MATRIX is Australia’s international, residential mathematical research institute. It facilitates new collaborations and mathematical advances through intensive residential research programs, each lasting 1-4 weeks. This book is a scientific record of the five programs held at MATRIX in its first year, 2016:

• Higher Structures in Geometry and Physics

• Winter of Disconnectedness

• Approximation and Optimisation

• Refining C*-Algebraic Invariants for Dynamics using KK-theory

• Interactions between Topological Recursion, Modularity, Quantum Invariants and Low- dimensional Topology

The MATRIX Scientific Committee selected these programs based on their scientific excellence and the participation rate of high-profile international participants. Each program included ample unstructured time to encourage collaborative research; some of the longer programs also included an embedded conference or lecture series.

The articles are grouped into peer-reviewed contributions and other contributions. The peer-reviewed articles present original results or reviews on selected topics related to the MATRIX program; the remaining contributions are predominantly lecture notes based on talks or activities at MATRIX.