Discrete Mathematics and Combinatorics

Discrete Math 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. 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.