Discrete mathematics refers to both finite and countable phenomena, including the two central topics combinatorics (advanced counting and arrangements) and graph theory ( the mathematics of networks) and important contemporary examples include the study of social networks, analysis of efficiency of algorithms, combinatorial design of experiments, as well as routing, assignment, and scheduling problems.

Discrete mathematics tools are essential in fields ranging from operations research, computer science and epidemiology to protein folding and telecommunications.

## Primary Areas of Research

Hopkins Engineering faculty research in discrete mathematics focuses primarily in the following areas:

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 including neuroscience, computer science, and routing. 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 computer science, functional analysis and probability theory.

Discrete Geometry describes 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.

**Related Courses**

Complete descriptions appear in the course catalog.

View the semester course schedule.

EN.553.171. Discrete Mathematics

EN.553.371 Cryptology & Coding

EN.553.4/620 Introduction to Probability

EN.553.4/629 Introduction to Research in Discrete Probability

EN.553.4/663 Network Models in Operations Research

EN.553.4/665 Introduction to Convexity

EN.553.4/671 Combinatorial Analysis

EN.553.4/672 Graph Theory

EN.553.730 Statistical Theory

EN.553.731 Statistical Theory II

EN.553.766 Combinatorial Optimization

## Learn More

Research and academic opportunities in discrete mathematics