## Final Report Summary - CCOSA (Classes of combinatorial objects: from structure to algorithms)

This project from pure mathematics aimed at analyzing complex structures from discrete mathematics using tools from other areas of mathematics and at applying the obtained results to design fast algorithms.

Discrete mathematics deals with objects that consists of well defined pieces. One may think that a basket of apples is a discrete object (we can talk about individual apples) while a glass of milk is not (milk is homogenous). The paradigm addressed in one part of the project is that if such a discrete object is very large, then it looses some of its discrete structure. Think of sand that consists of sand grains (so, it is discrete) but you would not think of a sand castle as built from bricks but rather as built from homogenous material. This is the phenomenon the project exploits.

Let us be more precise. One of the notions in discrete mathematics is that of graphs. Graphs can be used to model computer networks, railway systems, work flow diagrams, etc. If a graph is large, it can be approximated by a smooth structure (continuous function, measure) that captures its essential properties. Several of our results are based on this approach. A large graph is modeled by a smooth analytic object and semiautomatized methods (e.g., using semidefinite programming from operational research) are used to deduce some of its properties. This approach led to solving several old problems from extremal combinatorics, some of them popularized by famous Hungarian mathematician Paul Erdős who authored more than 1500 mathematics papers (in the 1990’s, a documentary film “N is a Number” was made about Erdős and his work), as well as to obtaining several algorithmic results.

Discrete mathematics deals with objects that consists of well defined pieces. One may think that a basket of apples is a discrete object (we can talk about individual apples) while a glass of milk is not (milk is homogenous). The paradigm addressed in one part of the project is that if such a discrete object is very large, then it looses some of its discrete structure. Think of sand that consists of sand grains (so, it is discrete) but you would not think of a sand castle as built from bricks but rather as built from homogenous material. This is the phenomenon the project exploits.

Let us be more precise. One of the notions in discrete mathematics is that of graphs. Graphs can be used to model computer networks, railway systems, work flow diagrams, etc. If a graph is large, it can be approximated by a smooth structure (continuous function, measure) that captures its essential properties. Several of our results are based on this approach. A large graph is modeled by a smooth analytic object and semiautomatized methods (e.g., using semidefinite programming from operational research) are used to deduce some of its properties. This approach led to solving several old problems from extremal combinatorics, some of them popularized by famous Hungarian mathematician Paul Erdős who authored more than 1500 mathematics papers (in the 1990’s, a documentary film “N is a Number” was made about Erdős and his work), as well as to obtaining several algorithmic results.