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Abstract: We define triangulated piecewise linear constant mean curvature surfaces using a variational characterization. These surfaces are critical for area amongst continuous piecewise linear variations which preserve the boundary conditions, the simplicial structures, and (in the nonminimal case) the volume to one side of the surfaces. We then find explicit formulas for complete examples, such as discrete minimal catenoids and helicoids.
As an application we study the index of unstable minimal surfaces, by numerically evaluating the spectra of their Jacobi operators. Our numerical estimates confirm known results on the index of some smooth minimal surfaces, and provide additional information regarding their area-reducing variations. In other applications show that these discrete geometries provide a good language for studying meshes used in computer graphics and mathematical visualization.
Colloquium - 16:00
Abstract: Suppose we are given a (simple) polytope in d-space defined by n linear inequalities, and an "upwards" direction (a linear objective function). If our goal is to climb towards the top by walking along edges, we can ask, "How many vertices might we have to visit in the worst case?" Behind this lies the decade-old question about giving upper bounds for the complexity of the simplex algorithm of linear programming. One obvious bound is, "At most as many vertices as a d-polytope with n facets could possibly have." In fact, today this is still the only known answer! McMullen's Upper Bound Theorem from 1970 explicitly evaluates this upper bound, and states that no simple d-polytope with n facets can have more vertices than the polar of the cyclic d-polytope C_d(n), which has O(n^{\lfloor d/2\rfloor}) of them, for fixed d.
In this talk, we will see that for d=4 and n=7 (the "first interesting case"), this bound is tight in some instances, i.e. there exist realizations of P=C_4(7)^Delta together with linear objective functions that yield strictly ascending Hamiltonian paths on the graph Gof P. Moreover, we will show that other orientations of G that admit Hamiltonian paths and satisfy two well-known necessary conditions for realizability actually are not realizable, thus providing the smallest known examples showing that these conditions are not sufficient.