國家理論科學研究中心(南區)數學組

Coordinator:

Xuding Zhu (NSYSU)

Combinatorics, Computer Science

zhu@math.nsysu.edu.tw

Tel: +886-7-5252000 Ext. 3827

 

Background, current status, and Modes of Operations:

Since the beginning of the twentieth contrary, tools and methods in discrete type of mathematics are developed and applied in various fields. The focus and the meaning of a new field become clear gradually. These related subjects combine together to form a big umbrella of the field of discrete mathematics. In particular, the invention of computer makes a break through. Since 1930s, the power of computer changes all ability on handling large amount of data. It leads human life into a fully new stage. It not only makes our living convenient, but also changes our thinking style and knowledge developing. Since computer can only deal with discrete data, in order to get help from computer, continuous mathematics needs be discredited. This not only shows the importance of the discrete phenomena, but also very often develops deep discrete problems. Computer helps handling large amount of numbers and finite structures. It also creates an entirely new world. The importance of discreteness is also showed in the recent studies of DNA sequences in biology, and quantum phenomena in physics…etc.

Recently, several significant progresses in the field of discrete mathematics include the following events. In 1998, Timothy Growers received his Fields Medal in ICM Berlin. One of his contribution cites is the generalization of Szemeredi's theory in combinatorics. In 2002, the Strong Perfect Graph Conjecture was proved after posed over forty years. Maria Chudnovsky, Neil Robertson, Paul Seymour and Robin Thomas finished the proof after a long march. This is a new victory on the analysis of graph structure and graph algorithm. In 2004, Journal of Combinatorics Series B published the last paper among Neil Robertson and Paul Seymour's 20 papers on graph minor. These establish the famous conjecture by Wagner. This series of papers start publishing from 1988 and end at 2004. They not only earthquake the study on the fundamental structure of graph, but also lead the progressing of graph algorithm. In 2004, Ben gree and Terry Tao proved that there is an arbitrary long arithmetic progress in the set of all prime number. This is a break through in combinatorial number theory. The tools they use include Szemeredi's theorem, ergodic theory and pseudo random number theory. Tao got his Fields Medal in 2006 ICM Madrid.

 

Mathematicians in Taiwan involve the research in discrete mathematics only for two decades. Among all topics of discrete mathematics, they most emphasize on graph theory and its algorithmic aspects. For the later topic, there are also theoretical computer scientists involved. Besides various workshops (mostly once per half year), regular seminars are held in Taipei , Hsinchu, and Kaohsiung . For the information of related workshops in Taiwan over the past two decades, please see the web page at http://www.math.ntu.edu.tw/~2003comb/ . In the past five years, the discrete mathematics group in Taiwan has been actively engaged in many academic activities sponsored by NCTU, and doing fruitful research in various areas. Each year, we have sponsored by NCTU, and doing fruitful research in various areas. Each year, we have regular seminars, organize workshops and conferences, host visitors, and have collaborations with researchers from France , Canade , US , Slovenia and Czech Republic . Through these activities, young researchers got opportunities to learn the newest development and problems in discrete mathematics, discuss problem with experts from abroad and promote their own research. The research topics include but not restricted to: graph coloring, competition graphs, domination in graphs, algebraic graph theory, graph algorithm, network theory, distance-two labelings of graphs, profile and bandwidth of graphs, distinguishing labelings of graphs and group actions, enumerative combinatorics, cluster combinatorics, geodetic graph theory and hamiltonicity of graphs. For the focus program in discrete mathematics at NCTS 2009-2014, we have the following main topics.

 

Main research focus of discrete mathematics group of NCTS(south) :

  1. Graph coloring and its applications.
  2. Labeling and weighting of graphs.
  3. Coxeter combinatorics.