48 research outputs found
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 (, 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 EilenbergâSteenrod axioms, and prove a discrete analogue of the MayerâVietoris 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 âfine enoughâ 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