GWCAWMMG

Abstracts:

All of the talks will take place in Bruininks Hall 230. The dessert social will take place in one of the conference rooms of the Days Hotel. For the schedule of events please check the schedule page.

Title: Applications of blowup algebras

Abstract: In this talk, we will define what is a blow up algebra. We will see all kind of applications of blow up algebras including chemical reaction network, geometry modeling, and permutation statistics. We will see that this topic has many wide open questions.


Title: Trace maps and trace modules over commutative rings

Abstract: The well-known trace map on matrices can be generalized to a map on any module over a commutative ring. The image of such a map is a trace module. In this talk, I will discuss some historical uses for and recent developments in the theory of trace modules over commutative Noetherian rings. This will include applications of trace modules in understanding free summands, endomorphism rings, and special classes of rings.
Title: Rings of characteristic \(p>0\), the Frobenius functor, and test ideals

Abstract: he study of commutative rings of characteristic \(p>0\) has played a major role in the study of singularities of commutative rings in all characteristics. The main tool used to study these rings is the Frobenius functor, which acts on ring elements by \(r \mapsto r^p\). We will discuss the action of the Frobenius functor on modules over commutative rings and use this to explore the test ideal, a tool for classifying singularities of commutative rings in characteristic \(p>0\).
Title: Hilbert functions in algebra and geometry

Abstract: In a famous 1890 paper, David Hilbert shook the world of algebra by proving that for any graded module over a polynomial ring the vector space dimensions of its graded components are eventually given by a polynomial function. The function that records these dimensions is nowadays known as the Hilbert function in his honor.

In this talk, we will explore the properties of Hilbert functions and discuss some open problems concerning them that have puzzled algebraists for a while. Many questions on Hilbert functions can be reduced to the zero-dimensional case, that is, to investigating the Hilbert functions of a finite set of points in projective space. This will lead us towards some fruitful connections to geometry.


Title: Projections of the Veronese Variety

Abstract: The degree \(\delta\) Veronese variety is the image of the map \(v_{\delta}: \mathbb{P}^{d-1} \rightarrow \mathbb{P}^{n-1}\) which sends \([x_1: \cdots : x_d]\) to all monomials of total degree \(\delta\). Let \(R=k[x_1, \ldots, x_d]\) and let \(I\) be the \(R\)-ideal generated be all monomials of total degree \(\delta\). We are interested in computing the coordinate ring of the image of \(v_{\delta}\), also known as the special fiber ring of \(I\). In this talk we will discuss a classical computation for this coordinate ring in the \(delta=2\) case. We will then discuss how to compute the coordinate ring of a projection of the degree 2 Veronese variety from a general point.
Title: Local Cohomology Modules

Abstract: The local cohomology modules associated to a ring are an indexed sequence of objects that give information, such as the dimension and the depth of the ring. However, they are also fascinating modules with rich structures in their own right. In this talk, we will commute several examples, some straightforward and some bizarre.
Title: Quantitative Properties of Ideals arising from Hierarchical Models

Abstract: We will discuss hierarchical models and certain toric ideals as a way of studying these objects in algebraic statistics. Some algebraic properties of these ideals will be described and a formula for the Krull dimension of the corresponding toric rings will be presented. One goal is to study the invariance properties of families of ideals arising from hierarchical models with varying parameters. We will present classes of examples where we have information about an equivariant Hilbert series. This is joint work with Uwe Nagel.
Title: Properties of Binomial Edge Ideals

Abstract: Binomial edge ideals are ideals that are attached to a simple graph, and they are a generalization of the ideal of 2-minors of a (2 x n)-matrix of indeterminates. After a review of free resolutions, I will show how the homological properties of binomial edge ideals relate to combinatorial invariants of the corresponding graph. Time permitting, I will also discuss what is known and some open questions in the special case of binomial edge ideals that come from closed graphs, which are equivalent to having a quadratic Gröbner basis with respect to lex order.