Graph theory provides the mathematical foundations for the study and analysis of networks. The group at Hopkins explores both structural and algorithmic aspects of this branch of mathematics, as well as its applications in fields such as neuroscience, computer science, routing, to name a few. Core topics of research include random graphs, spectral graph theory, efficient algorithms for computing properties of graphs.
Combinatorics is the art of counting finite structures and understanding set families over finite universal sets. A very simple example of situations where counting can be non-trivial is the following: How many rectangles can be formed using the squares in a standard chessboard? In more complex settings, closed form formulas for the size of certain sets, can aid computations immensely where a brute force approach to counting can be highly inefficient. Moreover, insights from combinatorics have proved invaluable in fields as diverse as complexity theory in commuter science, functional analysis and probability theory.
Discrete Geometry is usually used to describe the subfield of geometry that deals with the interaction between convex sets and lattices in Euclidean space. A major emphasis is on polyhedra within convexity. Structural and algorithmic aspects of discrete geometry have been core mathematical tools in optimization, theoretical computer science, and more recently in astronomy and machine learning. Although simple in their definitions, the ideas in discrete geometry have surprisingly powerful and deep reaching consequences that have made them highly relevant for modern problems in science and technology.