KPPY 80

Multiple zeta values are a multivariate analogue of special values of the Riemann zeta function. After giving a brief introduction to the theory of multiple zeta values, I shall present results related to a formula established by Bowman and Bradley.

The problem of determining the Krull dimension of the polynomial ring $R[X]$ and the power series ring $R[[X]]$ is one of the problems in commutative rings. When $\dim(R)$ is finite, $\dim(R[X])$ was completely determined. In power series case, there are some difficulties. For example, $R[[X]]$ may have uncountable Krull dimension even if $\dim(R) = 1$. In traditional power series extension, many results were showed. But in composite power series extension, only some results were discovered. In this talk, I will introduce some tools to construct prime chains and classify the Krull dimension of composite power series rings over valuation rings.

An $r$-dynamic $k$-coloring of a graph $G$ is a proper $k$-coloring $\phi$ such that for any vertex $v$, $v$ has at least $\min\{r,\deg_G(v) \}$ distinct colors in $N_G(v)$. The $r$-dynamic chromatic number $\chi_r^d(G)$ of a graph $G$ is the least $k$ such that there exists an $r$-dynamic $k$-coloring of $G$. The list $r$-dynamic chromatic number of a graph $G$ is denoted by $ch_r^d(G)$. Recently, Loeb, Mahoney, Reiniger, and Wise showed that the list $3$-dynamic chromatic number of a planar graph is at most 10. And Cheng, Lai, Lorenzen, Luo, Thompson, and Zhang studied the maximum average degree condition to have $\chi_3^d (G) \leq 4, \ 5$, or $6$. In this paper, we study list 3-dynamic coloring in terms of maximum average degree. We show that $ch^d_3(G) \leq 6$ if $mad(G) < \frac{18}{7}$, and $ch^d_3(G) \leq 7$ if $mad(G) < \frac{14}{5}$, and both of the bounds are tight. This is joint work with Boram Park.

Recently, there was a chance to look at again the definitions of p-adic integers, p-adic numbers, and Witt vectors. I thought (conjecturally) that these constructions can be unified in terms of (a generalization of) Ore's extension. I will provide also some history of embedding problem of a domain into its quotient skew field: When is a ring without zero divisors embedded in a division ring? This subject is my current research in progress.

A graph class $C$ is called chi-bounded if there is a function $f$ such that for every graph $G$ in $C$, the chromatic number of $G$ is bounded by $f(w(G))$ where $w(G)$ is the size of a maximum clique in $G$. Geelen conjectured that for every graph $H$, the class of $H$ vertex-minor free graphs is chi-bounded. We prove that this conjecture is true when $H$ is a fan graph. This is joint work with Choi and Oum.