Matrix Spaces and Graphs: Connections and Applications

Explore the fascinating connections between matrix spaces and graphs in this illuminating lecture from the Workshop on Tensors: Quantum Information, Complexity and Combinatorics. Delve into the intriguing question of determining invertible matrices from linear combinations and discover how this relates to proving circuit lower bounds. Examine the surprising equivalences between linear-algebraic properties of matrix spaces and corresponding properties of graphs. Learn how these connections reinterpret and generalize classical theorems in algebra by Dieudonné and Gerstenhaber, while also finding applications in complexity theory and quantum information theory. Follow along as the speaker covers topics such as matrix space support, determinantal varieties, inherited correspondence, and symmetries. Gain insights into concrete open problems in the field and understand the broader implications of this research across theoretical computer science, quantum information theory, abstract algebra, and algebraic topology.

Introduction
Warmup question
Matrix spaces
Graphs
Matrix space support
Determinantal varieties
Summary
Inherited correspondence
No potency
Properties of two graphs
Symmetries
Summarize
Concrete open problem
Conclusion