# KPPY 80

Shingo Saito, Min Jae Kwon, Seog-jin Kim, Kyoung-tark Kim, O-joung Kwon

Dec 17, 2016. 11am - 6pm at YNU.

Schedule | ||

11:00 - 11:50 |
Shingo Saito Kyushu University | Bowman-Bradley type theorems for multiple zeta values and analogues |

12:00 | Lunch | |

1:40 - 2:30 |
Min Jae Kwon Kyungpook National University | The Krull dimension of composite power series rings over valuation rings |

2:40 - 3:30 |
Seog-Jin Kim Konkuk University | List 3-dynamic coloring of graphs with small maximum average degree |

4:00 - 4:50 |
Kyoung-tark Kim Sogang University | Power series Ore's extension |

5:00 - 5:50 |
O-joung Kwon Technische Universitat Berlin | Coloring graphs without fan vertex-minors |

## Abstracts

Shingo Saito

Bowman-Bradley type theorems for multiple zeta values and analogues

Bowman-Bradley type theorems for multiple zeta values and analogues

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.

Min Jae Kwon

The Krull dimension of composite power series rings over valuation rings

The Krull dimension of composite power series rings over valuation rings

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.

Seog-Jin Kim

List 3-dynamic coloring of graphs with small maximum average degree

List 3-dynamic coloring of graphs with small maximum average degree

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. Kyoung-tark Kim

Power series Ore's extension

Power series Ore's extension

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.

O-joung Kwon

Coloring graphs without fan vertex-minors

Coloring graphs without fan vertex-minors

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.