Combinatorics After CC-Conjecture — Notions and Achievements

Abstract

As a powerful technique for holding relations in things, combina- torics has experienced rapidly development in the past century, particularly, enumeration of configurations, combinatorial design and graph theory. How- ever, the main objective for mathematics is to bring about a quantitative analysis for other sciences, which implies a natural question on combinatorics. Thus, how combinatorics can contributes to other mathematical sciences, not just in discrete mathematics, but metric mathematics and physics? After a long time speculation, I brought the CC conjecture for advancing mathematics by combinatorics, i.e., any mathematical science can be reconstructed from or made by combinatorialization in my postdoctoral report for Chinese Academy of Sciences in 2005, and then reported it at a few conferences of China. Clearly, CC conjecture is in fact a combinatorial notion and holds by a philosophical law, i.e., all things are inherently related, not isolated. The main purpose of this report is to survey the roles of CC conjecture in developing mathematical sciences with notions, such as those of its contribution to algebra, topology, Euclidean geometry and differential geometry, non-solvable differential equa- tions or classical mathematical systems with contradictions to mathematics, quantum fields after it appeared 10 years ago. All of these show the importance of combinatorics to mathematical sciences in the past and future.