Sunday, June 9

The first Mathematical Crystallography minisymposium was entitled “Geometric Foundations”.

Frank Morgan talked about Minimal Interface Structures. Just as soap bubbles in 3-space assume the shape of minimal area enclosing the fixed volume of air they contain (subject to constraints), so a soap bubble in a 3-torus will assue such a shape of minimal area within that torus. Using that 3-torus, one kind generate periodic structures minimizing surface area (subject to constraints).

Egon Schulte talked about Polyhedral Geometries and Symmetry. Branko Grunbaum’s “skeletal” approach to polyhedra presents a polyhedron as a graph embedded in 3-space: vertices are vertices and edges are edges and a distinguished family of cyclic subgraphs (“rings” to chemists) are faces. This approach includes several infinite (periodic) polyhedra.

Ma. Louise N. de las Penas talked about Nanostructures Arising from Crystallographic Tilings. Given a graphene sheet and two appropriate vectors lying on its plane, one to define a unit and one to define a gluing identity, one can construct a graphene tube. Similarly, one may construct a torus out of the tube. Other manifolds may be constructed similarly, and in the ensuing discussion, it was proposed that algebraic and other mathematical criteria might determine when a graphene (or other layer) sheet may be shaped into a given manifold.

Bernd Souvignier talked about Capturing the Essence of Infinite Graphs in Quotient Graphs. Given a graph G, and a group of automorphisms A, one may define a quotient graph G/A whose vertices and edges are the orbits of the vertices and edges of G. Given sufficient information about how G/A articulates with G, one may go the reverse direction, constructing G out of G/A (and the articulation information).

Then we went to lunch at Reading Terminal Market.