Title:  On a Class of Determinantal Ideals.

Abstract:  We will discuss a class of ideals determined by taking minors in a subregion of a matrix of indeterminates, called a skew tableau, and also in a reflected version of this considered as a subregion of a symmetric matrix.  We will use liaison-theoretic tools to investigate properties of these ideals, and study their liaison classification.

Title:  A sufficient condition for covering ideals.

Abstract:  The concepts of envelope and cover were introduced independently by Enochs and Auslander-Smalo  for classes of modules. Since then the definition has been applied to di fferent classes of categories. One of the recent application was introduced by Asensio, Herzog, Fu and Torrecillas where the theory of covers and envelopes is extended to ideals. In this talk, we will show how identifying an ideal I with a certain class of objects in the quiver A_2 can help us to obtain su fficient conditions for I to be a covering ideal.

After subspace codes were introduced in 2008 by Koetter and Kschischang most constructions involved the lifting of matrix codes. However, Rosenthal et al. introduced in 2011 a new method of constructing constant dimension subspace codes by using a group action of $\textup{GL}_n(\mathbb{F}_q)$ on $PG(q,n)$, called orbit codes. A specific subset of these codes, which have been studied more in depth, are irreducible cyclic orbit codes. In this talk, I will introduce the construction of an irreducible cyclic orbit code as well as explore a method to find the cardinality and distance of such a code. This is based on joint work with Heide Gluesing-Luerssen.

A way of constructing Gorenstein ideals from small Gorenstein ideals in local Gorenstein rings is shown by A. Kustin and M. Miller in 1983. In this talk, we show a variant of their construction for graded case and avoid the ring extension. We see that how the liaison theory helps us immensely to modify their construction.

Cellular resolutions have been an area of active interest in the study of monomial ideals in the past few years. A cellular resolution is a way of encoding the information of the free resolution of an ideal in a cell complex. We will study the Stanley-Reisner rings arising from the simplicial polytopes known as the cyclic polytopes. These polytopes have the interesting property that they maximize all entries in the f-vector of the set of polytopes with fixed dimension and vertices. I will explain a little of the background theory and then cover the main idea of the construction.