of sum of ideals.
Brian Laverty, West Virginia University
Title: Depth formula for modules of finite reducing projective dimension
Abstract: Assume R is a commutative Noetherian local ring, and M and N are finitely generated R-modules. A relatively new homological invariant is introduced, called reducing projective dimension (denoted red-pd), and is a generalization of reducible complexity. Our goal was to find conditions under which the derived depth formula holds for M and N. We found such results whose hypotheses include: red-pd_R(M) < infty and the vanishing of Tor_i(M,N).
Shiji Lyu, University of Illinois Chicago
Title: Formal lifting of dualizing complexes and consequences
Abstract: We show that for a Noetherian ring A that is I-adically complete for an ideal I, if A/I admits a dualizing complex, so does A. We discuss several consequences of this result. We also consider a generalization of the notion of dualizing complexes to infinite-dimensional rings and prove the results in this generality. In addition, we give an alternative proof of the fact that every excellent Henselian local ring admits a dualizing complex, using ultrapower.
Benjamin Oltsik, University of Connecticut
Title: Symbolic Defect of Monomial Ideals
Abstract: Symbolic powers of ideals in a Noetherian commutative ring have long been a topic of interest in commutative algebra. They are directly related to the associated primes, and always contain the ordinary powers. The symbolic defect function is a numerical function designed to measure the "closeness” of symbolic powers and the ordinary powers. In particular, symbolic defect measures the minimal number of generators of the quotient of symbolic powers with ordinary powers. This poster exhibits an interpretation of these concepts for monomial ideals via convex geometry. We also introduce a related invariant, the integral symbolic defect.
Abraham Pascoe, University of Kansas
Title: Local cohomology modules and simplicial homology
Abstract: The poster will discuss vanishing of local cohomology modules and in particular the cohomological dimension, i.e. the maximum index where the local cohomology modules are nonzero. A classical problem in local cohomology is determining when cohomological dimension is bounded by the depth of R/I. I will define a family of simplicial complexes whose simplicial homology can determine the vanishing of local cohomology modules in certain cases. Additionally, the poster will define numerical invariants of local cohomology modules, called Lyubeznik numbers (Bass numbers for local cohomology modules). The poster will provide alternate calculations of the Lyubeznik numbers, again utilizing the homology of this family of simplicial complexes.
Joseph Skelton, Clemson University
Title: Monomial Ideals of Linear Codes
Abstract: In coding theory, a linear code is a subspace of a finite dimensional vector space over a finite field. From the linear code we define a squarefree monomial ideal that is Cohen-Macaulay and matroidal. We investigate properties of symbolic powers of these ideals.