Model theory studies the ways in which mathematical objects can be defined in some restricted formal language, and what structural properties are implied by these definability assumptions. It provides methods of converting asymptotic questions about finite structures into qualitative questions about the shape, volume or dimension of certain limiting infinite objects. This method of study originated in questions on foundations of mathematics, but in recent years it has found important applications in the study of some central objects of classical mathematics and computer science. The project investigates further these connections, with the major motivation of extending the existing techniques from binary structures (graphs) to structures of higher arity (hypergraphs), which represent a mathematical way of describing more complex networks in which interactions happen not just between two nodes at a time, but between multiple nodes simultaneously. This study will both deepen and extend the scope for applications of the infinitary model-theoretic machinery to questions in combinatorics of geometrically or algebraically arising hypergraphs, and conversely for applications of combinatorics to open questions in model theory. The project will involve training of graduate and undergraduate students. Shelah's classification program isolates combinatorial dividing lines (stability, distality, NIP, etc.) separating mathematical structures exhibiting various degrees of Gödelian behavi