This research program originates from a pressing practical need and from a purely new geometric perspective of discrete mathematics..
Graphs play a key role in many application areas of mathematics, providing the perfect mathematical description of all systems that are governed by pairwise interactions, in computer science, economics, biology and more. But graphs cannot fully capture scenarios in which interactions involve more than two agents. Since the theory of hypergraphs is still too under-developed, we resort to geometry and topology, which view a graph as a one-dimensional simplicial complex. I want to develop a combinatorial/geometric/probabilistic theory of higher-dimensional simplicial complexes. Inspired by the great success of random graph theory and its impact on discrete mathematics both theoretical and applied, I intend to develop a theory of random simplicial complexes.
This combinatorial/geometric point of view and the novel high-dimensional perspective, shed new light on many fundamental combinatorial objects such as permutations, cycles and trees. We show that they all have high-dimensional analogs whose study leads to new deep mathematical problems. This holds a great promise for real-world applications, in view of the prevalence of such objects in application domains.
Even basic aspects of graphs, permutations etc. are much more sophisticated and subtle in high dimensions. E.g. it is a key result that randomly evolving graphs undergo a phase transition and a sudden emergence of a giant component. Computer simulations of the evolution of higher-dimensional simplicial complexes, reveal an even more dramatic phase transition. Yet, we still do not even know what is a higher-dimensional giant component.
I also show how to use simplicial complexes (deterministic and random) to construct better error-correcting codes. I suggest a new conceptual approach to the search for high-dimensional expanders, a goal sought by many renowned mathematicians.
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