Topology is the study of continuous functions between spaces, with broad latitude both for what qualifies as a space, and for which continuous functions are of interest. Of fundamental importance is the task of organizing spaces, treating two spaces as the same if, roughly speaking, each can be continuously deformed into the other. In this sense, the surface of a cube and the surface of a ball are the same, while neither is the same as the surface of a one holed doughnut. Topology has strong ties to abstract algebra, notably the study of homology, cohomology, and homotopy theory. Outside of mathematics itself, topology has real world applications in subjects such as biology, computer science, and robotics, (e.g. knot theory, topological data analysis, and motion planning in a configuration space).
The research group also focuses on graph theory and matroid theory. Graphs are fundamental combinatorial structures used to model pairwise relationships between objects, with applications across fields such as computer science, biology, social sciences, and beyond. In addition, the group investigates the structural properties of matroids, which are combinatorial structures that abstractly capture the key properties of dependence found in graphs and matrices. Matroids naturally arise in many combinatorial optimization problems, as they are precisely the simplicial complexes for which the greedy algorithm works. They also have connections to areas like coding theory, geometry, hyperplane arrangements, and more.
Faculty
- Paul Fabel - geometric and categorical topology
- Jagdeep Singh - matroid theory and graph theory
- Vaidyanathan Sivaraman - graph theory