Title: Khovanskii bases and rational, complexity $1$ $T$-varieties
Abstract: Khovanskii bases (introuduced by Kiumars Kaveh and Christopher Manon) are an important tool used to study $\mathbb{K}$-domains relative to a valuation, and in fact, are a vast generalization of the notion of a SAGBI basis. We begin by introducing Khovanskii bases and provide several examples. We are particularly interested in the case when a pair $(A, \mathfrak{v})$ has a finite Khovanskii basis. Next, we discuss $T$-varieties which are a natural generalization of toric varieties. We will see how affine $T$-varieties arise out of the quasi-combinatorial data of a polyhedral divisor via the construction of J"urgen Hausen and Klaus Altmann. Finally, we will see a result by Nathan Ilten and C. Manon: for any homogeneous valuation $\mathfrak{v}$ on the coordinate ring of a rational complexity $1$ $T$-variety, there is an embedding whose coordinates are a Khovanskii basis for $\mathfrak{v}$.
Title: Subspace Polynomials and Cyclic Subspace Codes
Abstract: Subspace codes are collections of subspaces of the finite vector space $\mathbb{F}^n_q$ under the subspace metric. Of particular interest due to their efficient encoding/decoding algorithms are constant dimension cyclic codes where each codeword is a subspace of the same dimension and which are invariant under an action of $\mathbb{F}^{*}_{q^n}$. We will see how to represent subspace codes using particular polynomials and deduce properties of the codes from the structure of these polynomials based on the work of Ben-Sasson et. al. in 2016. Using these results, we will give a construction of a constant dimension cyclic code containing multiple orbits under the $\mathbb{F}^{*}_{q^n}$ action with highest possible subspace distance.
Abstract: We generalize cyclic codes to skew θ-constacyclic codes using skew polynomial rings. We provide a useful tool for exploring these codes: the circulant. In addition to presenting some properties of the circulant, we use it to re-examine a theorem giving the dual code of a θ-constacyclic code first presented by Boucher/Ulmer (2011).
