Symmetries of Discrete Structures in Geometry (17w5015)

Arriving in Oaxaca, Mexico Sunday, August 20 and departing Friday August 25, 2017


(Université Libre de Bruxelles)

(Northeastern University)

(Universidad Nacional Autonoma de Mexico)


Our main goal is to nourish new and unexpected connections established recently between the theories of polytopes, polyhedra, maps, Coxeter groups and incidence geometries, focussing on symmetry as a unifying theme. In each case an important class of groups acts on a natural geometrical and combinatorial object in a rich enough way to ensure a fruitful interplay between geometric intuition and algebraic structure. Naturally their study requires a broad and long-range view, merging approaches from a wide range of different fields such as geometry, combinatorics, incidence geometry, group theory and low-dimensional topology. These connections are further enriched by bringing in new ideas and methods from computational algebra, where powerful new algorithms provide an abundance of inspiring examples and challenging conjectures.

The quest for a deeper understanding of highly-symmetric structures in geometry, combinatorics and algebra has inspired major developments in modern mathematics. In the past three decades there has been an outburst of activity in each of the areas represented by the proposal.

In polytopes and symmetry, the recent progress has centered around the modern theory of abstract polytopes and combinatorial symmetry. Highly-symmetric abstract polytopes are combinatorial structures with distinctive geometric, algebraic, or topological properties, in many ways more fascinating than traditional highly-symmetric polyhedra, polytopes or tessellations. Their automorphism groups are certain quotients of Coxeter groups (C-groups) satisfying an intersection condition on subgroups that ensures desirable combinatorial and geometric properties. The rapid development of abstract polytope theory has resulted in a rich theory featuring an attractive interplay of methods and tools from discrete geometry (classical polytope theory), group theory and geometry (Coxeter groups and their quotients, as well as reflection groups over the reals, complex numbers, or finite fields), combinatorial group theory (generators and relations), and hyperbolic geometry and topology (tessellations and their groups). Still, even after an active period of research, many deep problems have remained open and await solution.

Regular or chiral maps and hypermaps have been studied since the time of Felix Klein. Deep exciting connections exist between hypermaps and other branches of mathematics (hyperbolic geometry, combinatorial group theory, Riemann surfaces, number fields, Galois theory, and graph theory). For instance, maps (not necessarily regular) can be viewed as complex algebraic curves, defined over algebraic number fields. A central theme in the theory of maps is the problem of their classification, which is usually approached from one of the following three viewpoints: graph-theoretical, group-theoretical, and topological. In the past few years, great progress has been made in the computer-aided enumeration of maps by genus, exploiting new fast algorithms for finding low index normal subgroups in finitely-presented groups. These findings have lead to a host of challenging new conjectures.

Incidence geometries provide an overarching theme for most research activities in polytopes and hypermaps. In this vein, maps (usually) are abstract polytopes of rank 3, and abstract polytopes are thin residually-connected incidence geometries with a string diagram. Thus progress in either subject is likely to influence the others. Polytopes and maps stand to benefit greatly from new developments on buildings and diagram geometries. In a nutshell, buildings are the natural geometric counterparts of certain kinds of simple Lie groups where the classical regular polytopes or Coxeter complexes occur as fundamental structural components. On the other hand, diagram geometries provide geometric interpretation for the sporadic simple groups, and although the theories of buildings and diagram geometries are highly developed, the understanding of their interaction with polytopes and hypermaps is still limited at this point.

The following key problems can serve as focal points for the proposed workshop activities.

(1) Topological classification of chiral polytopes, including the classification of locally toroidal chiral polytopes; (2) Developing a theory of regular or chiral hypertopes (as hybrids of polytopes and incidence geometries);

(3) Polytope-theoretic interpretation for sporadic and other simple groups;

(4) Group-theoretical classifications of regular and chiral polytopes;

(5) Classification of highly symmetric skeletal polyhedral structures in Euclidean spaces.

We are planning to schedule a number of key lectures by international experts surveying the state-of-the-art in a particularly active area and addressing existing connections. Other participants will have the opportunity to propose 25-minute talks to present their work, but due to the time constraints we expect to be able to accept only a relatively small number of talks. Our workshop schedule would provide ample time and opportunity for participants to interact and engage in mathematical discussion.

A particular effort will be made to attract junior faculty, postdoctoral fellows, graduate students, and advanced undergraduate students to participate in the workshop. It is noteworthy that the proposed participant list contains an unusually large number of women researchers.