Researchers in combinatorics are interested in properties of finite (or countable) structures: graphs, networks, partially ordered sets, codes, integer sequences, finite groups and fields, finite geometries, polyhedra, and so on. Combinatorics has applications to virtually all branches of mathematics, pure and applied, including areas where the fundamental objects of study are uncountable sets such as real or complex numbers.
To give easily stated examples of questions that are asked in combinatorics, think of ways to properly color a map. Say that a coloring is proper if adjacent countries receive different colors. Typical combinatorial questions can concern structure ("can every map be properly colored using only 4 colors?"), enumeration ("in how many ways can a given mapbe properly colored using n colors?"), or optimization and algorithms ("what is the smallest number of colors that can properly color a given map, and what is the quickest way to find such a coloring?").
Combinatorics has a long history, tracing its roots to ancient India. Leibniz, Pascal and Euler were early pioneers in the modern development of the subject, but it was not until the 20th century that combinatorics truly emerged as a field on its own. The growth of the field has been quite rapid in recent decades, much thanks to the emergence of computer science, which relies heavily on combinatorics.
Modern research in combinatorics often interacts closely with other branches of mathematics and its applications. Thus, researchers in combinatorics at the SMC use techniques from such diverse fields as probability theory, group theory, linear algebra, convex geometry, algebraic topology, and complex analysis, and also provide combinatorial feed-back to help solve problems from such areas. There are also more applied lines of research at the center, pertaining to bioinformatics, error-correcting codes and election methods.