Title:  Enumerating Equivalence Classes of Rank-Metric and Matrix Codes

Abstract:  Due to their applications in network coding, public-key cryptography, and space-time coding, both rank-metric codes and matrix codes, also known as array codes and space-time codes over finite fields, have garnered significant attention.  We focus on characterizing rank-metric and matrix codes that are both efficient, i.e. have high dimension, and effective at error correction, i.e. have high minimum distance.  A number of researchers have contributed to the foundation of duality theory for rank-metric and matrix codes, which has demonstrated that the inherent trade-off between dimension and minimum distance for a code is reversed for its dual code; specifically, if a code has high dimension and low minimum distance, then its dual code will have low dimension and high minimum distance.  Thus, with an aim towards finding codes with a perfectly balanced trade-off, we study self-dual matrix codes.  In particular, we enumerate the equivalence classes of self-dual matrix codes of short lengths over small finite fields.  To perform this classification, we also examine the notion of equivalence for rank-metric and matrix codes and use this to characterize the automorphism groups of these codes.

Title:  Non-negative polynomials and Gorenstein ideals.

Abstract:  A homogenous polynomial of degree d in n variables is called non-negative if it is at least zero when evaluated at any point with real coordinates. The cone of such non-negative polynomials contains the cone of the homogeneous polynomials that are sums of squares. Hilbert characterized the pairs (n,d) such that the two cones are the same.
Recently, Blekherman strengthened Hilbert's results by describing the extremal rays of the cone that is dual to the cone of non-negative polynomials. These rays correspond to certain extremal Gorenstein ideals.
We will discuss these results.

Title:  (Flat) modules that are fully invariant in their pure-injective (cotorsion) envelopes

Abstract:  We introduce two concepts. A module M is said to be purely quasi-injective (resp. quasi-cotorsion) if it is fully invariant in its pure-injective envelope (resp. if it is flat and fully invariant in its cotorsion envelope). Endomorphism rings of both of the above types of modules are proved to be regular and self injective modulo their Jacobson radicals. If M is a purely quasi-injective (resp. quasi-cotorsion) module, then so is any finite direct sum of copies of M. Each of the above concepts is stronger than the well-known concept of quasi-pure-injectivity, but not equivalent. This solves, negatively, a problem of Mao and Ding's of whether every flat quasi pure-injective module is fully invariant in its cotorsion envelope. Certain types of rings are characterized in terms of purely quasi-injective modules. For example, a ring R is regular if and only if every purely quasi-injective R-module is quasi-injective, and is pure-semisimple if and only if every R-module is purely quasi-injective.

Title:  Codes over Frobenius rings and an extension theorem--Part 2

Abstract:  An important result in algebraic coding theory tells us that every Hamming weight-preserving isomorphism between subspaces in F^n, where F is a finite field, extends to a Hamming weight-preserving isomorphism on the entire F^n. This has led to a variety of generalizations, namely to submodules over certain rings and/or different weight functions.  In this talk, I will discuss the extension theorem for Frobenius rings and poset weights. Both notions, Frobenius rings and poset weights, will be introduced.

Title:  Codes over Frobenius rings and an extension theorem

Abstract:  An important result in algebraic coding theory tells us that every Hamming weight-preserving isomorphism between subspaces in F^n, where F is a finite field, extends to a Hamming weight-preserving isomorphism on the entire F^n. This has led to a variety of generalizations, namely to submodules over certain rings and/or different weight functions.  In this talk, I will discuss the extension theorem for Frobenius rings and poset weights. Both notions, Frobenius rings and poset weights, will be introduced.

Title:  Integer decomposition property of dilated polytopes

Abstract:  We say that P has the integer decomposition property (IDP, for short) if any integer point in mP can be written as the sum of m integer points in P, where m is an arbitrary positive integer. In this talk, we discuss the problem when an integral convex polytope without IDP has IDP by integral dilation.

Title:  An algebraic study of Cameron-Walker graphs

Abstract:  Given a finite simple graph G, two commonly studied invariants in graph theory are the matching number, m(G), and the induced matching number of a graph, i(G). These combinatorial invariants provide upper and lower bounds, respectively, for the (Castelnuovo-Mumford) regularity of the quotient of the edge ideal associated to the graph, R/I(G). Cameron and Walker characterize all graphs where the matching number is the same as the induced matching number and therefore the regularity can be explicitly calculated. In this talk we will examine other algebraic and combinatorial properties of R/I(G) where G satisfies m(G)=i(G), such as Cohen-Macaulayness, shellability, and vertex decomposability.

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.