# Algebra Seminar

Date:

09/26/2018 - 2:00pm to 3:00pm

Speaker(s) / Presenter(s):

Joseph Cummings, University of Kentucky

**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 $\mathfrak{v}$.

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