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# Brief Description of Combinatorics

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Combinatorics is one of the oldest branches of mathematics. It has been influenced by almost all areas of mathematics, including number theory, algebra, topology, mathematical logic, and many more. In recent years, stimulated by the development of computer science, combinatorics has found new applications in algorithm analysis, network designs, and so on. Now it is one of the fastest growing areas in mathematics. The research of the LSU combinatorics group focuses on the study of graphs, matroids, and hyperplane arrangements.

The origin of graph theory can be traced back to Euler when he studied the Königsberg bridges problem in 1736. However, the development of graph theory as a separate subject occurred predominantly in the twentieth century. In the last twenty years, the study of graph structures has attracted more and more researchers. In this study, the objective is to decompose a large graph into small pieces according to certain very specific rules. Results of this study often have a wide range of applications in computer science and operations research. Several members of the LSU Combinatorics group are currently working in this area.

Matroid theory examines and answers questions such as the following. Why do cycles and minimal edge cuts in a graph exhibit many similar properties? What is the essence of the similarity between linearly independent sets of columns of a matrix and forests in a graph? Why does the greedy algorithm produce a spanning tree of minimum weight in a connected graph? Can one test in polynomial time whether a matrix is totally unimodular?

Over half a century of study of matroids has seen the development of a rich theory that links many disparate areas of combinatorics. Matroids arise in the study of graphs, lattices, codes, transversals, and projective geometries. They are of fundamental importance in combinatorial optimization and their applications extend into electrical engineering and statics. Several members of the LSU combinatorics group work in matroid theory.

Some more details of the research of the combinatorics group can be found on the home pages of individual faculty members.

LSU offers a variety of combinatorics courses from introductory courses in both combinatorics and graph theory to more advanced courses such as matroid theory and combinatorial optimization. Since 1990, many students have received LSU Mathematics Ph.D.s after writing dissertations in combinatorics.