The large scale geometry of high dimensional expansion

• Applicant: Professor Dr. Federico Vigolo
• Subject Area: Mathematics
• Term: since 2026
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 574320459

The main goal of this project is to develop a large-scale geometric approach to phenomena of high-dimensional expansion. The "large scale" or "coarse" geometric paradigm aims to uncover key global geometric features of spaces by ignoring confounding local structure. This is a very powerful approach, which lies at the basis of a great deal of contemporary mathematics. Expander graphs are sequences of finite graphs that remain highly connected despite becoming increasingly sparse. These objects play a fundamental role in several aspects of both pure mathematics and computer science. Crucially, the relevant notion of well-connectedness (or "expansion") is a large-scale geometric property, and this enables one to use coarse geometric techniques to construct and study expander graphs. Higher-dimensional analogues of expansion have recently found prominence (e.g., for the construction of good locally testable codes in computer science): here the role of graphs is played by simplicial complexes, and the expansion condition is expressed in terms of spectral properties of discrete Laplacian operators. This higher-dimensional setup has not yet been investigated from a coarse geometric perspective, and one of the goals of this project is to address this deficiency. The first concrete goal is to adopt a large-scale geometric approach as a guiding principle to construct higher-dimensional expander complexes by discretizing certain "highly mixing" group actions. This also represents a first step in a study of high-dimensional dynamics for group actions. A second goal is to contrast this construction with the so-called coarse Baum-Connes Conjecture. One very important achievement of the last century was to construct connections between the topological and the analytical worlds, as most famously witnessed by the Atiyah-Singer Index Theorem. One such connection is the construction of an assembly map from the coarse K-homology of a metric space to the K-theory of its Roe algebra. It was initially conjectured that this homomorphism should always be an isomorphism, a fact that would have tremendously deep consequences. This statement is alas false in full generality, but it is true for a very wide class of metric spaces - the main counterexamples coming from expander graphs. One of the second main goals of this proposal is to investigate how high-dimensional expansion interacts with the above mentioned assembly map. The objectives are twofold: on the one hand, I aim to better understand the phenomena underlying the known failure of the coarse Baum-Connes Conjecture. On the other, the proposed construction of high-dimensional expanders seems to be a valid candidate to produce the first example of spaces for which the assembly map is not injective.

• DFG Programme: Research Grants