Combinatorics is the study of discrete structures, and related algorithms. We might be interested in these things for their own sake, or because of potential applications to real world problems. Optimization deals with determining the values of variables that maximize or minimize an objective.
Research and teaching in our department emphasizes six areas: algebraic combinatorics, combinatorial/discrete optimization, continuous optimization, cryptography, graph theory, and quantum computing. At other universities, these subjects would lie in mathematics, computer science or operation research departments, but at Waterloo we find that they fit together very well, and cross fertilize each other in ways you might not at first expect. Thus students in quantum computing may use tools from continuous optimization, while effective algorithms for combinatorial optimization can depend on sophisticated ideas from graph theory.
Research topics pursued by graduate students in our department can be found in the research in Combinatorics and Optimization page. Companies and other enterprises that hire C and O graduates (both those with graduate and undergraduate degrees) can be found in the Why C&O page for undergraduates. Finally, valuable cross-cutting research is also showcased within the Faculty of Mathematics research information for graduate students.