Graph C*-algebras are analytical objects blessed with a tangible structure and classification theory derived from their combinatorial origins. Through the analysis of directed graphs, including higher-rank graphs or quantum graphs, one can visualize and explore them in intuitive ways lacking elsewhere. They serve as strikingly efficient models for key open problems in noncommutative geometry and topology, as well as in C*-dynamical systems. They also provide a focal point for the much-needed extension of the celebrated Elliott classification program to non-simple C*-algebras.

The main objectives of the research project behind the proposed Simons Semester concern the classification, symmetry, structure and noncommutative metric geometry of operator algebras. The innovation in our approach comes from our emphasis on the ubiquitous class of graph C*-algebras, and through combination of ideas from many fields of research: unbounded operator theory, metric topology, classification by K-theory, KK-theory, and combinatorial and topological properties of graphs and groupoids. Here the key observation is that graphs provide convenient models for C*-algebras which one can visualize and explore in intuitive ways not generally avaliable, and yield manageable K-theoretical invariants. Such invariants are crucial for the celebrated Elliott classification program and studying the structure of corresponding operator algebras. This program was famously completed for simple C*-algebras in recent years, but the much-needed progress in extending the classification program to the non-simple cases relevant here requires deeper understanding of the relationship between combinatorics and invariants of operator algebras.

The very notion of an invariant relies heavily on a notion of functoriality with respect to an admissible class of morphisms. Therefore, working out right notions of morphisms between graphs is necessary to relate invariants of operator algebras and their combinatorial models. In particular, invariants satisfying the Mayer-Vietoris principle should be put into the context of pushouts of graphs transforming into pullbacks of corresponding graph C*-algebras. Higher-rank graphs or quantum graphs are here a natural extension.

There are other important natural constructions to be related with C*-algebraic invariants. The join construction, deeply rooted in algebraic topology, can be related to such important classical results as the Borsuk-Ulam theorem, the most celebrated classical result about equivariant maps between spheres. In dimension 2, assuming that both temperature and pressure are continuous functions, one can infer from this theorem that there always exist two antipodal points on Earth with exactly the same pressure and temperature.

The join construction admits a C*-algebraic generalization allowing the construction of the celebrated Jiang-Su C*-algebra (integral to the Elliott program) as an inductive limit of joins of matrix algebras, and a formulation of a far reaching generalization of the Borsuk-Ulam context to not necessarily commutative C*-algebras. The latter takes the form of the noncommutative Borsuk-Ulam-type conjecture of Baum, Dąbrowski and Hajac that is formulated via an equivariant noncommutative join construction for unital C*-algebras equipped with a free action of a compact quantum group. One of the principal objectives here is the proof of the conjecture for important special cases. In this noncommutative context, a conjecture widely generalizing the classical Borsuk-Ulam theorem is strictly related to local triviality dimensions, a new invariant of quantum-group actions on unital C*-algebras. In the classical case, the Borsuk-Ulam-type conjecture implies the weak Hilbert-Smith conjecture, which is a long-standing open problem in the theory of actions of locally compact groups on topological manifolds.

Every graph C*-algebra is a groupoid C*-algebra of an appropriate groupoid related to the graph. Therefore, it is a fascinating and highly suggestive fact that the most interesting classifiable C*-algebras admit a groupoid model. Exploring the relationship between combinatorial and topological properties of the groupoid and invariants of its C*-algebra is consequently of fundamental importance. Indeed, graph C*-algebras have a rich internal structure, and come naturally equipped with a circle action holding important geometric information, and hence an equivariant analysis is often necessary. They also come equipped with a canonical dense subalgebra intensely studied in noncommutative algebra, opening the door to purely algebraic methods. Fully capitalizing on these structures requires the development of new methods.

Graphs afford us strikingly efficient models to attack key open problems in noncommutative geometry and topology, as well as in C*-dynamical systems. Better still, researchers using graph C*-algebras for applications in adjacent fields are likely to provide groundbreaking insights making impact way beyond graph algebras themselves.