Combinatorial Representation Theory

We attempt to survey the field of combinatorial representation theory, describe the main results and main questions and give an update of its current status. We give a personal viewpoint on the field, while remaining aware that there is much important and beautiful work that we have not been able to mention

Group size effect on cooperation in one-shot social dilemmas II. Curvilinear effect

In a world in which many pressing global issues require large scale cooperation, understanding the group size effect on cooperative behavior is a topic of central importance. Yet, the nature of this effect remains largely unknown, with lab experiments insisting that it is either positive or negative or null, and field experiments suggesting that it is instead curvilinear. Here we shed light on this apparent contradiction by considering a novel class of public goods games inspired to the realistic scenario in which the natural output limits of the public good imply that the benefit of cooperation increases fast for early contributions and then decelerates. We report on a large lab experiment providing evidence that, in this case, group size has a curvilinear effect on cooperation, according to which intermediate-size groups cooperate more than smaller groups and more than larger groups. In doing so, our findings help fill the gap between lab experiments and field experiments and suggest concrete ways to promote large scale cooperation among people.Comment: Forthcoming in PLoS ON

Discrete cubical homotopy groups and real KK(¥pi¥pi, 1) spaces (Women in Mathematics)

In this talk we wish to demonstrate how a theory, developed entirely for the purpose of solving problems stemming from search-and-rescue missions, gave rise to one that in turn has applications to fundamental mathematics. Discrete cubical homotopy theory is a discrete analogue of (singular) simplicial homotopy theory, associating a bigraded sequence of groups to a simplicial complex, capturing some of its combinatorial structure. The motivation for this construction came initially from the desire to find invariants for dynamic processes that were encoded using (combinatorial) simplicial complexes. The invariants should be topological in nature, but should also be sensitive to the combinatorics encoded in the complex, in particular to the level of connectivity among simplices. Over the last few years similar notions have arisen from several areas of mathematics (e.g., geometric group theory, coarse geometry, computer science) signaling both the pressing need for such a theory as well as its universal nature. As an illustration, we will provide a real analogue of Brieskorn's result on complex Eilenberg-MacLane spaces associated with Coxeter groups

Discrete homology theory for metric spaces

We define and study a notion of discrete homology theory for metric spaces. Instead of working with simplicial homology, our chain complexes are given by Lipschitz maps from an n n -dimensional cube to a fixed metric space. We prove that the resulting homology theory satisfies a discrete analogue of the Eilenberg–Steenrod axioms, and prove a discrete analogue of the Mayer–Vietoris exact sequence. Moreover, this discrete homology theory is related to the discrete homotopy theory of a metric space through a discrete analogue of the Hurewicz theorem. We study the class of groups that can arise as discrete homology groups and, in this setting, we prove that the fundamental group of a smooth, connected, metrizable, compact manifold is isomorphic to the discrete fundamental group of a ‘fine enough’ rectangulation of the manifold. Finally, we show that this discrete homology theory can be coarsened, leading to a new non-trivial coarse invariant of a metric space